Bayesian Models for Longitudinal Data - QUT · Bayesian Models for Longitudinal Data Margaret Rolfe ... Lyndon Brooks (SCU) for their encouragement, support and guidance throughout

Bayesian Models for Longitudinal Data

Margaret Rolfe

Bachelor of Science, Master of Statistics

University of New South Wales

Submitted in fulfilment of the requirements

of the degree of Doctor of Philosophy

March 11, 2010

Discipline of Mathematical Sciences

Faculty of Science and Technology

Queensland University of Technology

Principal supervisor: Professor Kerrie Mengersen, Queensland University of Technology

Associate supervisors: Dr Lyndon Brooks, Southern Cross University

Dr Helen L Johnson, Queensland University of Technology

Abstract

Longitudinal data, where data are repeatedly observed or measured on a temporal basis of time

or age provides the foundation of the analysis of processes which evolve over time, and these

can be referred to as growth or trajectory models. One of the traditional ways of looking at

growth models is to employ either linear or polynomial functional forms to model trajectory

shape, and account for variation around an overall mean trend with the inclusion of random

effects or individual variation on the functional shape parameters. The identification of distinct

subgroups or sub-classes (latent classes) within these trajectory models which are not based on

some pre-existing individual classification provides an important methodology with substantive

implications. The identification of subgroups or classes has a wide application in the medical

arena where responder/non-responder identification based on distinctly differing trajectories

delivers further information for clinical processes.

This thesis develops Bayesian statistical models and techniques for the identification of

subgroups in the analysis of longitudinal data where the number of time intervals is limited.

These models are then applied to a single case study which investigates the neuropsychological

cognition for early stage breast cancer patients undergoing adjuvant chemotherapy treatment

from the Cognition in Breast Cancer Study undertaken by the Wesley Research Institute of

Brisbane, Queensland.

Alternative formulations to the linear or polynomial approach are taken which use piece-

wise linear models with a single turning point, change-point or knot at a known time point and

latent basis models for the non-linear trajectories found for the verbal memory domain of cog-

nitive function before and after chemotherapy treatment. Hierarchical Bayesian random effects

models are used as a starting point for the latent class modelling process and are extended with

the incorporation of covariates in the trajectory profiles and as predictors of class membership.

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The Bayesian latent basis models enable the degree of recovery post-chemotherapy to be

estimated for short and long-term followup occasions, and the distinct class trajectories assist

in the identification of breast cancer patients who maybe at risk of long-term verbal memory

impairment.

Declaration of Original Authorship

The work contained in this thesis has not been previously submitted for a degree or diploma

at any other higher educational institution. To the best of my knowledge and belief, the the-

sis contains no material previously published or written by another person except where due

reference is made.

Signed:

Date:

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List of Publications

This thesis is comprised of four published or submitted for publication papers and are listed

below.

Title: Latent Class Piecewise Linear Trajectory Modelling For Short-Term Cognition

Responses After Chemotherapy For Breast Cancer Patients. Accepted for publication by the

Journal of Applied Statistics.

Authors: Margaret Rolfe, Kerrie Mengersen, Geoffrey Beadle, Katharine Vearncombe,

Brooke Andrew, Helen Johnson, Cathal Walsh.

Title: Bayesian Analysis Of Longitudinal Cognition Models: Verbal Memory Performance In

Women Undergoing Adjuvant Chemotherapy Treatment For Breast Cancer. Submitted for

publication to the journal Biostatistics.


Brooke Andrew.

Title: Impact of Chemotherapy on Verbal Memory in Breast Cancer patients. Who is at Risk?.

Submitted for publication to the journal Memory.


Brooke Andrew.

Title: Bayesian Estimation Of Extent Of Recovery For Aspects Of Verbal Memory In Women

Undergoing Adjuvant Chemotherapy Treatment For Breast Cancer. Submitted for publication

to the Journal of the Royal Statistical Society C Applied Statistics.

Authors: Margaret Rolfe, Kerrie Mengersen, Katharine Vearncombe, Brooke Andrew,

Geoffrey Beadle.

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Acknowledgements

I am extremely grateful to Professor Kerrie Mengersen, my supervisor, for guidance,

understanding, encouragement and can-do attitude which enabled my research to be

completed, even in the required time frame. I understand the struggle of timing and discipline

that is often overlooked in the PhD journey, and am extremely grateful for her ability to

facilitate all of this. The promptness and thoroughness of Kerrie’s responses to any of my

written material has been indeed remarkable. Thank you Kerrie and QUT for making

available the ARC Discovery Scholarship which ensure this endeavor was at all possible, and

the flexibility of being able to study as an external student.

I would also like to extend my deep appreciation to the other members of the Cognition in

Breast Cancer Study team at the Wesley Research Institute Brisbane, especially my

psychologist co-authors Katharine Vearncombe and Brooke Andrew who were instrumental in

conducting the neuropsychological assessments, data entry and management, and patiently

answering all my questions about the neuropsychological instruments used for assessing

cognitive function. A special acknowledgement must also be directed to my other co-author,

oncologist Dr Geoff Beadle for his clinical expertise and medical perspective required for the

study.

Special thanks is extended to my associate supervisors Dr Helen Johnson (QUT) and Dr

Lyndon Brooks (SCU) for their encouragement, support and guidance throughout this

research endeavor.

I would also like to express my gratitude to the Division of Research at Southern Cross

University Lismore in agreeing to host me for the duration of my candidature, and for the

interest, support and encouragement of all staff within the section and to the other academic

staff throughout the university who voiced their encouragement.

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This research also benefited tremendously from the fellow PhD students and staff in the

School of Mathematical Science at Queensland University of Technology, and especially from

the BRAG group who assisted me such a variety of ways (social, technical, administrative,

collegiate, and friendship). A special tribute needs to directed to Drs Pat Rowe and Michael

Christie for enabling me to have a home-way-from-home for my Brisbane visits and for

providing support, encouragement, guidance and stats problems along the way.

Finally, I would like to direct my heartfelt thanks to my husband Tony and son Tuk for their

endless patience and encouragement throughout this PhD journey, and who endured both my

physical and mental absences. Without whom I would have struggled to find the inspiration

and motivation needed to complete this dissertation, and I dedicate this thesis to them.

Contents

1 Introduction 19

1.1 Overall Objectives of the Research . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Research Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3 Structure of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Case Study - Cognition in Breast Cancer . . . . . . . . . . . . . . . . . . . . 24

2 Literature Review 35

2.1 Introduction to Longitudinal Data Analysis . . . . . . . . . . . . . . . . . . . 35

2.2 Approaches to Analysis of Longitudinal Data . . . . . . . . . . . . . . . . . . 37

2.2.1 Repeated Measures Analysis of Variance . . . . . . . . . . . . . . . . 37

2.2.2 Multilevel or Hierarchical Longitudinal Analysis . . . . . . . . . . . . 38

2.2.3 Structural Equation Modelling and Latent Growth Curve Modelling . . 42

2.2.4 Piecewise Linear Growth Models . . . . . . . . . . . . . . . . . . . . 44

2.2.5 Latent Basis Growth Models . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.6 Bayesian Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.2.7 Growth Mixture Models (GMM) . . . . . . . . . . . . . . . . . . . . . 51

2.2.8 Bayesian Growth Mixture Models . . . . . . . . . . . . . . . . . . . . 56

2.3 Aspects of Bayesian Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.3.1 Markov Chain simulation . . . . . . . . . . . . . . . . . . . . . . . . 58

2.3.2 Priors for Bayesian Hierarchical Models . . . . . . . . . . . . . . . . . 60

2.3.3 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.3.4 Assessing Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.3.5 Goodness of Fit or Model Checking . . . . . . . . . . . . . . . . . . . 64

9

10 CONTENTS

2.4 Missing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3 Latent Class Piecewise Linear Trajectory Models 85

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.2.1 Study Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.2.2 Primary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.2.3 Supplementary Analyses . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.3.1 Latent Class Growth Analysis . . . . . . . . . . . . . . . . . . . . . . 96

3.3.2 Results of K-means clustering . . . . . . . . . . . . . . . . . . . . . . 100

3.3.3 Discriminant Analyses and Logistic Regressions . . . . . . . . . . . . 100

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4 Bayesian Longitudinal Models 111

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2.1 Study design and participants . . . . . . . . . . . . . . . . . . . . . . 117

4.2.2 Bayesian piecewise linear latent growth model . . . . . . . . . . . . . 119

4.2.3 Bayesian latent class growth mixture models . . . . . . . . . . . . . . 121

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.3.1 Results for Bayesian piecewise linear growth models . . . . . . . . . . 123


4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5 Bayesian Estimation of Class Predictors for Latent Class Growth Models 141

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.2.1 Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.2.2 Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.2.3 Verbal Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.2.4 Medical indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

CONTENTS 11

5.2.5 Statistical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6 Bayesian Estimation Of Extent Of Recovery 175

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.2.1 Study design and participants . . . . . . . . . . . . . . . . . . . . . . 182

6.2.2 Bayesian random effects latent basis growth models . . . . . . . . . . . 184


6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.3.1 Bayesian single class random effects latent basis growth models . . . . 187

6.3.2 Latent class growth mixture models . . . . . . . . . . . . . . . . . . . 190

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7 Conclusion 203

7.1 Research Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.2 Limitations of the Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

7.3 Possible Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Appendices 213

A Cognition Joint Paper 1 215

B Cognition Joint Paper 2 235

C Cognition Joint Paper 3 255

D Cognition Joint Poster 265

Full Reference List 269

List of Figures

1.1 Sample mean scores for Learning, Immediate Retention and Delayed Recall

for four times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.2 Sample mean scores by class membership for Learning, Immediate Retention

and Delayed Recall for the three class latent basis model . . . . . . . . . . . . 30

2.1 Linear Latent Growth Curve Model for three time points. . . . . . . . . . . . . 43

2.2 Plots of some different trajectory profiles for latent basis models . . . . . . . . 47

3.1 Piecewise Linear Latent Class Growth Model. . . . . . . . . . . . . . . . . . . 93

3.2 Two and three class trajectory models for Learning, Immediate Retention and

Delayed Recall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.1 Plots of mean scores for Learning, Immediate Retention and Delayed Recall

from before chemotherapy to 18 months post-chemotherapy. . . . . . . . . . . 123

4.2 Plots of growth trajectories for Learning, Immediate Retention and Delayed

Recall for combinations of years of education (10, 13, 16), and Stage (I, II/III)

for Model C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.3 Plots of two and three class trajectories for learning, Immediate Retention and

Delayed Recall using Model B for years of education=13 . . . . . . . . . . . 129

5.1 Two Class Trajectory Plots for Learning, Immediate Retention and Delayed

Recall from Latent Class Growth Models with core (AES) Predictors. . . . . . 159

5.2 Probability surface plots of Low class membership for Learning, Immediate

Retention and Delayed Recall. . . . . . . . . . . . . . . . . . . . . . . . . . . 161

13

14 LIST OF FIGURES

6.1 Plots of some different trajectory profiles for latent basis models . . . . . . . . 180

6.2 Sample mean scores for Learning, Immediate Retention and Delayed Recall

for four times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.3 Posterior means and 95% credible intervals for the estimated degree of recov-

ery at 18 months estimated under the three class latent growth mixture model

for the three verbal memory outcomes . . . . . . . . . . . . . . . . . . . . . . 193

6.4 Sample mean scores by class membership for Learning, Immediate Retention

and Delayed Recall for the three class latent basis model . . . . . . . . . . . . 194

B.1 Interaction between menopausal status, endocrine treatment and time (pre-

menopausal data not shown). . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

B.2 Significant time by menopausal group interaction (across time) for the tele-

phone search subtest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

B.3 Performance on the Auditory Verbal Learning test (total recalled over 5 trials)

in different treatment groups (over time). . . . . . . . . . . . . . . . . . . . . . 248

C.1 Standardised distribution of change on DKEFs Card Sorting Task for the three

different change methods (PCS, RCI, RCIp, SRB). . . . . . . . . . . . . . . . 261

List of Tables

1.1 Complete and missing data numbers and percentages with patterns of missing-

ness in addition means and se for baseline verbal memory scores and age . . . 28

1.2 Summary Statistics for Learning, Immediate Retention and Delayed Recall . . 29

3.1 Sociodemographic and clinical data for participants with complete data for

three measurement occasions, n = 130. . . . . . . . . . . . . . . . . . . . . . . 96

3.2 Summary statistics for Learning, Immediate Retention, Delayed Recall, Anxi-

ety, Depression and FACT scores for participants with complete data for three

measurement occasions, n = 130. . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.3 Results of two/three class models of Latent Class Growth analysis for Learning,

Immediate Retention and Delayed Recall . . . . . . . . . . . . . . . . . . . . 98

3.4 Standardized Coefficients for resultant predictors of stepwise discriminant anal-

ysis for Learning, Immediate Retention and Delayed Recall. . . . . . . . . . . 101

3.5 Means for demographic variables, quality of life scores, and numbers for stage

of cancer [I/II&III] by classes for the Learning, Immediate Retention and De-

layed Recall outcomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.6 Results of stepwise logistic/multinomial analysis for Learning, Immediate Re-

tention and Delayed Recall with age, NART, years of education, FACT Gen-

eral, Fact Breast, FACT fatigue, stage of cancer. . . . . . . . . . . . . . . . . . 103

4.1 Sociodemographic and clinical data for participants with complete data for 4

measurement occasions n=120. . . . . . . . . . . . . . . . . . . . . . . . . . 124

15

16 LIST OF TABLES

4.2 Summary Statistics for Learning, Immediate Retention and Delayed Recall for

four measurement occasions n=120. . . . . . . . . . . . . . . . . . . . . . . . 125

4.3 Assessment of Bayesian Model Fit with Deviance Information Criterion . . . . 125

4.4 Posterior parameter estimates for Bayesian Piecewise Linear Growth Model C

education and stage adjusted intercept; posterior standard deviation in brackets 126

4.5 Posterior mean parameter estimates for Bayesian Piecewise Linear Growth for

Model E for Learning and Model D for Immediate Retention and Delayed Re-

call; posterior standard deviation in brackets . . . . . . . . . . . . . . . . . . 126

4.6 DIC, Posterior estimates for probabilities of class membership and numbers in

classes Models B, C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.7 Means and SD of parameters of Bayesian Latent Class Piecewise Linear Growth

Models - Education adjusted intercept Model B for two and three classes . . . . 130

4.8 Class means for demographic variables, quality of life scores, and numbers for

stage of cancer [I/II&III]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.9 Class means for demographic variables, quality of life scores, and numbers for

stage of cancer [I/II&III]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.1 Sociodemographic and clinical data for participants with complete data for 4

measurement occasions n=120. . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.2 Comparison Of Verbal Memory Scores By Ten Year Age Classes With Pub-

lished Norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.3 Summary Statistics for Learning, Immediate Retention and Delayed Recall,

Fatigue, Depression, Anxiety and Estrogen Producing ability over four mea-

surement occasions n=120. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.4 Posterior Estimates For Logistic Regression Parameters For Predictors Used

Singly With Probability Of Class Membership (for the Low class of the two

class model). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.5 Posterior Estimates For The Latent Class Growth Full Unconditional Model. . . 157

5.6 Posterior Estimates for the LCGM full trajectories with the predictor Age-

Education-Stage model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

LIST OF TABLES 17

5.7 Posterior Estimates for Logistic regression parameters for Predictors for the

core model (Age, Education and Stage) plus additional single covariates for

the probability of class membership. . . . . . . . . . . . . . . . . . . . . . . . 160

5.8 Changes in class membership composition with the addition of covariates. . . . 160

5.9 Posterior Estimates for substantive time-varying trajectory covariates added to

the AES class membership predictor models. . . . . . . . . . . . . . . . . . . 162

6.1 Summary Statistics for Learning, Immediate Retention and Delayed Recall . . 188

6.2 Posterior parameter estimates for Bayesian latent basis using Wishart df=2,

Wishart df=3 and Uniform priors . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.3 Posterior parameter estimates for Bayesian latent basis two class mixture model;

classes are denoted as Low and High . . . . . . . . . . . . . . . . . . . . . . . 191

6.4 Posterior parameter estimates for Bayesian latent basis three class mixture

model; classes are denoted as Low, Mid and High . . . . . . . . . . . . . . . . 192

6.5 Posterior Probability estimates for Bayesian latent basis two and three class

mixture models, for each class, assigned class and proportion of participants

with average posterior probabilities less than 0.7 and 0.6 . . . . . . . . . . . . 195

A.1 Neuropsychological and self-report measures and outcome variables . . . . . . 221

A.2 Demographic and treatment related characteristics of the study sample . . . . . 224

A.3 Means, standard deviations, and reliability estimates for Time 1 and Time 2

cognitive variables in the chemotherapy and non-chemotherapy groups . . . . . 225

A.4 Classifications of impaired, no change, and improved after chemotherapy . . . 225

A.5 Means and standard deviations for cognitive change (T2-T1) in the chemother-

apy group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

A.6 Baseline and change (T2-T1) means and standard deviations for the psycho-

logical, health, and treatment factors in the chemotherapy group . . . . . . . . 227

A.7 Pearson correlations between change in cognitive measures (T2-T1) and health

and psychological measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

A.8 Correlations between psychological and clinical change variables (T1-T2) . . . 228

B.1 Neuropsychological and self-report measures and outcome variables . . . . . . 241

18 LIST OF TABLES

B.2 Demographic and treatment related characteristics of the menopausal groups . . 243

B.3 Means (M) and standard deviations (SD) for cognitive functioning measures at

baseline (Time 1: T1), 1 month post chemotherapy (Time 2: T2) and 6 months

post chemotherapy completion (Time 3: T3). . . . . . . . . . . . . . . . . . . 244

B.4 Demographic and treatment related characteristics for each of the systemic

treatment groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

B.5 Means (M) and standard deviations (SD) for cognitive functioning measures at

baseline (Time 1: T1), 1 month post chemotherapy (Time 2: T2) and 6 months

post chemotherapy completion (Time 3: T3). . . . . . . . . . . . . . . . . . . 248

C.1 Means and standard deviations for each of the cognitive measures for both the

chemotherapy and non-chemotherapy groups. . . . . . . . . . . . . . . . . . . 260

C.2 Number of participants classified as not impaired (no decline) and impaired

(decline) by the three methods using the 90% confidence interval cut-off. . . . . 261

Chapter 1

Introduction

Longitudinal data, where data are repeatedly observed or measured on a temporal basis of

time or age provides the foundation of the analysis of processes which evolve over time, and

these can be referred to as growth or trajectory models. One of the traditional ways of looking

at growth models is to employ either linear or polynomial functional forms to model trajectory

shape, and account for variation around an overall mean trend with the inclusion of random

effects or individual variation on the functional shape parameters. The identification of

distinct subgroups or sub-classes (latent classes) within these trajectory models which are not

based on some pre-existing individual classification provides an important methodology with

substantive implications. The identification of subgroups or classes has a wide application in

the medical arena where responder/non-responder identification based on distinctly differing

trajectories delivers further information for clinical processes.

1.1 Overall Objectives of the Research

The overall focus of this thesis is the identification of subgroups in longitudinal data with few

time points.

The objectives are thus two-fold; first to develop Bayesian statistical models and techniques

for the identification of subgroups in analysis of longitudinal data where the time intervals are

limited in number and secondly to apply these methodologies to a single case study which

investigates the neuropsychological aspects of cognition for early stage breast cancer patients

undergoing adjuvant chemotherapy treatment.

19

20 CHAPTER 1. INTRODUCTION

Identification of subgroups of respondents has been increasingly used in the medical area with

latent class models being used to identify differential responder classes for medical treatment,

medication usage, and emotional mood tendencies in longitudinal studies, and for differential

symptom responses in disease diagnosis for migraine, Parkinson disease and fetal growth with

gestational age. These applications are explored further in Chapter 2.

Bayesian growth latent class or mixture models for longitudinal data will be the culmination

of incorporating the latent variables defining trajectory shapes and possible random effects

with the mixture of a finite number of trajectories. The identification of a finite number of

subgroups or mixtures can be used to identify outlier subjects and/or sensitive subject groups.

1.2 Research Aims

There are two main research aims addressed by this thesis:

1. To identify, tailor and extend state-of-the-art Bayesian latent class identification in the

context for longitudinal studies with a limited number of temporal observations and

target to the application in hand.

2. To apply the statistical methodologies developed to a single case study to gain

substantive insight.

In order to model the non-linear trajectories of the cognitive processes observed in the

Cognition in Breast Cancer Study, alternatives to the traditional family of polynomial

trajectories were investigated. Piecewise linear models were investigated with a single turning

point or knot at a known time point as Bayesian latent growth models with the inclusion of

random effects for the single class situation and as Bayesian latent class growth mixture

models for subgroup analysis. In order to explore the characteristics of the latent class

membership a range of methods investigating the role of predictor variables was employed

and encompassed predictor profiles, predictors included in the trajectory process and

predictors of class membership.

Bayesian latent basis growth models were also employed to assess non-linear response over

time and estimate the degree of recovery at set times. The parameterisation of nonlinear

growth trajectories by latent basis model where, rather than fixing the basis coefficients for the

1.3. Structure of Thesis 21

slope to some predetermined values, the optimal shape is estimated from the data [8, 9, 10].

The latent basis growth model in the way the weights or basis coefficients are set, either fixed

or partially estimated, ensures flexibility so that different trajectory shapes can be estimated.

Bayesian latent basis latent class growth mixture models were used to estimate the extent of

recovery for identified sub-classes of women demonstrated different patterns of response.

1.3 Structure of Thesis

This thesis has been written as a series of papers included in Chapters 3 to 6 which have been

submitted to journals and have been left in their entirety and in doing so exhibit some degree

of overlap. Chapters 3 to 6 each address both of the research aims pivotal to this thesis. Each

chapter has its own bibliography and is replicated in the comprehensive bibliography at the

end of the thesis.

Chapter 2 comprises a Literature Review on latent variable hierarchical models, growth

models and growth mixture models from both the frequentist and Bayesian perspective. This

literature review provides the background and foundation for the methodological component

of the thesis and gives more detailed grounding than is incorporated in the papers.

Chapter 3 addresses the two research aims in a frequentist framework for 130 Cognition in

Breast Cancer Study participants who had completed all of the first three assessments, at

baseline, one month and six months post-chemotherapy. The aims of this paper was to

characterize the responses of subjects over time, in regard to the potential decline and recovery

process as a two-part piecewise linear model of verbal memory change with chemotherapy

treatment and the identification of possible subgroups based on different patterns of this

process. Posterior probability-based classifications were used to determine class membership,

and therefore provided the basis to obtain profiles of trajectory group members with respect to

demographic, quality of life and cancer severity measures. These classifications also provided

the ability to use binomial and multinomial regression models, dependent on whether two or

three latent classes were identified, to identify a range of class predictors from baseline

mediating variables. The models used here provide a base and set the scene for Bayesian

models developed in the subsequent chapters of this thesis, where the models over three time

points (baseline, one and six month post-chemotherapy) presented in this chapter are extended


to a fourth assessment time of eighteen months post-chemotherapy. This research has been

written as a journal article, for which I am first author and has been accepted for publication in

the Journal of Applied Statistics (January, 2009) and is presented verbatim in Chapter 3.

Chapter 4 uses Bayesian piecewise latent class growth models to address the two research

aims of identification of subgroups for the components of verbal memory from a two-part

linear piecewise process over the four assessment occasions (baseline, one, six and eighteen

months post-chemotherapy). The changepoint was set to one month post-chemotherapy or

time 2, so with the first linear segment assessed the potential decline from baseline to one

month post-chemotherapy and the second linear component modelled the possible recovery or

non-recovery phase from one month to eighteen months post-chemotherapy. The single class

model was initially used to determine sets of predictors which were more influential when

added to the trajectories themselves. These predictors included age, years of education,

intellectual functioning (NART), stage of cancer, and quality of life scales including fatigue,

depression and anxiety. Two and three class latent class growth models were fitted with two

sets of predictors (education, and education plus stage of cancer) where possible. For all

Bayesian analyses non-informative prior distributions were used for all model parameters.

This work has been submitted to Biostatistics for review with myself as first author, and an

earlier version of this research was presented as a poster at the International Symposium for

Bayesian Analysis (ISBA), Hamilton Island, July 2008.

Chapter 5 addresses primarily the second research question with the identification of predictor

or mediating variables in the determination of the probability of class membership for

subgroups or classes of subjects who demonstrated distinct trajectory patterns using Bayesian

latent class growth models. In addition, the impact of time-varying factors of fatigue,

depression, anxiety and estrogen producing status on the trajectory outcomes is assessed. This

work has been submitted for review in the peer-reviewed journal, Memory (December 2009)

and has been included as submitted in Chapter 5.

Chapter 6 addresses the research aims whereby Bayesian latent basis models are employed to

estimate the degree of recovery at six and eighteen months post-chemotherapy. Both Bayesian

random effects and Bayesian latent class growth models are used to address the overall degree

of recovery for participants undergoing adjuvant chemotherapy treatment, and the differing

1.3. Structure of Thesis 23

recovery rates with subgroups of participants who exhibit distinct trajectory responses. Latent

basis models are a different parameterisation of time in order to model a non-linear trajectory

response with a linear configuration (intercept or level and slope). The parameterisation used

in this paper sets the baseline or the initial time to one and the time 2 (one month

post-chemotherapy) to zero, thus setting a scale, and times 3 and 4 are estimated from the data

for the recovery response. This paper with myself as first author, has been submitted to the

Journal of the Royal Statistical Society C: Applied Statistics (December, 2009).

Most of the Bayesian models used vague proper prior distributions, which can also be

described non-informative, whereas the prior distributions for the estimated degree of

recovery parameters α3 and α4 used weakly informative priors. Scores at time 1 have been

referred to as both ‘baseline’ or ‘initial’ scores.

Declarations of contributions of authors of the research articles presented in chapters 3 to 6

can be viewed in the first few pages of each chapter.

The work included in Chapters 3 to 6 was undertaken as part of a larger project, the

“Cognition in Breast Cancer” study which is described in detail in Section 1.4. As part of the

project team engaged in collaborative interdisciplinary research, I provided statistical support

to other members of the team, which was within the scope of the larger project but outside the

scope of this thesis. The papers awaiting publication and submitted for publication presented

in Appendix A to C are co-authored with other members of the project team and are included

to demonstrate the meeting of the secondary aim of the PhD which focuses on training and

interdisciplinary collaboration. These papers cover predictors of cognitive decline after

chemotherapy in breast cancer patients where impairment is assessed over multiple cognitive

domains using Reliable Change Index; assessment of the cognitive effects of chemical

menopause as hormonal outcome of chemotherapy and adjuvant endocrine treatment and the

evaluation of different methods used to detect cognitive change, namely percentage change, a

reliable change index, a reliable change index with practice effect corrections and a

standardised regression-based approach. My contribution to these research papers is detailed

at the commencement of each Appendix. Similarly Appendix D presents the abstract for a

poster presented by the oncologist from the “Cognition in Breast Cancer” study at the 30th

San Antonio Breast Cancer Symposium 2007. My contribution as second author,


demonstrates further the extent of interdisciplinary collaboration.

1.4 Case Study - Cognition in Breast Cancer

Even though Bayesian longitudinal models were the primary motivation for this research,

being part of the Cognition in Breast Cancer Study conducted by the Wesley Research

Institute and the questions of interest arising from the study became the driver of the direction

my research and the formulation of research questions.

Success in the treatment of breast cancer, especially when diagnosed in early stages of the

disease has resulted in an increased relative 5 year survivorship rate in the last quarter century

from 74% in 1982 to nearly 90% in 2006 [3] in Queensland, Australia while the annual

incidence rate increased from 86.3 to 116.4 cases per 100,000 population for the same period.

Similar increases in incidence rates and 5 year survival rates have been experienced both

nationally and internationally in Western countries. With this rapid increase in the number of

breast cancer survivors, quality of life becomes an area of primary attention. Decline in

cognitive function also known as ’chemo-brain’ is a frequently reported side-effect for women

undergoing adjuvant chemotherapy treatment for breast cancer, with estimates of women

suffering from cognitive impairment after chemotherapy in the short term varies between

studies and ranges from 20% or 25% [5, 6] to 50% for women with moderate or severe

impairment [2]. The level of cognitive dysfunction has been shown to improve over time

[6, 18] with a subset of women still below baseline levels at 12 months post-chemotherapy

[7, 22] and with others suffering longer term effects for up to 10 years [1, 16]. However in

these studies the nature of this cognitive impairment has been described as subtle [2, 21] with

deficit levels better than those required for a clinical impairment diagnosis.

Many cognitive domains of attention, concentration, verbal and visual memory, processing

speed [1, 2, 16, 17, 22, 23] and executive function [14] have been specifically indicated as

areas of functional deficit. But the domain of verbal memory was consistently identified by

several studies [1, 12, 13, 17, 19, 23, 24] as suffering compromise from chemotherapy

treatment. This research concentrated on this area of verbal memory and on the identifications

of subgroup of women with differential responses over time.

The Cognition in Breast Cancer (CBC) study, is a prospective longitudinal study with the aim

1.4. Case Study - Cognition in Breast Cancer 25

of examining the causes of variation in cognitive functioning, health and well-being in women

up to 2 years post-chemotherapy undertaken during the period. The study commenced in early

2004 with recruitment of participants starting in May 2004 and continuing until April 2007,

with followup assessments being concurrent and ongoing up until early 2009. Eligible

participants were required to be between 18 and 70 years old, proficient in English, have no

previous history of cytotoxic drug treatment, neurological or psychiatric symptoms or current

use of medications that might affect neuropsychological test results. All participants provided

written, informed consent, and the conduct of this study was approved by the following ethics

committees; the Queensland Institute of Medical Research, the University of Queensland, and

all participating hospitals (the Wesley Hospital, Royal Brisbane and Women’s Hospital,

Redcliffe Hospital, Princess Alexandra Hospital, the Mater Hospital, St Vincent’s Hospital,

and St Andrew’s Hospital).

Three groups of early breast cancer patients were recruited from hospitals across south-east

Queensland, Australia; patients who were scheduled for chemotherapy treatment (with or

without endocrine treatment and post-operative radiotherapy), patients scheduled for adjuvant

endocrine treatment (with or without post-operative radiotherapy but no chemotherapy) or no

further treatment post-surgery. Patients were approached by their oncologist or a research

nurse after definitive surgery, and those who initially agreed to participate received a phone

call from a psychologist, who discussed the purpose and procedures of the study. The

psychologist also discussed the eligibility criteria, and those patients who were eligible and

willing to participate were scheduled to sign informed consent forms and complete the

neuropsychological assessment battery (approximately 2.5 hours in duration).

Neuropsychological testing was administered before commencement of chemotherapy (but

after definitive surgery), and at one month, six and eighteen months after completion of

chemotherapy, or at similar timepoints for non-chemotherapy participants. This thesis is

confined to investigation of participants who underwent adjuvant chemotherapy treatment.

One hundred and fifty four participants scheduled to receive adjuvant chemotherapy treatment

were recruited, with withdrawals during the course of the study and missing assessments,

resulting in 120 participants with complete data for all four measurement occasions, and 130

with complete data over the first three measurement times.


The neuropsychological cognitive battery of tests assessed a variety of different cognitive

domains, i.e. verbal learning/ memory, visual memory, cognitive and motor processing speed,

as well as different aspects of attention and executive function. The instruments used for these

assessments are detailed in Tables A.1 and B.1 of Appendices A and B.

Quality of life (QOL) was measured using the Functional Assessment of Cancer Therapy -

Breast (FACT-B), along with the fatigue subscale. The FACT-B is a combination of the

Functional Assessment of Cancer Therapy - General (FACT-G) and additional ten questions

relating to breast cancer concerns [4]. The FACT-G comprises 27 items covering four QOL

domains, specifically physical, emotional, social/family, and functional well-being. The

fatigue subscale [25] comprises 13 items measuring the disruptiveness and intensity of

fatigue, e.g. ”I feel listless (washed out)”. A higher score indicates more satisfaction/

well-being and less fatigue on the QOL and fatigue scale respectively. Details of the FACT-B

instrument and fatigue subscale are presented in Appendix D.

Self-reported depression and anxiety was measured using the Hospital Anxiety and

Depression Scale (HADS), a 14-item rating scale assessing the presence and prominence of

depressive and anxious symptoms over the week prior to test administration. Separate scores

for depressive and anxious symptomatology were calculated, with higher scores indicating

higher levels of depression or anxiety. Details of the Hospital Anxiety and Depression Scale

[26] questions are presented in Appendix E. Age, education level (maximum 20 years) and

general intellectual functioning ability (Predicted Full Scale IQ NART) were collected as

covariate information because these variables have been previously found to affect

performance on objective neuropsychological tests [15]. Intellectual functioning ability

(NART) was estimated using the National Adult Reading Test, version 2 [11], which is a

validated reading test. Participants are required to read 50 irregularly spelt words, and

accuracy of pronunciation is used to predict IQ [20].

Time-invariant treatment and health related information were also collected, with

time-invariant data including stage of cancer, oestrogen receptor status (positive or negative),

type of surgery (breast conserving or mastectomy), and number of chemotherapy courses. In

addition, women were classified as pre-, peri-, or post-menopausal based on the four

assessments of the larger study. Women were classified as pre-menopausal if they had regular,


active menstruation throughout chemotherapy or recovered cycles prior to the 18 months post

completion assessment. Women were regarded as postmenopausal if they had not menstruated

within the past 3 months prior to diagnosis, and peri if neither pre-menopausal nor

post-menopausal.

Participants were interviewed in a quiet room at a participating hospital or in their homes.

Participants completed a demographic interview and neuropsychological assessment battery at

three time points: at baseline (after surgery but prior to commencement of chemotherapy -

T1), approximately 1 month (T2), 6 months post chemotherapy completion (T3), and eighteen

months. Each of the neuropsychological assessments was individually administered by

psychologists (trained at the postgraduate level) and all participants completed the test battery

in the same order. Clinical information was collected before chemotherapy and at

chemotherapy completion by clinical research nurses at the participant’s hospital. In order to

reduce practice effects alternative forms of test were used if available.

Missing data due to attrition is a common occurrence with longitudinal data, attrition means

that the participant drops out or withdraws from the study, thus data is missing for subsequent

measurement times. The data in the Cognition in Breast Cancer study were missing

predominately from attrition with 7, 10 and 13 participants withdrawing after T1, T2 and T3

respectively, giving a total of 30 (19.5%) lost by attrition. Another four (2.4%) participants

missed intermediate non-consecutive times with one at T2, two at T3 and 1 at both T2 and T4.

The patterns of missingness (numbers and percentages) for participants undergoing

chemotherapy treatment are presented in Table 1.1, together with baseline means and standard

errors of means (se) for the verbal memory outcomes (learning, immediate retention and

delayed recall) and age at baseline. Attrition after T3 (6 months post-chemotherapy) was

broken into two subsets; dropout from unknown causes and dropout from medical causes

which included further cancer diagnosis and treatment or cancer related death. In considering

all participants with missing data together (n=34 or 22.1%), the missing group had

significantly lower mean baseline scores for learning (p <0.0005), immediate retention

((p=0.004) and delayed recall (p=0.003), but did not differ with age. However, participants

missing due to attrition after T3 did not differ from the complete data participants on any

baseline scores.


All papers used complete data, with Chapter 1 considered 130 subjects with complete

measurements at times 1 to 3 and Chapters 4, 5, 6 focused on 120 participants who

participated in the study at all four observational times.

Table 1.1 Complete and missing data numbers and percentages with patterns of missingness inaddition means and se for baseline verbal memory scores and age

Learning Immediate Delayed AgeRetention Recall

n % mean se mean se mean se mean se

Complete 120 77.9 53.05 0.64 11.33 0.22 11.34 0.22 49.52 0.71Missing all 34 22.1 46.71 1.43 9.94 0.48 9.91 0.48 49.90 1.64

Attrition after T1 7 4.5 45.29 3.90 8.71 1.29 8.86 1.50 53.12 4.21Attrition after T2 10 6.5 46.30 2.99 9.50 0.98 9.50 0.91 52.08 3.11Attrition after T3 8 5.2 49.38 2.21 10.75 0.45 10.62 0.53 48.99 2.77Attrition after T3 medical 5 3.2 48.40 4.26 10.80 1.39 11.60 1.17 48.68 5.10Missing mixed times 4 2.6 42.75 2.32 10.50 1.44 9.25 1.03 42.19 2.80Total 154

The sample means over the four measurement occasions for complete data (n=120) for the

three outcome measures of interest: learning, immediate retention and delayed recall,

appeared to follow a similar pattern with highest scores before chemotherapy, lowest at one

month after chemotherapy, and increasing improvement over the third and four measurement

occasions. For all three outcome measures, higher scores are indicative of better verbal

memory. Table 1.2 presents the sample means, standard deviations and score minima and

maxima for the outcome variables for the four occasions, and Figure 1.1 graphically depicts

the mean verbal memory trajectory patterns.

One of the aims of this dissertation is to identify subgroups of participants with distinct

trajectory patterns, and as was introduced in an earlier part of the introduction, one of the

methods used to identify such subgroups was with Bayesian latent basis latent class growth

mixture models which utilised estimated standardized recovery parameters to assess degree of

recovery at six and eighteen months post-chemotherapy (T3, T4). The resultant identified

subgroups from the three class models for each of the verbal memory outcomes were used to

construct the class trajectory plots in Figure 1.2. Each participant has an average posterior

probability for belonging to each of the three trajectory groups, and the posterior median for


Table 1.2 Summary Statistics for Learning, Immediate Retention and Delayed Recall

Variable Occasion Mean SD Min Max

Learning T1 53.11 6.97 33 68T2 49.82 7.96 25 66T3 50.54 8.46 32 70T4 52.98 8.52 33 69

Immediate Retention T1 11.32 2.35 6 15T2 10.15 2.46 4 15T3 10.26 2.68 3 15T4 10.87 2.60 4 15

Delayed Recall T1 11.33 2.37 6 15T2 9.77 2.52 4 15T3 9.87 2.82 3 15T4 10.71 2.80 3 15

3040

5060

70

Learning

Before 1 mth 6 mths 18 mths Chemo Post−Chemotherapy

Mea

n W

ord

Cou

nt

46

810

1214

Immediate Retention

Before 1 mth 6 mths 18 mths Chemo PostChemotherapy

46

810

1214

Delayed Recall


Figure 1.1 Sample mean scores for Learning, Immediate Retention and Delayed Recall for fourtimes


an individual participant was used to determine the group allocation of each participant, so

each participant has a probabilistic indicator for group membership.

3040

5060

70

Learning Low Class


Mea

n W

ord

Cou

nt

3040

5060

70

Learning Mid Class


3040

5060

70

Learning High Class


46

810

1214

Immediate Retention Low


Mea

n W

ord

Cou

nt

46

810

1214

Immediate Retention Mid


46

810

1214

Immediate Retention High


46

810

1214

Delayed Recall Low


Mea

n W

ord

Cou

nt

46

810

1214

Delayed Recall Mid


46

810

1214

Delayed Recall High


Figure 1.2 Sample mean scores by class membership for Learning, Immediate Retention andDelayed Recall for the three class latent basis model

BIBLIOGRAPHY 31

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Chapter 2

Literature Review

2.1 Introduction to Longitudinal Data Analysis

In the broadest sense a longitudinal study can be defined to be the situation when information

is collected on measurement units at more than one time point. Examples of measurement

units can be individuals, sites, organisations or anything which is able to be measured multiple

times. For most part in this document, measurement units will be referred to as subjects.

Longitudinal data can be referred to as repeated measures data, where repeated measurements

in the most obvious sense are observations of the same response variable or characteristic

taken at two or more points in time or on multiple occasions.

Longitudinal studies can be classified into two types “retrospective” or “prospective”. A

retrospective study has measurements made only at the final occasion but at that time gathers

information about previous occasions. Information for previous occasions can come from

recall or from archives or records. Prospective studies have measurements made at all

occasions and so ensure that the quality of the information remains comparable [53]. Clinical

trials are an example of prospective longitudinal studies, whereas retrospective studies are

often used to look at the impacts of major events. Similar statistical methodologies can be

used for both types of studies.

Longitudinal data can also be considered as hierarchical in nature, with multiple observations

made over time on each subject. In the analysis of variance framework repeated observations

can be considered as blocked on each subject; and in the multilevel framework observations

35

36 CHAPTER 2. LITERATURE REVIEW

over time are nested in or clustered by subject. Observations on any one subject may be more

alike than observations between subjects giving rise to correlated residuals. Modelling of

growth or modelling change are terms which are regularly used to describe longitudinal data

analysis.

As background to Bayesian growth mixture models, statistical methods used to analyse

longitudinal data in the non-Bayesian and Bayesian frameworks will be discussed.

Restrictions imposed by the data such as unequal intervals between measurement occasions,

missing data, non-normality of the response and a multiplicity of response variables

(multivariate data) can impact on statistical methods. The aims of the statistical analysis

include determining the best suited statistical methodology.

Repeated measures analysis of variance is a traditional way to analyse repeated measures data

and is discussed in the next section. The history of modelling longitudinal data in the

frequentist or non-Bayesian framework includes the concurrent development of multilevel,

hierarchical, mixed effects and structural equation growth models. These are discussed in later

sections.

Modelling of longitudinal data can be considered to have three components:

• Outcome measures repeated over time can be singly or multiply measured, and can be

continuous or discrete in their measurement properties .

• The structure or model of the repeated measures:

– modelling means of times [54].

– linear or polynomial time trajectory profiles [33, 97].

– differences between times [91, 92].

– change-point or piecewise profiles.

– non-linear trajectories with latent basis coefficients.

– the inclusion of both time-invariant and time varying covariates.

• The dependencies or variance covariance structure over time [30].

Inherent in Bayesian analysis is the estimation of the posterior distribution from the

combination of a prior distributions and the likelihood function with Markov chain simulation

2.2. Approaches to Analysis of Longitudinal Data 37

sampling of the posterior distribution until stationarity or convergence is obtained. All models

developed in the frequentist framework can be analysed in the Bayesian framework by the

addition of prior distribution assumptions on the parameters of the models, and by using

combinations of Gibbs, Metropolis-Hasting and reversible jump Markov Chain Monte Carlo

(MCMC) sampling methods to generate posterior probability distributions for parameters.

MLwiN [123] can perform multilevel regression Bayesian analysis using its MCMC

procedure but has limitations with respect to complexity of models and options for prior

distributions. WinBugs [144] is one of the most used softwares for fitting Bayesian models

and has greater flexibility in models and specification of prior distributions.

Raudenbush and Bryk [124, Ch 13] provide an introduction to Bayesian inference for

hierarchical or multilevel models with an application of the Gibbs Sampler to vocabulary

growth data for small numbers (22) of children measured on either six to seven occasions or

three occasions. Longitudinal methods using the Bayesian statistical framework have been

collated more recently in the well regarded texts of Gelman et. al. [44] and Congdon [22, 23].

Further details of Bayesian longitudinal models are presented in a later section.

2.2 Approaches to Analysis of Longitudinal Data

The history of trajectory models can date back to the early 19th century with the work of

Gompetz, Verhurst and Quetelet in the estimation of non-linear trajectory models for

mortality, population growth and growth of the human body over time, where a single

trajectory was developed for a group. Wishart 1938 [157] changed the focus from group

trajectories to estimating specific trajectories for each individual in the group and the

examination of differences in individual trajectories.

2.2.1 Repeated Measures Analysis of Variance

A distinct form of analysis of longitudinal data has been the analysis of variance approach to

repeated data, where both the univariate and multivariate forms handle correlated residuals of

repeated measures data. The univariate repeated measures analysis of variance or

within-subject analysis of variance [145, 147] is the least complex but most restrictive

approach, where the null hypothesis of means at each occasion being equal is tested. The


univariate situation restricts the variance structure to that of sphericity, of which compound

symmetry is a specific form. Compound symmetry requires equal variances at each time point

and equal covariance between any two time points [63]. Other assumptions for the response or

outcome variable are independence of observations between subjects and multivariate

normality. With the multivariate repeated measures analysis of variance approach, the

repeated measures are transformed into adjacent differences so the null hypothesis of means

being equal becomes the hypothesis that the differences of pairs of means are equal to zero.

The assumption of sphericity is not required for the multivariate approach [63, 145]. The

repeated (within) factor can be considered by polynomial (linear, quadratic, cubic) contrasts

or trends. This can be a useful method to understand the trajectory of the mean response by

occasion. This method can be extended to multiple variables being measured over time in

doubly-multivariate repeated measures analysis of variance with a correlational structure

within each variable group, as we had in the multivariate situation previously and also a

correlational structure over the measured variables [145, 147]. The restrictions on the data for

these methods are fixed measurement occasions for all subjects and complete data for each

subject.

2.2.2 Multilevel or Hierarchical Longitudinal Analysis

Multilevel models are mixed effects models in that they involve both fixed and random

components from data which are hierarchically structured. That is, the data are not

independent in the manner of its collection, or the data can be considered to have a nested

structure. A typical example of nested or hierarchical data are measurements on students,

from classes, within schools, within districts, and for longitudinal data measurements at

different times nested in the subject.

Advantages of the multilevel or hierarchical approach to repeated measures are the flexibility

to handle unbalanced structures (incomplete data) or when subjects are measured at

differently spaced time points [124, 141]. This section will predominately consider the

two-level hierarchical model, with a level 1 model of each person’s development represented

by an individual’s change or trajectory over time and level 2 consider changes across

individuals [72, 124, 136].


For longitudinal or repeated measures data the hierarchical structure is measurement occasion

nested in subject, with subjects as the level 2 units denoted by subscript i, and time occasions

within subject as the level 1 units (subscript t) and a single response variable denoted as yti for

occasion t on subject i.

The simplest linear unconditional (no covariates) growth model using a variation of

Raudenbush and Bryk [124] notation with fixed times t = 1,. . . ,T for i = 1,. . . n subjects, can

be depicted in two ways by using level 1 (Equation 2.1) and level 2 sub-models (Equation 2.2

and 2.3) or by a single composite model (Equation 2.4) by algebraically combining equations

2.1, 2.2 and 2.3. The composite representation is required by many of the multilevel statistical

software programs [136] but the sub-model representation is easier to understand the

differentiation between including time-invariant and time varying covariates.

Level 1 Repeated Observation Model: yti = η0i + η1iati + eti (2.1)

Level 2 Subject Model: η0i = β00 + u0i (2.2)

η1i = β10 + u1i (2.3)

Combined Model: yti = η00 + η10ati + u0i + u1iati + eti (2.4)u0i

u1i

∼ N

0

0

,σ2

u0 σu01

σu01 σ2u1

(2.5)

eti ∼ N(0, σ2e) (2.6)

For time point t and subject i: yti are the repeated outcome measures; ati are the time related

variables; η0i and η1i the random intercept and slopes; eti are the level 1 residual where

eti ∼ N(0, σ2

e

)and u1i, u1i the level 2 random effects with zero mean and variances σu0 and

σu1 respectively. For the simplest situation for a linear trajectory ati takes the values

t − 1 = 0, 1, 2, 3, 4 . . . T − 1 for all i and t times.

There are two main forms of the unconditional model: the unconditional means model (or null

model) which has no predictors at any level where the partitioning of variation predominates,

and the unconditional growth model with only the time predictor included [136]. The

unconditional linear growth model interprets the level 1 residual variance σ2e as the deviations


around an individual’s trajectory and level 2 residuals as between-subject variability in initial

status (intercept) σ2u0 and rate of change (slope) σ2

u1.The response over time can be modelled

most simply by a linear regression equation which can have different regression coefficients

for each individual. So each subject may have their own regression curve with the intercepts

and slopes varying randomly by subject, around a common trajectory specified by intercept

β00 and slope β10. For the linear model the following illustrate the three types of subject level

error structures. These can be generalised for more complex growth models.

Random Intercept

u0i

u1i

∼ N

0

0

,σ2

u0 0

0 0

Random intercept random slope

u0i

u1i

∼ N

0

0

,σ2

u0 0

0 σ2u1

Full random effects

u0i

u1i

∼ N

0

0

,σ2

u0 σu01

σu01 σ2u1

in all cases eti ∼ N(0, σ2

e

)

More General Models

More general models can include more complex growth trajectory shapes with higher order

polynomials, piecewise linear or latent basis coefficients determining non-linear trajectory

response. The addition of time varying and subject varying (time invariant) covariates can also

be part of the modelling process [54, 56, 63, 113].

Consider the level 1 component of the unconditional two level polynomial growth model of

degree p, for i = 1,. . . , n subjects, ati is function of time for person i at time t and ηpi is growth

trajectory parameter for subject i associated with the pth power of the time variable ati.

Level 1: yti = η0i + η1iati + η2ia2ti + . . . + ηpia

pti + eti (2.7)

This formulation permits the number and spacing of measurement occasions to vary across

subjects where each person is observed on Ti occasions. This can be generalised to functions


of time other than polynomials denoted as apt for the (p + 1)th growth parameter ηpi.

Conditional models with the inclusion of time-varying covariates (TVC) and subject-varying

or time invariant covariates (TIC) can be denoted by the level 1 and level 2 equations

respectively. Using the linear trajectory model with a single time varying covariate wti and a

single time invariant covariate xi the equations 2.1 and 2.2 becomes

Level 1 Repeated Observation Models: yti = η0i + η1iati + γtwti + eti (2.8)

Level 2 Subject Model: η0i= β00 + β01xi + u0i

η1i= β10 + β11xi + u1i (2.9)

For each of the p + 1 individual growth parameters, multiple subject specific covariates can be

included as follows:

Level 2: ηpi = βp0 +

Qp∑

q=1

βpqxqi + upi (2.10)

where xqi is the qth measured subject level characteristic (factor or time invariant covariate

like initial age, IQ or education), βpq is effect of time invariant covariate xqi on (p + 1)th

growth parameter, upi is the (p + 1)th random effect with mean 0, the set of P + 1 random

effects for person i are assumed to be multivariate normally distributed with full covariance

matrix T of dimension (p + 1) ∗ (p + 1). Similarly multiple time varying covariates wsti can be

included in the level 1 equation 2.8 [136] with γst in place of γt.

A common assumption for the error structure eti is that each eti is independently normally

distributed with mean 0 and constant variance σ2e . However the level 1 error variance can take

more a complex form with the inclusion of correlated times and time specific variances.

Multivariate outcomes can be modelled with the inclusion of dummy variables indicating each

of the multiple outcome variables hence adding another level to the model [55, 90, 141]. An

alternative approach to the handling of multivariate outcomes is the factor analytic approach

which utilises a principal component analysis [55]. Uncorrelated linear functions of the

multivariate outcomes are obtained which sequentially maximize variances. In the multilevel

framework this can be conducted on the covariance or correlation matrix of residuals at a


specific (or at each) measurement level. This forms the basis of multilevel structural equation

modelling.

2.2.3 Structural Equation Modelling and Latent Growth Curve Modelling

Multilevel modelling of longitudinal data is closely related to latent curve or latent growth

curve models of Duncan et al. [34], Meredith and Tisak [97], Willett and Sayer [155]. The

latent growth curve model uses time to specify the latent variable structure, so that

consecutive measurements are modelled by the latent variables of intercept and slope of the

latent growth curve. Curran and Bauer [4, 29] indicate the close connection and often

equivalence of methods of multilevel/hierarchical modelling and structural equation

modelling, in particular that the multilevel and structural modelling longitudinal growth

curves under ‘broad conditions are empirical and analytically identical’ [29].

Latent growth curve models are special cases of multilevel regression and can also be

considered as a special application of structural equation modelling [19]. Latent growth curve

models are special cases of structural equation models as means are estimated and the loading

parameters λ are fixed instead of being estimated in the usual structural equation modelling

situation. The latent variable modelling and SEM framework is considered a single level

analysis with y = (y1 y2 y3)′ as a multivariate outcome, where the growth factors of the

intercept, slope and quadratic latent variables are measured by the multiple indicators of yt

(measurement part) and the structural part relating the growth factors together [7, 33, 136].

Models for the linear latent growth curve models for continuous outcomes yti with

i = 1, 2, . . . , n and t = 1, 2, . . . ,T can be described by two random effects and time specific

residuals and are presented in the following equations.

yti = η0i + η1iat + εti

η0i = ν0 + ζ0i

η1i = ν1 + ζ1i

where η0i, η1i are growth factors with η0i the initial level factor (intercept), η1i the growth rate

factor or slope, at is a function of time generally at = 0, 1, 2, . . . ,T − 1, εti is the time specific


residual, ν0, ν1 are the means of the growth factor parameters, and ζ0i, ζ1i the variances of the

growth parameters.

The random errors for the growth parameters ζ0i, ζ1i are assumed to be normally distributed

with mean zero and covariance matrix Ψ, which takes the form

Ψ =

ψ00 ψ10

ψ10 ψ11

.

In matrix terms the latent growth curve models can be expressed as y = ηΛ + ε for individual i

with elements

y1i

y2i

...

yTi

=

1 0

1 1...

...

1 T − 1

η0i

η1i

+

ε1i

ε2i

...

εTi

Multilevel growth models can be represented graphically by SEM growth curve path models

[160] with Figure 2.1 showing a graphical representation of a linear latent growth model for

three measurement occasions.

Figure 2.1 Linear Latent Growth Curve Model for three time points.

The latent growth curve approach to analysing change maps the multilevel model for change

to the structural equation approach [154], and is similar to the random intercept random slope

models of earlier sections. Heterogeneity over time can be easily be modelled with latent


growth curve models, and complex models for residual variances can be handled [90, 101].

Structural equation software packages like AMOS 4 or MPLUS can be used to model the

latent growth curve approach. If only 3 time points are available then the highest order model

would be linear, unless other constraints are employed. The more time points available, the

fewer the restrictions that are needed to retain degrees of freedom to assess model fit.

2.2.4 Piecewise Linear Growth Models

An option for modelling curvilinear growth trajectories is to break the model up into separate

linear components. Piecewise linear functions are continuous functions which have change in

slope at a number of nodes but are linear or have constant slope between these nodes

[124, 141]. The basic piecewise function is linear on a given interval and a constant outside

this interval. Or a two-piece linear growth model can be a combination of two linear segments

of the functional form yti = η0i + η1ia1t + η2ia2t + eti where a1t = t for t ≤ τ and τ for t > τ,

and a2t = (t − τ) for t > τ and 0 for t ≤ τ with the change point or knot at τ.

An example of a two-piece linear growth model in the multilevel modeling framework or as a

mixed effect model for six time points (1, 2, 3, 4, 5, 6) [124] with the node at time 3 (τ = 3 has

the first time component linear from time 1 to time 3 a1t = (012222)′, and the second time

component linear from time 3 onwards a2t = (000123)′.

This is useful in the comparison of growth rates in different periods [124], with the periods

being marked by a transitional change [156] and with the discontinuity, changepoint, node or

transition point occurring at a known or unknown time point. Multiple nodes or

discontinuities for more piecewise segments can be specified and spline functions extend

piecewise linear functions into smooth piecewise polynomials with nodes defining the range

of each polynomial function [141].

Piecewise linear trajectory models have been used in modelling developmental processes

primarily with fixed transition points in the frequentist multilevel [124, 136, 141] and

structural equation modelling frameworks [7, 33, 100]. Known changepoint models have been

used for substance and alcohol usage for schoolchildren with the changepoint at the transition

from middle school (grades 6 to 8) to high school (grades 9 to 10 or grades 9 to 12) [9, 79, 80],

and to assess the effectiveness of an intervention program for low birth weight pre-term infants


on intellectual development [74]. Educational assessment [135], wage patterns for high school

dropouts [136] and effectiveness of intervention processes [149] were other applications of

piecewise trajectory models. Multiphase models with three distinct components, baseline and

two non-linear segments with known changepoints for cortisol trajectory patterns [122]

demonstrate the flexibility of changepoint models for complex longitudinal profiles.

The monitoring of longitudinal biological markers, where sudden changes of level or

gradients of marker trajectories are important indicators of status change obviously utilises

unknown changepoint models and have been used to determine changes in cognitive decline

as an indicator of dementia and Alzheimer’s disease [58, 66]. Other applications for estimated

changepoint models include non-verbal performance data from childhood to adolescence [28]

and for responses of phosphate elimination for two participant groups (control and obese)

from a glucose challenge [28].

Bayesian estimation methods have been used for known changepoint models of longitudinal

symptom profiles in Chronic Prostatitis chronic pelvic pain syndrome (single fixed

changepoint) [76] and daily menopausal symptom trajectories with multiple known

changepoints [71]. However, Bayesian estimation methods are demonstrated primarily with

the random coefficient estimated change points for multiple phase longitudinal problems

especially for disease biomarkers and include the monitoring of CA125 levels for ovarian

cancer screening [138], prostate-specific antigen (PSA) level changes as biomarkers for

prostate cancer onset [116, 158], non-compliance measured by changes in mean corpuscular

volume (MVC) levels for HIV patients [117] and glaucoma progression [67].

Comparisons of traditional maximum likelihood estimation methods and Bayesian methods

were undertaken by McArdle and Wang [95, 150] and present a range of piecewise segmented

models with an estimated turning point or change point, including linear-linear (two piece

linear), quadratic-linear change point mixed effects models for longitudinal life-span growth

curves of cognition.

2.2.5 Latent Basis Growth Models

The work of Meredith and Tisak on latent growth models [97] introduced the modeling of

curvilinear trajectories with basis functions which could be either fully or partially known.


Completely specified basis functions can result in the standard linear or polynomial trajectory

response. However with partially known latent basis coefficients the optimal trajectory shape

can be estimated from the data [93, 94, 97] in a similar manner as estimating factor loadings

of the measurement part of a structural equation model, and thereby ensures flexibility in

fitting non-linear forms.

The latent basis growth model is flexible in the way the weights or basis coefficients are set in

being either fixed or partially estimated. For example a linear growth model over four

measurement occasions can be specified with fixed basis coefficients of (0, 1, 2, 3) or (0, 0.33,

0.666, 1) [94, 160] where the latter case shifts the units of the slope to a proportion of the time

range while retaining the linear trajectory. A monotonic increasing non-linear growth model

can be modelled with the first and last basis coefficients being fixed to zero and one, but with

the intermediate latent basis coefficients being estimated from the data [94, 160], so as to

obtain estimated change relative to the overall change. An alternative model would have the

first two basis coefficient fixed to 0 and 1 and have subsequent coefficients estimated

[7, 94, 97, 148], where the estimated change is relative to the initial change. Nonlinear

decline, and fluctuating change trajectories are a few of the possible other options handled by

latent basis coefficient models [160]. Bayesian latent basis growth models were introduced by

Zhang et al. [160] for the analysis of readings scores of children at four measurement

occasions over a six year period.

Figure 2.2 presents a range of trajectory profiles which different latent basis models can

represent, with linear decline α=(1,0.66,0.33,0), nonlinear decline with α=(1,0.33,0.17,0),

decline followed by a flat response α=(1,0,0.03,0.03), and decline with recovery

α=(1,0,0.16,0.5) for four measurement occasions.

Although latent basis models have been proposed for some time, there has been limited

implementation of these models. Recent research has utilised latent basis models for

nonlinear monotonically increasing responses, generally fitted in a non-Bayesian framework.

Applications with the first and last coefficients fixed to zero and one included modelling of

cortisol responses over 8 measurement occasions [122], the assessment of the five individual

learning trials which when summed produce an overall verbal memory learning score for the


1 2 3 4

45

67

89

10

Time

Out

com

e

1 2 3 4

45

67

89

10

Time

Out

com

e

1 2 3 4

45

67

89

10

Time

Out

com

e

1 2 3 4

45

67

89

10

Time

Out

com

e

Figure 2.2 Plots of some different trajectory profiles for latent basis models

Rey Auditory Verbal Learning Test [159], and the developmental trajectories of body mass

index (BMI) measurements for girls from childhood to adolescence over six measurement

occasions [148].

The latent basis growth model was written as a random effects model which is equivalent to a

hierarchical model in which the variability at each level is specified separately. This can be

presented in a similar way as equations 2.1 and 2.2 where ati the function of time is denoted

by the latent basis coefficients αt.

To describe the model, let yti be the response of individual i (i = 1,2,. . . n) at time t (t

=1,2,3,4). Then

yti ∼ Normal(µti, σ2ε ) (2.11)

where µti = η0i + η1iαt (2.12)η0i

η1i

∼ Normal

β0

β1

,σ2

0 σ01

σ01 σ21

(2.13)


where η0i represents the expected intercept or level; η1i represents the linear slope; and the

latent basis coefficients α1 and α2 are fixed and α3 and α4 are estimated. The random effects

η0i and η1i are considered to be random effects with means β0 and β1 and variances σ20 and σ2

1,

respectively, and covariance σ01 which can also be expressed as ρσ0σ1.

In the Bayesian context, the random effects for the intercept or level, slope and intercept/slope

interaction can be estimated using a range of different prior distributions and include the

inverse-Wishart or Wishart, Uniform and half-Cauchy distributions [43, 44, 45]. The

inverse-Wishart distribution was initially considered appropriate to estimate the

variance/covariance parameters of a multivariate normal distribution, but can be problematic

with variances close to zero, whereas the Uniform, or half-Cauchy or scaled-Wishart [43, 45]

reduces this problem. The latent basis coefficients of α3 and α4 used mildly informative priors

Normal(0,4) in order to be non-influential but sufficiently well defined for enhanced parameter

estimation.

2.2.6 Bayesian Growth Models

It has already been discussed that growth models can be considered as hierarchical or

multilevel models, and so are easily interpreted in the Bayesian context, where the posterior

distribution is a function of the likelihood of the model based on the data L(y) and prior

information on the parameters of the model θ.

p(θ|y) ∝ (θ)p(y|θ) = p(θ)L(θ; y) (2.14)

There are extensive applications of longitudinal data in the Bayesian context. The main

differences between the Bayesian and non-Bayesian approaches are the inclusion of prior

information for the modelling of the growth and variance/covariance parameters. Several

Bayesian hierarchical (random effects or mixed) models or latent growth curve for

longitudinal data based on continuous outcomes and/or discrete outcomes have been

presented in recent papers. Continuous or normally distributed univariate outcomes with

Bayesian linear growth curve models, that is, two level models with subject as the

second-level variable and time as the level-1 variable have been presented by Choi and Seltzer

[17], Choi et al. [18], Seltzer et al. [134], Sithole and Jones [137], Zhang et al. [160] and with


Choi and Seltzer introducing a third level [17]. The recent paper by Zhang et al. [160]

compares Bayesian and maximum likelihood estimates for a latent basis trajectory model, and

considers a range of prior distributions for the intercept and slope random effects which

includes noninformative, half-informative and fully informative priors under the Bayesian

approach. These models have assumed a simple homogenous variance model for time with

mention of a more complex time possibility.

Dissertations by Choi [16], Leiby [76], Patil [115] demonstrate the recent interest in Bayesian

approaches to hierarchical or latent growth trajectory models. Discrete outcomes have also

been addressed in the Bayesian analysis of longitudinal data with binary outcomes being

presented in the papers of Carlin et al. [13], Erkanli et al. [39], O’Brien and Dunson [112] and

ordinal outcomes with Dunson and Colombo [35], Pettitt et al. [118].

Structural equation modelling in the non-Bayesian framework is based on the covariance

structure of response and predictor variables, whereas the Bayesian approach has its focus on

the raw observations. The traditional structural equation modelling approach uses maximum

likelihood estimation and requires asymptotic normality assumptions and is therefore only

valid for large sample sizes. Bayesian analysis is less dependent on asymptotic assumptions

and is able to produce reliable results with smaller sample sizes [75] with prior distributions

having a significant role when samples sizes are small or moderate [75]. As the latent growth

curve is a special case of structural equation modelling, the ability to utilise small sample

sizes in the Bayesian framework is most useful.

Longitudinal change in cognitive performance for individuals with mild cognitive impairment

was analysed using linear mixed effects or multilevel growth models based on age [2]. The

neuropsychological battery of 22 tests was reduced by structural factor analysis to four

common factors of general knowledge, episodic memory, spatial skill and executive function.

Baseline individual factor score weights were used to generate factor scores for subsequent

repeated time of measurement. The impact of age, education, an apriori four level grouping

of cognitive function - normal, stable, decliner, converters (diagnosed with Alzheimer’s

disease), and the presence of the APOE ε4 allele was assessed.

Although this paper did not use a Bayesian approach, it is indicative of the interest in

multivariate outcomes and the grouping of trajectories into subgroups and the assessment of


possible predictors in the area of cognition.

Returning to a simple linear multilevel model of longitudinal data with normal outcomes yti,

ati is time related variable (time or age scores), for i individuals measured at t = 1, . . . ,T

occasions a random effects model can be denoted by

yti = η0i + η1iati + eti

η0i = β0 + u0i

η1i = β1 + u1i

eti ∼ N(0, σ2e)

u0i

u1i

∼N

0

0

,σ2

0 σ01

σ01 σ21

.

This is specified in a Bayesian framework as

yti ∼ N(µti, σ

2e

)

µti = η0i + η1ia1ti

η0i ∼ N(β0, σ

20

)

η1i ∼ N(β1, σ

21

)

or as

η0i

η1i

∼N

β0

β1

,σ2

0 σ01

σ01 σ21

with prior distributions to be specified on the hyper-parameters β0, β1 and for the level specific

variances σ2e , σ2

0, σ21 or covariance σ01.

Non-informative prior distributions for the hyper-parameters β0, β1 can be set as

N(0, 100, 000)) or N(0, 1000). Traditionally for the level 2 variances σ20, σ2

1 (collectively

denoted as σ2β) the σβ are set to have Inverse −Gamma(ε, ε) prior distributions where ε is

small (0.01 or 0.001), since this prior distribution has conditional conjugacy properties.

However an infinite mass can occur if σβ → 0. A Uniform prior distribution on σβ has a finite

integral near σβ = 0 [43]. Similarly, the variance components of a multivariate normal

distribution were given inverse-Wishart priors with degrees of freedom ω where ω are the


number of varying coefficients but later work by Gelman [43], indicates the use of

half-Cauchy or Uniform distributions on each of the components of variance of a bivariate

normal (ω = 2) with σ0, σ1 as Uni f orm(0, 100) or Uni f orm(0,U) where U is appropriate to

the scale of the variance and σ01 as Uni f orm(−1, 1). Gelman indicates the inverse-Wishart

with ω + 1 degrees of freedom has the effect of setting a Uniform distribution on the

individual correlation parameters [43, 45]. The inverse-Wishart with ω+ 1 degrees of freedom

is reasonable for the correlations but constrains the diagonal components σ0, σ1 . . . and a

scaled inverse-Wishart model is suggested to overcome this problem [45]. A Uniform prior on

σβ can have difficulties when the number of level 2 groups (in our case subjects) is small, but

for most purposes this is not a problem.

The likelihood of the observed data Y or yti for the level 1 model given the parameters η and

σ2 for normally distributed Y is

f (Y |η0, η1, σ2) ∝ 1

σexp

−n∑

i=1

Ti∑

t=1

(yti − η0i − η1iati)2/(2σ2)

(2.15)

2.2.7 Growth Mixture Models (GMM)

The most general multilevel longitudinal model for growth has the ability for individual

trajectories for each subject for outcome yti over time to be modelled. Any heterogeneity in

these trajectories is absorbed into the random effects. An underlying assumption with this

model is that all subjects come from a single population with common parameters. The main

goal of finite mixture modelling is to identify two or more latent classes that represent

sub-populations that are hypothesised to exist but which were unable to be observed by direct

measurement. Growth mixture models are a special application of finite mixture models

where parameter differences across unobserved sub-populations are a result of latent

trajectory classes. Instead of subject variation about a single mean growth curve, the growth

mixture model has different classes of individuals varying around different mean growth

curves [99, 100, 101, 106, 107].

If Yi = (y1i, y2i, . . . , yTi) denotes a longitudinal sequence of measurements on subject i over T

measurement occasions, then let P(Yi) denote the unconditional probability of of the sequence

of measurements Yi, and Pk(Yi) the probability of Yi given membership in class k and πk


denotes the probability of a randomly chosen population member belonging to class k. So by

aggregating the K conditional likelihood functions Pk(Yi) the unconditional probability of the

data Yi is:

P(Yi) =

K∑

k=1

πkPk(Yi), (2.16)

where the group membership probabilities πk as constrained by∑K

k=1 πk = 1. This is the sum

across all K classes of the probability of Yi given subject i′s membership in class k weighted

by the probability of membership in class k. This equation defines a ”finite mixture model”

since it sums across a finite number of discrete classes that compose the population. Growth

curve models are also a type of mixture model but the mixing distribution is not finite.

The likelihood of the sample of N subjects is the product of the individual likelihood

functions of the N individuals specified by equation 2.16: L =∏N P(Yi).

Categorical latent variable ci represents the unobserved subpopulation membership for subject

i, with ci = 1, 2 . . . ,K and where c is the latent class or trajectory class variable. A covariate x

is also included which influences class membership c, and the growth parameters.

A multinomial logistic regression model is used to predict the latent class variable c by the

covariate x for K classes.

P(ci = k|xi) =eδ0k+δ1k xi

K∑

s=1

eδ0s+δ1s xi

(2.17)

The growth mixture model considers separate growth models for each of the K latent classes,

with differences across classes being found in the fixed effects of the intercept and slopes.

Differences in classes may also occur with the effect of the time invariant covariate x.

The growth mixture model can also be extended to predict a categorical (distal) outcome,

where the latent trajectory class variable is used to predict the distal outcome together with the

time-invariant covariate x by way of a logistic regression. This extension is referred to as

general growth mixture model (GGMM) in the overview paper by Muthen [101] stemming

from original work presented by Muthen and Shedden [104] using the Expectation

Maximisation EM algorithm.

A latent class growth model (LCG) is a special type of growth mixture model, where growth


factor variances and covariances are set to zero [108]. The LCG model is considered to be

semi-parametric due to the absence of random effects in the underlying growth models

[106, 110]. This method has also been recommended as a way to set starting values for growth

mixture modelling [105]. Generalisation of latent class growth models can include predictors

of group membership, predictors of trajectories and the inclusion of dual or related trajectories

[107].

The impact of predictor variables or covariates can be considered in a number of ways in

growth mixture or latent class growth models: as part of a profile description, as predictors of

the parameters of the class trajectories or as predictors of class membership.

Firstly, in determining characteristics of trajectory group members as distinct from members

from other trajectory groups, group or class classification can be based on posterior

probabilities. The posterior probability of class membership is a measure of an individual’s

likelihood of belonging to each of the k trajectory groups or classes. The reference to

’posterior’ probability is because they are computed “postmodel estimation using the model’s

estimated coefficients” [107], as distinct from Bayesian posterior probabilities obtained after

simulations. Given an estimated model, each individual obtains a posterior probability

estimate for each class computed as a function of the model parameter estimates and the

individual’s observed values by way of Bayes’ theorem. The class with the highest posterior

probability will determine the most likely class membership for the individual [107]. The

posterior probabilities of group membership can determine the ability of the model to clearly

differentiate between subjects. An average posterior probability of group membership equal to

1 demonstrates the optimal or ideal situation, with Nagin [107] specifying a rule of thumb of

at least 0.7 for all groups as an acceptable measure. Profiles of group membership, for

subject-level characteristics which may be associated with group membership can be

determined.

Secondly, covariates can be included in the trajectory specifications as adjustments to the

intercept and/or as interactions with other slope parameters. The growth mixture models of

Muthen (GMM) [100, 102, 104] and Nagin (LCGA) [107] include both time-invariant and

time-varying covariates as part of the trajectory models where the models for the kth group

are:


Level 1 or occasion level

ytik = η0ik + η1ikg1(t) + η2ikg2(t) + κtkwti + εtik

where the Level 2 or subject level part of the model is

η0ik =β0k + γ0kxi + ζ0ik



where g1(t) and g2(t) are functions of time, wti is a time-varying covariate and xi is a time

invariant covariate, with εtik as time-specific residuals with zero means and covariance matrix

Σ for subject i,occasion t and class k. This can be generalised for multiple time-varying and

time-invariant predictor variables. Both Nagin [107] and Muthen [101] use only binary

covariates in the trajectory parameterisation, and for time-varying covariates Muthen

advocates time specific κtk coefficients whereas Nagin uses a common κk for all measurement

occasions. The paper presented in chapter 5 uses the time-specific κtk for all time-varying

covariates, both continuous and binary.

The notations of Muthen and Nagin differ, in that Nagin uses the class k as ykti and Muthen

uses ytik, and the Nagin latent class growth mixture model has all ζk = 0. The Nagin model

and notation are used in chapters 3, 4, 5 and 6.

Thirdly, covariates can be predictors of trajectory group membership. A multinomial logistic

regression model can specify the functional relationship between the probability of class

membership πk for the kth group, where (k = 1, . . . ,K), and set of M covariates xmi, where

m = 1, . . . ,M, and is estimated simultaneously with the trajectory parameters [102, 104, 107].

The probability of class membership for class k can be defined as:

P(ci = k|xmi) = πk(xmi) =eδ0k+δ1k x1i+...+δmk xmi+···+δMk xMi

∑Ks=1 eδ0s+δ1s x1i+...+δms xmi+···+δMs xMi

, (2.18)

where the logistic regression parameters δmK are set to zero for last class K being set as the

reference class.


For K = 2

log[π1(xmi)π2(xmi)

]= δ01 + δ11x1i + . . . + δm1xmi + · · · + δM1xMi. (2.19)

Muthen indicates that the same time-invariant covariates can be used in both the trajectory

part of the model as well as the logistic prediction of group membership part , this often

results in the covariate being a significant contributor to the trajectory part of the model and

non-significant contributor to group membership [104, 107], whereas if used in only the

logistic part of the model results as a significant contributor to group membership. Nagin

[107] restricts covariates in the predictor part of the model to those variables available at the

initial measurement occasion.

There are examples of inclusion of covariates in frequentist literature for all of the three ways

of incorporating covariates described previously. Covariates assessed after the allocation of

subjects (level 2 variables) to classes by maximizing the posterior probability of group

membership, are evident in studies of social research with BMI change [148], post-traumatic

stress disorder [37], adolescent behaviour [110] and alcohol consumption [60]. The inclusion

of covariates in predicting the probability of group membership has been used in areas of

education [68, 70, 151], substance and alcohol use [20, 31, 61, 126], PTSD in Gulf War

veterans [114], delinquent behaviour [101, 104, 106, 107, 109, 152], psychiatric studies [69]

and in the medical area of cancer biomarker research [81]. Examples of the inclusion of

time-varying covariates are in alcohol usage studies [31, 61, 78], smoking ban studies [57]

and education [68, 70]. Time-invariant covariates have been used as predictors of trajectory

parameters in the areas of alcohol research [79, 104], smoking bans [57], criminality studies

[69], marketing [26] and education [103, 151].

Growth mixture models have been used with the latent basis parameterisation of time for BMI

change in young females [148] aged from 5 to 15 years, where four distinct non-linear

trajectory groups were determined. In this model the latent basis coefficients αt were set as

α1 = 0 and α8 = 1 and the remainder estimated from the data. A two-piece linear trajectory

growth mixture model has been used for adolescent smoking, where the changepoint varied

depending on the class membership [21].

The methods used to determine the number of classes which best represent the data for

frequentist models are varied and include the likelihood ratio test [102], Bayesian Information


Criteria BIC [133] which adjusts the likelihood by the number of parameters and sample size,

the Lo-Mendell-Rubin likelihood ratio test [89], and a bootstrap likelihood ratio test [96].

Simulation studies have indicated for growth mixture models that BIC and the bootstrap

likelihood ratio test (BLRT) are the most reliable [111]. There has been evidence of bias with

assuming level-1 residual variances to be constant across classes when they do vary in each

subgroup, and worsening with disparate class proportions and increased numbers of classes

[38].

2.2.8 Bayesian Growth Mixture Models

There are limited journal articles using Bayesian growth mixture models, namely those of

Elliott et al.[36], Mohr [98], Slaughter et al.[140] and Leiby et al. [77]. However recent PhD

theses of Leiby, Patil, Slaughter [76, 115, 139] have used these Bayesian growth mixture

methods.

An example of the Bayesian approach to growth mixture models is presented in a paper which

extends the finite mixture model (mixture of classes) of trajectories to consider joint

modelling of continuous and discrete trajectories into a general growth mixture model with

the inclusion of a binary covariate to assist in the prediction of latent classes [36]. A mixture

of finite latent classes was estimated as the driver of an underlying quadratic trajectory for a

daily reported continuous variable (daily reported positive affect scores) and a linear trajectory

for daily reported binary variable (presence or absence of negative events) for subjects

following a myocardial infarction [36]. This Bayesian model built on the general growth

mixture models of Muthen [99, 100, 101]. The paper looks at daily observations for a period

35 days after the myocardial infarction for 35 subjects. One, two and three class models were

fitted both jointly and separately to the trajectories and a binary covariate of presence-absence

of baseline clinical depression was used as predictor of classes. These models considered the

two longitudinal processes to be independent, but could be extended to correlated processes.

A Bayesian approach to multivariate growth curve latent class models is presented by Leiby

[76, 77] where components of factor analytic models, linear and generalised linear mixed (and

piecewise linear) effects models and latent class models are combined.

The modelling of the multivariate aspects of this paper are of interest. The factor analytic


modelling of multiple continuous outcomes assumes the multiple outcomes are characterised

by a single latent factor outcome, and each outcome is characterized by its own intercept, its

loading on the single latent factor and measurement error correlated over time (similar to a

factor analysis model with an intercept term) as presented by [128]. The mixed effect and

latent class components followed the previously described growth mixture modelling of

Muthen.

Several extensions to this work were identified:

• trajectory extending to a nonlinear curve

• inclusion of multiple latent variables underlying the multiple observed outcomes

• inclusion of covariates in the factor analytic model

• relaxation of the time-invariant relationship between the multiple observed outcomes

and their underlying single latent trait variable

• avoidance of the starting value problems with Gibbs samplers

Longitudinal change-point mixture models were fitted using the Bayesian techniques of

Gibbs, Metropolis-Hasting and reversible jump steps to assess noncompliance in treatment of

patients with HIV by Pauler and Laird [117]. The low numbers of noncomplying patients (8

out of 187) identified with these methods prevented the assessment of associations with

demographic or clinical covariates.

A Bayesian reversible jump MCMC approach to model finite mixtures of linear changepoint

time-to-event models [65], was used for regularly observed lung function measurements

FEV1 following lung transplant operations. Additional work was suggested to incorporate

serial correlation into the model.

Bayesian finite mixture models were used to identify from up to three to five trajectory classes

based on ten knot spline functions of a continuous time-based covariate (age between 4 to 26

years) for the biometrical measurement of triceps skinfold for young Gambian females [130].

The interest in identifying differentially responding sub-groups (responder and non-responder;

improvers and mild improvers) is evidenced in the dissertation of Leiby [76] the application of

Bayesian multiple outcome latent growth mixture models to two clinical datasets. Firstly, to


interstitial cystitis sufferers from a multi-centre trial assessing the efficacy of a specific

treatment with the aim of identifying a subset of true responders with multiple continuous

outcomes. Secondly to data collected for the Chronic Prostatitis Cohort Study where the

multiple outcomes were a combination of continuous, ordinal and binary measures assessed

over time where binary predictors of the improver subgroup were assessed using odds ratios.

The finite mixture part of the growth mixture models for K components with the probability of

class membership πik, for class k = 1, . . . ,K, subject i, i = 1, . . . n where∑K

k=1 πk = 1 follow a

multinomial distribution. For the unconditional mixture model, no covariates included as

predictors of the probability of group membership, the πik = πk the natural conjugate prior

distribution is the Dirichlet distribution π ∼ Dirichlet(α1, α2, . . . , αK) where αk = 1 for all k

which sets equal densities for πk [44]. When covariates are included as predictors of the

probability of class membership as in equations 2.18 and 2.19 the prior distribution for the δmk

can take various forms. Non-informative Normal prior distribution N(0,1000) have been used

for logistic multinomial regression models [24, 25]. The prior distribution N(0, 9/4) has been

used by Leiby [77] and Elliott [36] in following the relatively flat proper priors of Garrett and

Zeger [40]. Weakly informative Cauchy distribution priors with mean zero and scale 2.5 have

also been used for the coefficients of logistic regression models [46], where non-binary

variables were rescaled to mean zero and standard deviation 0.5 and binary variables set to

have a mean zero and differ in their upper and lower condition by 1.

2.3 Aspects of Bayesian Analysis

2.3.1 Markov Chain simulation

Markov chain simulation or Markov chain Monte Carlo (MCMC) is a method of drawing

values of θ from approximate distributions and then correcting those draws to better

approximate the target posterior distribution p(θ|y). The samples are drawn sequentially with

the distribution of the sampled draws depending on the last value drawn; hence the draws

result in a Markov chain. The key to the success of the method is that the approximate

distributions are improved by each step of the simulation resulting in convergence to the target

distribution. Both the Gibbs sampler and the Metropolis algorithms [52] are special cases of

2.3. Aspects of Bayesian Analysis 59

Markov chain simulation.

The Gibbs sampler is a Markov chain algorithm useful for multidimensional problems, and is

described as alternating conditional sampling, defined in terms of sub-vectors of θ. The joint

posterior distribution is decomposed into a sequence of simpler conditional distributions,

where the goal is to generate a data point from the conditional distribution of each parameter,

conditional on the current values of the other parameter [50]. Let θ = (θ1, . . . , θq) with q

unknown parameters in the model of interest. The conditional distribution

p(θi|θ1, . . . , θi−1, θi+1, . . . , θq; y) for θi can be obtained using Bayes theorem. The following

scheme is used to sample the parameters from the conditional distribution at the (i + 1)th

iteration with current value θ(i) = (θ(i)1 , θ

(i)2 , . . . , θ

(i)q ; y), update θ(i+1) = (θ(i+1)

1 , . . . , θ(i+1)q ) by

sequentially generating

θ(i+1)1 from p(θ1|θ(i)

2 , θ(i)3 , . . . , θ

(i)q ; y)

θ(i+1)2 from p(θ2|θ(i+1)

1 , θ(i)3 , . . . , θ

(i)q ; y)

...

θ(i+1)q from p(θq|θ(i+1)

1 , θ(i+1)2 , . . . , θ(i+1)

q−1 ; y)

This iteration process can be repeated B times, and for sufficiently large B, θ(B) can be viewed

as simulated observations from the posterior distribution p(θ|y); the simulated observations

after B are recorded for further analysis. Thinning, that is retaining every ath observation, is

used to reduce autocorrelation and computer storage space. The posterior mean as a point

estimate

θ̄ =1N

N−1∑

m=0

θ1+ma

with posterior variance

Var(θ) =1

N − 1

N−1∑

m=0

(θ1+ma − θ̄)(θ1+ma − θ̄)T .

Credible intervals 100 × (1 − α)% can be constructed from the 100 ×α/2 percentile as the

lower bound and (1-α/2) percentile upper bound.

The Metropolis-Hastings algorithm can increase efficiency with the utilisation of a random


work process through the parameter space.

2.3.2 Priors for Bayesian Hierarchical Models

A comparison of Bayesian and frequentist methods of fitting variance component and

random-effects logistic regressions, resulted in the endorsement of Bayesian approaches [10].

Simulation results with respect to bias in level 2 variances indicate the impact of prior

distributions (inverse gamma and uniform) under certain conditions. Gelman [42] investigates

this more closely and recommends the use of a uniform prior instead of the traditional inverse

gamma for non-informative priors, and the use of the half - t distribution (absolute value of

Student-t distribution centred at zero) or half-Cauchy for the variance of level 2 variables.

The effect of 13 different priors on the scale parameter for simulated random effects

meta-analysis data was investigated [73] and found that the choice of prior was more

important when the number of level 2 subjects were limited. Although biases were limited,

the precision of estimates varied greatly and hence credible intervals and statistical inferences

could differ.

2.3.3 Model Selection

In selecting between several models from the same data set, comparisons can be made using

summary measures of fit. The deviance is a statistic which is available in both the frequentist

and Bayesian modelling frameworks, and is equal to minus twice the log-likelihood

deviance = −2log(L) where the likelihood is the probability of the data given the estimated

parameters of the model L = p(y|θ). In the frequentist methodology the addition of parameters

to the model is expected to improve the fit. Even if an additional single parameter does not

contribute to the model, it will reduce the expected deviance by one, so the addition of k

predictors is expected to reduce the deviance by k. If k predictors are added and the deviance

is reduced significantly more than k then the observed improvement is statistically significant.

As the Adjusted deviance = deviance + number of predictors, so the difference between the

deviance and adjusted deviance is tested against the χ2 distribution with degrees of freedom

set as the number of additional predictors. These models are nested, that is where the

specification of one model is a result of placing constraints on the parameters of another,


usually with the setting of one or more parameters to zero. For non-nested models the Akaike

information criterion can be a model comparison alternative where AIC = deviance + 2

(number of predictors), with a decreasing AIC indicates a model with reduced prediction error

and thus better fit [1, 11]. Another related measure of model selection is the Bayesian

information criteria where the BIC = −2log(L) + klog(n) where k is the number of estimated

parameters and n the sample size [133].

The concepts of deviance and AIC apply to multilevel models, but the number of parameters

are not as easily defined. The number of parameters relates to the amount of pooling, where

ignoring the higher level data structure is one extreme (complete-pooling) and considering

higher level structure separately is the other (no-pooling), whereas a multilevel analysis

results in something in between. In considering a random-intercept model with n level-2

individuals and if there were less than n different estimated intercepts then if the model is

improved the effective independent parameters are also reduced, being related to the variance

of the level-2 or group-level parameters.

In Bayesian analysis a measure of the mean posterior deviance D̄ = E[D] has been suggested

as a measure of model fit [143] where, the deviance is D(θ) = −2logp(y|θ) for a likelihood

defined by p(y|θ). However the mean posterior deviance does not account for the

improvement of fit with increasingly complex models. A Bayesian model comparison criteria

known as the deviance information criteria (DIC) [143] combines goodness of fit and model

complexity with the complexity measured by and estimate of the “effective number of

parameters” pD. The DIC is defined in an analogous way to the AIC as

DIC= D(θ̄) + 2pD = D̄ + pD where models with smaller DIC are considered to be better

supported by the data and are preferred. The number of effective parameters pD can be

computed as the difference between posterior mean deviance and the deviance for posterior

means pD = D̄ − D(θ̄) [143]. The deviance information criteria (DIC) is the hierarchical

modelling generalisation of the AIC and the BIC [45].

An alternative to pD has been suggested, as half the posterior variance of the deviance

pV = var(D)/2 [44] which is invariant to parameterisation, and has the properties of

robustness and accuracy [142] and is directly estimated from the posterior simulations. The

DIC using pD is implemented in WinBUGS program but is not available for mixture models,


as the class membership is not well estimated by its posterior mean; whereas pV is used in the

R2WinBUGS package [146]. Celeux et al has provided eight different DIC alternatives to

deal with missing data and mixture models [15], and recommend DIC3 and DIC4 as the most

reliable of the DICs studied, where DIC3 = −4Eθ[logp(y|θ)|y] + 2logp̂(y). However, these are

not without problems where the DIC4 did not indicate the correct number of underlying

components for simulated mixture models [15] and DIC3 being questionable in the estimation

of pD [15]. Carlin favours the DIC7 criterion [15] and indicates that for model selection ‘there

exists several possible solutions, but no consensus choice’ [12].

Bayes factors are another way to compare models, where the ratio of marginal likelihood of

two competing models is assessed. If there are two competing models H1 and H2 then the

ratio of their posterior probabilities is

p(H2|y)p(H2|y)

=H2

H1× Bayes factor (H2; H1)

where

Bayes factor(H2; H1) =p(y|H2|p(y|H1)

=

∫p(θ2|H2)p(y|θ2,H2)dθ2∫p(θ1|H1)p(y|θ1,H1)dθ1

Computational difficulties and improper posterior distributions were reasons for these

methods not being used as a matter of course [48]. Applications of Bayes factors for Bayesian

binary mixture models have used the addition of hyper-priors for the parameters of Beta prior

distributions in determining the best number of classes [5]. This was done to ensure the prior

distributions did not contradict the data, due the sensitivity to Bayes factors to priors

distributions [48].

2.3.4 Assessing Convergence

The difficulties of inference from MCMC simulations can be that the simulations are not

representative of the target distribution, the influence of starting values in the early part of the

chain and the inefficiencies of the within chain serial correlation [44]. These difficulties are

handled by simulating multiple chains with starting points distributed through the parameter

space, the monitoring of convergence, and discarding early iterations of simulation runs

(burnin). Thinning can be used to reduce the serial correlation effect by only retaining every


ath simulation draw from each chain and discarding the rest.

Gelman [41] discusses possible ways to compare parallel chains, with the parameter-iteration

plots of overlayed chains for each estimated model parameter used to visually assess their

degree of separation. The Gelman-Rubin statistic which is used to assess or monitor

convergence [41, 44, 47] takes a quantitative approach which separately monitors the

convergence of all parameters of interest. Convergence is monitored by estimating the factor

by which the scale parameter might shrink if sampling were to continue indefinitely. So for

each estimated scalar parameter ψ the simulation draws are labeled as

ψi j(i = 1, . . . , n; j = 1, . . . ,m) for m,m > 1 simulated chains of length n (after discarding

burnin) and where the between- and within-sequence variance are B and W respectively.

B =n

m − 1

m∑

j=1

(ψ̄. j − ψ̄..)2 where ψ̄. j =1n

n∑

i=1

ψi j

and ψ̄.. =1m

m∑

i=1

ψ. j

W =1m

n∑

j=1

s2j where s2

j =1

n − 1

n∑

i=1

(ψi j − ψ̄. j)2

The marginal posterior variance of the estimated parameter ψ is denoted as ̂var(ψ) as a

weighted average of B and W, so ̂var(ψ) =

(n − 1

n

)W +

(1n

)B which overestimates the

marginal posterior variance assuming over-dispersion of the starting distribution but unbiased

under stationarity. For any finite n the ’within’ variance W should be an underestimate of

var(ψ|y). The potential scale reduction is estimated by R̂ =

√v̂ar(ψ)

Wwhich tends to 1 as

n→ ∞. If the potential scale reduction is high then further simulations are indicated, but if R̂

is near 1, that is, R̂ < 1.1 is an indicator of convergence. A corrected version of the R̂ statistic

was devised by Brooks and Gelman [8] R̂c =d + 3d + 1

R̂, where d is the estimate of the degrees of

freedom for the pooled posterior variance estimate.

Theoretical aspects of the convergence of MCMC simulations for Bayesian sampling have

been described by Robert [125] with the importance of the process being ergodic to a target or

stationary distribution, geometric convergence and the properties of the Ergodic Theorem in

determining the Markov chain length presented.


Raftery and Lewis’s diagnostic [120, 121] determines the minimum number of iterations

based on minimal autocorrelation, the required sample size and length of burnin for a single

chain. The Geweke diagnostic [51] is based on a test for equality of means of the first and last

part of a Markov chain.

At least 10 other diagnostic tools have been proposed to assess convergence have been

reviewed and compared by Cowles and Carlin [27], who recommend a two-stage process,

with model specification and sampling being separated from convergence diagnostics.

Convergence diagnostics, sample correlations and plots of model parameters sampled were

recommended as a worthwhile check on the modelling process.

A range of convergence diagnostics have been implemented in the CODA package written for

S Plus [6]and later adapted as a package for R [119]. CODA provides four diagnostic tests

suggested by Geweke [51], Gelman and Rubin [47], Raftery and Lewis [120] and

Heidelberger and Welch [59]. The R2WinBugs package produced the Gelman-Rubin

diagnostic R̂ and effective n as part of the summary statistics.

In this thesis visual assessment of multi-chain parameter simulation plots and the

Gelman-Rubin diagnostic R̂ < 1.1 were the primary methods of assessing convergence.

2.3.5 Goodness of Fit or Model Checking

Posterior predictive checking uses a replicated data set generated by the model in question to

compare with the observed data. For y the observed data and θ the vector of all parameters,

yrep is defined as the replicated data that could have been observed. Any covariates or

explanatory variables would be identical for both y and yrep. The distribution of yrep or the

posterior predictive distribution [44] is

p(yrep|y) =

∫p(yrep|θ)p(θ|y)dθ. (2.20)

A test quantity or discrepancy measure T (y, θ) is a scalar summary of the parameters and data

as a standard when comparing data and predictive simulations. T (y) is a test statistic

dependent on the data and model parameters under their posterior distribution. The Bayesian

p-value os defined as the probability that the replicated data could be more extreme than the

observed data as measured by the test quantity: pB = Pr(T (yrep, θ) ≥ T (y, θ|y) [44, 45, 49].

2.4. Missing Data 65

Posterior predictive checks are demonstrated for mixture models with the introduction of a

subject defined discrepancy measure Di which is determined for each simulation for observed

yi and yrepi , with posterior means of Di and Di plotted against each other [5]. The test quantity

Di is only one of many possible test quantities.

Bayesian residuals can also be used to assess model fit, where the predicted value

g(xi, θ) = E(yi|X, θ) for vector of predictors X, the residual=yi − g(xiθ̂) and can be used to

graphically or otherwise to assess fit [44].

A summary measure of fit known as the χ2 discrepancy resembles the classical χ2

goodness-of-fit measures where T (y, θ) =∑

i(yi − E(yi|θ))2

var(yi|θ) [44, 49]. A related option is the

deviance defined as T (y, θ) = −2logp(y|θ) [44].

Sensitivity analysis considers several probability models for the same problem. These models

can differ in the specification of their prior distributions with an example of using a Student-t

distribution in place of a normal distribution [44]. Similarly a range of priors distributions:

non-informative, half-informative and fully informative were used to show the stability of the

Bayesian modelling process and the impact of informative priors on parameter standard errors

for latent basis growth models on small numbers of subjects (n =34 and n =20) for reading

recognition for young children Zhang et al. [160].

2.4 Missing Data

In this section, we review the handling of missing data, but although Chapter 7 discusses

missingness, this topic is generally t outside the scope of this thesis.

Missing values commonly occur with longitudinal data. The usual pattern for missingness is

dropout or attrition (monotone missingness), where subjects are lost to followup or dropout

prematurely and so remain missing for the duration of the study, as distinct from missing data

by omission, where intermittent observational times are missed (non-monotone missingness).

The mechanisms for handling missing data depend on the knowledge of why the missingness

occurred and come under three major types of ‘missingness mechanisms’ based on the work

of Rubin and Little [84, 85, 86, 129] and Schafer [131] in identifying missing-data

mechanisms.

Missing completely at random (MCAR): A variable is missing completely at random (MCAR)


if the probability of missingness is not related to the outcome response or on any other

measured variable or covariate. The subset of complete data can be considered as a simple

random sample from the full data set. For this missingness condition the exclusion of missing

cases does not bias inferences. For longitudinal data this holds when the pattern of

non-response is independent of the response level or covariate level. These assumptions may

hold for data missing data by omission, or failure to complete a subset of recorded items at

any one time, so analyses based on complete data will be unbiased [33, 85].

Missing at random (MAR): A variable is said to be missing at random if its probability of

being missing depends on observed information, that is observed covariates and/or observed

outcomes. For missing at random, the inclusion of variables which determine the missingness

are required control for this mechanism, thus the missingness is ignorable [84]. For this

missing data mechanism, also denoted as “covariate-dependant dropout” the analysis of

complete data is not biased but however is subject to loss of efficiency [84].

Missing not at random (MNAR): Missing not at random covers any violation of the missing at

random (MAR) condition, and is considered to be non-ignorable. Two types of missing not at

random (MNAR) are discussed by Gelman and Hill [45], where missingness is differentiated

as being dependent on unobserved predictor variables, or dependent on the missing outcome.

If missingness is due to attrition, where a participant drops out prematurely, and if this is

related to the response level, this type of missing data is often missing not at random (MNAR)

or non-ignorable [85], so the analysis of complete data (deletion of incomplete cases or

listwise deletion) may be biased.

A number of methods to handle missing data have been discussed by Little [83, 86] which

included:

• methods which discard data: complete-case analysis or listwise deletion, available-case

analysis, weighted procedures where the inverse of predicted probabilities of response

used to weight the complete-case responses.

• simple approaches which retain all data: mean imputation, last value carried forward,

information from related observations, indicator variables for missingness, imputation

based on logical rules

2.4. Missing Data 67

• random imputation of a single variable; random imputation; regression predictions;

random regression imputation, matching and hot-deck imputation

• imputation of several variables with multivariate imputation; iterative regression

imputation

• model based imputation

• full likelihood methods introduce binary indicators for missingness, and are regarded as

additional observations for the full set of outcome data, both observed and missing, and

use iterative expectation-maximization (EM) algorithm. These can result in the

approaches of selection modeling [64, 84] where the hypothetical complete data are

modelled and a model for the missing data process conditional on the hypothetical

complete data is appended or pattern mixture modelling [84, 87]. For pattern mixture

modelling the sample is stratified by the pattern of dropouts and implies the model for

the whole population is a mixture over the patterns. For a three wave longitudinal data

set, the possible patterns can be OOO (complete data), OOM, OMO, OMM for the

situation where baseline scores observed. Selection and pattern mixture models can be

specified for non-ignorable missngness where the distribution of missingness

(probability of dropout)in addition to the model for complete data must be specified

[64, 84, 132].

• multiple imputation where several simulated imputed values are obtained for each

missing value, which also reflect sampling variability, thus forming several completed

data sets, on which standard analyses are run and estimates are averaged for parameter

estimates with variation within and between imputations estimated. Multiple imputation

can be implemented by non-Bayesian methods or Bayesian methods, although Bayesian

methods predominate.

In order to use multiple imputation methods, the imputation model is a device to preserve the

features of the joint distribution for the variables in the model. Details of considerations for

choosing the imputation model are problem specific with Schafer and Graham presenting

some details on how this can be achieved [132]. While good performance for parameters can


be achieved with MAR missingness, the problem is more difficult for MNAR, where

imputation model specification is critical and the ”performance may be poor unless the sample

is very large” [132].

Listwise deletion is robust when data are missing completely at random (MCAR) [3] but is

biased under other missingness regimes. The problems of missingness with a focus on both

MCAR and MNAR for longitudinal data are addressed specifically by Little [84]. Here the

model based methods are classified by either random-coefficient selection or

random-coefficient pattern-mixture models which use likelihood-based procedures which are

either maximum likelihood or Bayesian. Longitudinal models with ignorable dropout have

been specified by Liu [88] and Hogan [62], and with non-ignorable dropout use Gibbs

sampling (Bayesian) approaches by several authors [14, 32, 48, 82, 153]. The inclusion of a

latent dropout class [127] was also been used to assist in resolving imputation model

misspecification.

Expectation maximization EM, Bayesian and multiple imputation methods have been

indicated as superior strategies [83] with benefits of efficiency, lack of bias and accurate

standard errors [3]. Although advances in missing data methodologies and implementation

methods have improved the ability to handle missing data, especially with MNAR data, all

methods are depended on unverifiable assumptions as the missing data is unobservable [64].

Sensitivity analyses are recommended to understand the impact of missing-data extrapolation

[64].

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Chapter 3

Latent Class Piecewise Linear

Trajectory Models

This chapter has been written as a journal article, for which I am the first author and is

presented in its entirety. This article was first submitted to the Journal of Applied Statistics in

May 2008 and after revision was accepted in January 2009.

Title: Latent Class Piecewise Linear Trajectory Modelling For Short-Term Cognition

Responses After Chemotherapy For Breast Cancer Patients

Authors: Margaret Rolfea, Kerrie Mengersena, Geoffrey Beadleb, Katharine Vearncombecd,

Brooke Andrewcd, Helen Johnsona, Cathal Walshe

aSchool of Mathematical Sciences, Queensland University of Technology, GPO Box 2434,Brisbane, QLD 4001, AustraliabTranslational Research Laboratory, Queensland Institute of Medical Research, Brisbane,QLD, AustraliacSchool of Psychology, University of Queensland, Brisbane, QLD, AustraliadWesley Research Institute, The Wesley Hospital, Brisbane, QLD, AustraliaeTrinity College Dublin, Dublin 2, Ireland

In this paper we aim to demonstrate the applicability of mixtures of piecewise linear

trajectories to the short term responses for the cognitive domain of verbal memory (learning,

immediate recognition and delayed recall) with non-Bayesian methodologies. Three

measurement occasions were used, namely prior to chemotherapy treatment (post surgery),

one month and six months following completion of chemotherapy. Two piecewise linear

85

86 CHAPTER 3. LATENT CLASS PIECEWISE LINEAR TRAJECTORY MODELS

segments are used to model the growth or change process, so allowing for the identification of

two temporal responses: during chemotherapy treatment and subsequent recovery. Latent

class models using this trajectory parameterisation identified two and three subgroups of

participants who although demonstrating parallel treatment profiles, differed in their recovery

rate. Differences in subgroup composition were shown to include baseline age, predicted

intellectual ability (NART), severity of cancer (stage) and the baseline score for the Breast

subscale of the Functional Activity Cancer Test (FACT-B).

Contributions: Margaret Rolfe as first author was responsible for the concept of the paper,

data analysis, interpretation, writing all drafts and addressing the reviewers’ comments.

Professor Kerrie Mengersen was responsible for general advice and editorial comment.

Katharine Veancombe and Brooke Andrew in their role of neuropsychologists were

responsible for neuropsycholocial testing, data entry and integrity, advice on cognitive and

self-report measures and editorial comment. Dr Geoffrey Beadle as principal researcher in the

Cognition and Breast Cancer Study retained an overall advisory and editorial role. Dr Helen

Johnson contributed in an editorial role and Dr Cathal Walsh contributed in early conceptual

discussions and in a later editorial capacity.

87


Latent Class Piecewise Linear Trajectory Modelling For Short-Term

Cognition Responses After Chemotherapy For Breast Cancer Patients

Abstract

This paper investigates the impact of chemotherapy on cognitive function of breast cancer

patients, and whether this response is homogeneous for all patients. Latent class piecewise

linear trajectory (growth) models were employed to describe changes and identify subgroups

in three Auditory Verbal Learning Test measures (learning, immediate retention and delayed

recall) in 130 breast cancer patients taken at three time periods: before chemotherapy, one

month and six months post-chemotherapy.

Two distinct subgroups of women exhibiting different patterns of response were identified for

learning and delayed recall, and three for immediate retention. The groups differed in level

(intercept) at one month post-chemotherapy and patterns of decline and recovery.

Binomial and multinomial logistic regressions on the latent classes found that age, initial

NART Predicted IQ, stage of cancer and the initial FACT-Breast subscale (or subsets thereof)

to be significant predictors of classes.

Keywords: latent class, piecewise linear, trajectory, cognition, breast cancer,

chemotherapy, growth models, mixtures.

3.1. Introduction 89

3.1 Introduction

In the clinical setting, there are many examples where heterogeneity in longitudinal response

has been recognised within a population. For example, numerous authors have divided their

groups into ‘responders’ and ‘non-responders’ to treatment [11, 14, 22, 23]. The objective, in

this instance, is to use statistical methods to identify subgroups within the clinical population,

based on the trajectories of the clinical measurements over time.

This paper utilises data collected from a study that was designed to assess the nature, degree

and duration of changes in cognitive functioning associated with cytotoxic drug treatment

(chemotherapy) for early breast cancer. This study used a prospective, longitudinal design to

assess cognitive functioning before chemotherapy, one month after completion of

chemotherapy and six months post-chemotherapy.

Recent papers [5, 15] on breast cancer indicate that only a subset of women show a

measurable reduction in cognitive function after chemotherapy treatment. Estimates of

percentages of affected women range from 25% [15] to 50% with moderate or severe

impairment [3]. The existence of subgroups motivates the utilisation of finite mixtures of

trajectories or growth mixture models [25, 26, 27]. A special class of these models is the

latent class growth model [29, 30].

The aims of this paper are twofold: first, to characterise the response of subjects over time, in

particular the potential decline and recovery process of cognitive change, with the

identification of possible subgroups of breast cancer patients based on different patterns of

cognitive change after adjuvant chemotherapy and, secondly, to identify predictors of these

latent classes.

Piecewise linear growth models can be used to break up a nonlinear or curvilinear growth

trajectory into separate linear components. This is useful in the comparison of growth rates in

different periods [33]. Often these periods are marked by a transitional change or at an

(experimental) intervention point [41] with the discontinuity occurring at a known time point.

A piecewise model is also often preferable to a more general nonlinear continuous model

(such as a polynomial) if the number of time periods is small.

Piecewise linear trajectory models have been used in modelling the developmental process in


both the multilevel [33, 36, 37] and structural equation modelling methodologies [1, 7, 24],

with applications in the areas of educational assessment [32, 35], alcohol use in adolescents

[4, 17, 18], wages developments for high school dropouts [36] and assessment of the

effectiveness of intervention processes [16, 40].

Latent class growth models were fitted to the piecewise linear changes in three cognitive

measures of verbal learning and memory. This allows the identification of two temporal

responses: during treatment, through the comparison of measures before and one month after

chemotherapy, and during recovery, through the comparison of measures one month and six

months after chemotherapy.

The restricted number of time-points in the model required the assumption of linearity

between adjacent measurement times. Despite this constraint, the latent classes were able to

encompass the possibilities of increase, decrease or no change in cognitive ability between

adjacent time points.

Possible predictors were identified in two ways, using discriminant analysis and logistic

regression on the most likely latent classes membership of subjects, where predictors included

demographic, self-report assessments and medical information.

This paper is structured as follows. Details of the breast cancer study and the statistical

methodology are provided in Section 3.2. Results are presented in Section 3.3, followed by

conclusions and discussion in Section 3.4.

3.2 Methods

3.2.1 Study Design

Participants were recruited from community hospitals in south east Queensland and were

required to have histologically proven breast cancer treated initially by definitive surgery.

Eligible participants were required to be between 18 and 70 years, proficient in English since

early childhood, geographically accessible for assessment, and have a Karnofsky performance

status index of equal to or greater than 80% (indicating normal activity with minor

disruption). Participants were also required to have no recent history of cancer, no previous

history of cytotoxic drug treatment, neurological or psychiatric symptoms or current use of

3.2. Methods 91

medications that may lead to deviant neuropsychological test results.

Adjuvant chemotherapy was administered after surgery in all cases and patients were eligible

for participation if they were also receiving adjuvant endocrine treatment or post-operative

radiation treatment. Approval for this study was provided by the Human Research Ethics

Committees of all the participating hospitals as well as the Queensland Institute of Medical

Research.

Demographic data collected included age, marital status, family cancer history, menopausal

status, use of hormone replacement therapy, current and previous medications. Planned

treatments including drug regimens used in chemotherapy, endocrine and radiation treatments

as well as the site and extent of the cancer were also recorded. Self-report measures included

depression/anxiety using the Hospital Anxiety and Depression Scale (HADS) [44], and

quality of life as determined by the Functional Assessment of Cancer Therapy General

(FACT-G), Breast (FACT-B), and Fatigue (FACT-F) scores [2, 6]. NART predicted IQ and

full-scale pre-morbid intellectual functioning via the National Adult Reading Test [31, 42]

were also assessed.

Participants undertook an individually administered, comprehensive battery of neurological

tests comprising assessments on numerous domains, namely attention, visual and verbal

memory, speed of information processing and executive function.

The cognition measures considered in this paper are the scales of verbal learning and memory

measured by the Auditory Verbal Learning Test (AVLT) as prescribed in Geffen and Geffen[8]

and utilised in other papers by the same authors [9, 10]. The primary response variables for

this paper were the learning score which was derived from the sum of the words recalled in

Trials 1-5 (Learning AVLT Trials 1-5), the immediate retention score after a distractor list

(AVLT Trial 7), and a delayed recall score comprising the total number of words recalled after

a 30 minute delay (AVLT Trial 8).

Complete data for three measurement occasions for the AVLT measures were available for

130 participants.


3.2.2 Primary Analysis

In modelling the cognitive verbal learning and recall aspects for women undergoing

chemotherapy, a two-piece linear growth model was specified. The first piece L1 covers the

period from before chemotherapy, baseline or initial time (time 1) to one month after

completion of chemotherapy (time 2), taking values -1 and 0 respectively. The second piece

L2 covers the period from one month after completion of chemotherapy (time 2) to six months

after chemotherapy (time 3), taking values 0 and 1, respectively.

It is more typical for a two-piece linear growth model to include more than two time points in

each piece, with five points in total enabling a full growth model (fully random) to be

estimated [1]. The use of fewer time points restricts the ability to estimate the full set of

random effects (variances and covariances for intercepts and two slopes). The following

model presents the single class random effects piecewise linear growth model with random

effects for the variances of the growth parameters, which when extended to two or more

classes becomes the growth mixture model (GMM) as described by Muthen [25, 26].

A single class random effects piecewise linear growth model with each of the n individuals

having an individual growth trajectory can be specified as follows. The restricted number of

time points only permits the estimation of the variances of the growth parameters, with

covariances among the growth parameters set to zero.

Let yti, i = 1,2,. . . n, t =1,2,3, denote the response of individual i at time t. The data are

described by a within-individual (level 1) model and between-individual model with a

maximum of three random effects on the growth parameters and time specific residuals, as in

the following set of equations.

yti = λti + eti

λti = η0i + η1iL1t + η2iL2t

η0i = β0 + u0i

η1i = β1 + u1i

η2i = β2 + u2i

3.2. Methods 93

Figure 3.1 Piecewise Linear Latent Class Growth Model.

where λti is the expected value of yti for the ith individual at time t; η0i represents the expected

response at time 2; η1i represents the linear change in response over the first time interval; η2i

represents the linear change over the second time interval; L1 = (-1,0,0) and L2 = (0,0,1); the

residuals eti have mean zero and variance σ2t and u0i, u1i, u2i are the growth factor residuals

with mean zero and variances ψ0, ψ1, ψ2.

The latent class growth analysis (LCGA) or grouped trajectory approach introduced by Nagin

and Land [30] used the identification of a finite number of distinct groupings of individual

trajectories, instead of the n individual trajectories from a random effects growth model, to

fully account for all heterogeneity. Hence for latent class growth analysis (LCGA) all growth

factor variances and covariances are set to zero, thus becoming a special case of the growth

mixture models (GMM) of Muthen [25, 26].

If there are K groupings the LCGA model can be written as

λkti = βk

0 + βk1L1ti + βk

2L2ti

for the kth group where k = 1, . . . ,K.

Figure 3.1 illustrates the Structural Equation Model representation of the piecewise linear

latent class growth model for three measurement occasions.

Mplus Version 4.21 [28] was used to fit one, two, three and four class two-piece linear LCGA


models to the learning, immediate retention and delayed recall data for the three measurement

occasions. One hundred sets of initial values and twenty iterations for each of the starting sets

were used in order to reduce the problem of sensitivity to local minima [13, 27].

A four-dimensional approach was used to assess the goodness of fit of the models, with regard

to the number of classes. Firstly, the adjusted Vuong-Lo-Mendell-Rubin likelihood ratio test

LMR LRT [19] which extends the likelihood ratio criteria for model assessment to non-nested

comparisons, namely k class model compared to the k − 1 model for k > 1 [26]. The test was

implemented in MPLUS [28]. Secondly, the Bayesian Information Criteria (BIC) [34] defined

by −2logL + plog(n) with n as sample size and p number of parameters, was used to assess

model complexity, with a smaller value indicating a better fit. Thus, on disagreement between

the number of classes using the LMR LRT test and the BIC, the conclusion determined by the

LMR LRT was accepted. Thirdly, a nominal minimal class size of 10% was a requirement to

ensure reasonable class sizes, and fourthly, convergence of the estimation algorithm was a

requirement for model choice. Recent papers [12, 39, 43] have shown the LMR LRT to be

more accurate than the BIC in determining the number of classes under simulated conditions.

Posterior probability of class membership is a measure of an individual’s likelihood of

belonging to each of the k trajectory groups or classes. Given an estimated model, each

individual obtains a posterior probability estimate for each class computed as a function of the

model parameter estimates and the individual’s observed values by way of Bayes’ theorem.

The class with the highest posterior probability will determine the most likely class

membership for the individual [29]. The posterior probabilities of group membership can

determine the ability of the model to clearly differentiate between subjects. An average

posterior probability of group membership equal to 1 demonstrates the optimal or ideal

situation, with Nagin [29] specifying a rule of thumb of at least 0.7 for all groups as a

acceptable measure.

3.2.3 Supplementary Analyses

A K-means cluster analysis was undertaken to confirm the subgroups identified by the LCGA

model. Although this approach to identifying subgroups has been described as inferior to the

LCGA approach by Magidson and Vermunt [21], it has value as part of a robustness and

3.3. Results 95

sensitivity assessment.

Magidson and Vermunt [21] argue that latent class methods extend the K-means approach in a

number of ways, including the replacement of an ad-hoc distance measure for classification,

model based posterior membership probabilities of clusters, provision of diagnostics statistics,

which can be useful in determining the number of clusters, and elimination of the requirement

of the K-means method for variables to be on the same scale.

The cluster analysis was undertaken using SPSS [38] which uses the MacQueen algorithm

[20]. Different orderings of the data were utilised to investigate the stability of clusters.

Stepwise discriminant analysis was used with the continuous time invariant demographic

variables of age, years of education and baseline measures of NART-predicted IQ level,

together with baseline self report measures of mood, HADS anxiety and depression and cancer

quality of life scales, FACT General, Breast and Fatigue subscales to determine significant

predictors for the latent classes for learning, immediate retention and delayed recall.

Similarly backwards stepwise binomial and multinomial logistic regression analyses were

used to determine significant predictors of the latent class groupings with continuous variables

as detailed for the discriminant analysis and categorical time-invariant attributes of surgery

type (mastectomy yes/no), menopausal status (pre, peri, post, unknown), marital status

(partnered, non-partnered) and a two category stage of cancer (stage I coded as 0/ stage II and

III coded as 1).

The class which maximizes an individual’s posterior probability becomes an individual’s

allocated class (most likely class allocation). As this probability is usually less than 1, there is

some some error in allocation process, which is different from knowing the true class

membership (no variability).

3.3 Results

Participants were 25 to 68 years of age (with mean 49.1 (SD 7.8), mean of 13 (SD 3.4) years

of education and a baseline NART predicted IQ ranging from 86 to 126 with mean 110.4 (SD

8.9). The majority of participants were married or living with a partner (84.6%), were

pre-menopausal (52.3%) and with 44.6% having undergone a mastectomy and the remainder

(54.6%) having local excision surgery. As being in early stages of breast cancer was a


Table 3.1 Sociodemographic and clinical data for participants with complete data for threemeasurement occasions, n = 130.

Variable Mean ± SD Min-Max

Age in Years 49.21 ± 7.84 25.2 − 67.9Education as FTE in Years 13.04 ± 3.38 6 − 26NART Predicted IQ Baseline 110.42 ± 8.93 86 − 126

Frequency Percent

Marital status Single/never married 7 5.4Married/living with partner 110 84.6Separated, Divorced, Widowed 13 10.0

Menopausal status Pre-menopausal 68 52.3Peri-menopausal 22 16.9Post -menopausal 38 29.2

Surgery Undergone Local Excision 71 54.6Mastectomy 58 44.6

Stage of Cancer I 31 23.8II 84 64.6III 8 6.8

requirement of the study very few participants were in Stage III of the disease (6.8%) with the

majority in stage II (64.6%) and fewer in stage I (23.8%).

The means over the three measurement occasions for outcome measures of interest: learning,

immediate retention and delayed recall appeared to follow a similar pattern with highest

scores before chemotherapy, lowest at one month after chemotherapy (occasion 2) and slight

improvement at the third time point. Higher scores on these measures imply more words

being learnt or recalled. Hence, higher scores indicate better cognitive ability.

Table 3.1 and Table 3.2 present, respectively, summaries of demographic and clinical data of

study participants with complete data for three measurement occasions.

3.3.1 Latent Class Growth Analysis

All two, three and most of the four class models satisfied the convergence criteria, with the

four class model for immediate retention being the exception. The two class piecewise linear

model improved on the single class model for all three responses (learning, immediate

retention and delayed recall), based on the LMR LRT test (p ≤ 0.001) and by the large

reduction in BIC values from 2728 to 2651 (77), 1854 to 1753 (101) and 1844 to 1785 (59).

3.3. Results 97

Table 3.2 Summary statistics for Learning, Immediate Retention, Delayed Recall, Anxiety, De-pression and FACT scores for participants with complete data for three measurementoccasions, n = 130.

Response Occasion Mean SD Min Max

Learning 1 52.50 7.04 33 682 49.59 7.78 25 663 50.11 8.43 32 70

Immediate Retention 1 11.20 2.34 6 152 10.05 2.51 4 153 10.13 2.63 3 15

Delayed Recall 1 11.26 2.35 6 152 9.68 2.50 4 153 9.78 2.73 3 15

Baseline Scores for:Anxiety (HADS) 1 6.56 3.76 0 21Depression (HADS) 1 3.03 2.39 0 13FACT-General 1 86.32 12.38 48 107FACT-Breast subscale 1 23.75 5.61 10 35FACT-Fatigue subscale 1 38.76 8.90 16 52

The four class models resulted in one of the classes for each outcome failing to achieve the

adequate sample size of at least 10%.The choice of models based on the LMR LRT test was

supported by the BIC criterion for immediate retention and delayed recall. For immediate

retention, three classes were preferred to two (LMR LRT p=0.001 for the additional class,

smallest BIC of 2643 for three classes). For delayed recall, two classes were preferred to three

(LMR LRT p=0.124 for the additional class, smallest BIC, 1785 for two classes). For

learning, the non-significance of the LMR LRT statistic (p = 0.118) for the addition of a third

class and the relatively small reduction in the BIC, from 2651 to 2643 (8) for the estimation of

an additional four parameters, resulted in preference of the two class model.

Thus the two class models were adopted for learning and delayed recall, and the three class

model for immediate retention.

Table 3.3 presents the results of the adopted models together with the two class model for

immediate retention for comparative purposes. Arranged by decreasing order of intercept

scores, the estimated sets of class percentages (proportion of subjects in each class) for the two

class models were 46%:54% for learning, 47%:53% for immediate retention and 50%:50%

for delayed recall, and the three class model 33%:50%:17% for immediate retention.


Table 3.3 Results of two/three class models of Latent Class Growth analysis for Learning, Im-mediate Retention and Delayed Recall

Class 1 Class 2 Class 3Response Estimate SE Estimate SE Estimate SE

Learning Intercept 54.899** 0.919 44.698** 1.233Slope 1 -2.428** 0.615 -3.512** 0.987Slope 2 1.232 1.211 -0.145 1.012n 60 70Percent 46 54Aver Post Prob 0.937 0.912

Immediate Intercept 12.025** 0.355 8.376** 0.322Retention Slope 1 -0.843* 0.322 -1.404** 0.301

Slope 2 -0.033 0.295 0.170 0.281n 61 69% 47% 53%Aver Post Prob 0.913 0.941

Intercept 12.469** 0.276 9.276** 0.277 7.441** 0.397Slope 1 -0.907* 0.348 -1.611** 0.314 -0.306 0.540Slope 2 0.032 0.364 0.368 0.314 -0.657 0.550n 43 65 22Percent 44 50 17Aver Post Prob 0.953 0.915 0.923

Delayed Intercept 11.392** 0.284 8.123** 0.387Recall Slope 1 -1.557** 0.372 -1.595** 0.411

Slope 2 0.160 0.327 0.045 0.348n 65 65Percent 50 50Aver Post Prob 0.899 0.944

∗∗ indicates p < 0.001,∗ indicates p < 0.05Aver Post Prob denotes Average Posterior Probability

3.3. Results 99

4045

5055

60

Learning

Before At Completion 6 Months After

Class 1 Class 2 Class 36

810

1214

Immediate Retention


68

1012

14

Delayed Recall


Figure 3.2 Two and three class trajectory models for Learning, Immediate Retention and De-layed Recall.

Table 3.3 also reports average posterior probabilities of class membership for the models

considered. These were all greater than the threshold of 0.7, indicating a high degree of model

agreement. Figure 3.2 presents the estimated trajectories for learning, immediate and delayed

recall.

Parameter estimates from the latent class growth models across all classes indicated that there

were significant intercept word scores (one month after chemotherapy). All but the third class

of immediate retention showed a significant decline in word score for the first linear

component, from before chemotherapy to one month after treatment. And all classes across all

three outcomes indicated a non-significant recovery for the period of 6 months post

chemotherapy.

For learning, the first class indicated a less rapid rate of decline (mean -2.43 (se 0.62),

p <0.001) than for the second class (mean -3.51 (se 0.99), p <0.001) but with a

non-significant faster rate of recovery (mean 1.232 (se 1.21)) than for the second class (mean

-0.15 (se 1.012)).

With the optimal 3 class model for immediate retention, the first two classes showed a similar

pattern to learning with an initial significant decline for class 1 (mean -0.91 (se 0.35),

p < 0.05) and class 2 (mean -1.61 (se 0.31), p < 0.001)), followed by a non-significant

recovery. The third class with the lowest word count at the change point of 7.44 (se 0.40)


showed a non-significant decline for both linear components (mean -0.31 (se 0.54) and mean

-0.66 (se 0.55) respectively).

The two classes of delayed recall appeared to have parallel response trajectories, only

differing by intercept or level (mean 12.47 (se 0.28) and mean 8.12 (se 0.28) for each class

respectively), with similar initial declines (mean -1.56 (se 0.35), p < 0.001; mean -1.60 (se

0.41), p < 0.001), followed by a non-significant improvement (or flat) recovery (mean 0.16

(se 0.33) and mean 0.45 (se 0.35)).

For all outcomes the difference in word count at the one month post chemotherapy, between

adjacent classes was significant with differences for learning of 10.2 words (p < 0.001),

immediate retention 3.2 (p < 0.001) and 1.9 words (p < 0.001), and 3.2 words (p < 0.001) for

delayed recall.

3.3.2 Results of K-means clustering

Differences in classification between the K-means clustering and the resultant LCGA classes

were minimal, when the number of clusters (2, 3, 2) in the K-means analysis were set to

match the adopted number of classes from the latent class analyses. The centroids of the

clusters generated by the K-means analysis and the class means from the LCGA were in close

agreement, with 66% being within 0.5, and 100% differing by a maximum of one. Class sizes

under the two methods also differed by a maximum of two for learning and delayed recall.

However the class sizes were more disparate for the three classes of immediate retention with

differences ranging from 1 to 6 between the two methods. The levels of agreement for

intra-class membership was also high, ranging from 90 to 100% with the maximum number of

unmatched subjects being six.

3.3.3 Discriminant Analyses and Logistic Regressions

The discriminant functions designed to optimally separate the classes derived from the LCGA

models were constructed as a linear combination of the demographic variables (age, baseline

NART-predicted IQ and years of education), self-reported scores for FACT fatigue, breast,

general and stage of cancer. A stepwise approach was used to retain only those variables that

significantly contributed to discrimination between the classes.

3.3. Results 101

Table 3.4 Standardized Coefficients for resultant predictors of stepwise discriminant analysisfor Learning, Immediate Retention and Delayed Recall.

Predictor Learning Immediate Retention Delayed RecallDiscriminant Function: 1 1 2 1

p value 0.000 0.007 0.635 0.004

Age -0.577NART-Predicted IQ 0.544 0.793 -0.620 0.719FACT Breast 0.472Stage of Cancer (I=0, II & III=1) 0.612 0.704 0.719 0.711

As the number of discriminant functions produced is one less than the number of classes, one,

two and one discriminant function(s) were used for learning, immediate retention and delayed

recall respectively. For each discriminant function the Wilks’ lambda and its χ2 statistic tests

for a true ability to differentiate between the classes.

For learning (Wilks lambda=0.812, χ24 = 25.4, p < 0.0005) and delayed recall (Wilks

lambda=0.913, χ22 = 11.2, p = 0.004) the first and only discriminant function was able to

significantly differentiate between the two classes generated from the LCGA. For the three

classes derived for immediate retention the first discriminant function was significant (Wilks

lambda=0.890, χ24 = 14.2, p = 0.007), with the second discriminant function being

non-significant (Wilks lambda=0.998, χ21 = 0.23, p = 0.635). The standardised coefficients on

variables retained by the stepwise procedure (Table 3.4) can be used to assess their relative

importance, with the sign showing the direction of the relationship.

Thus for learning, four variables were identified as equally important and the two classes were

best separated by the difference between age and an average of NART, FACT Breast and stage

of cancer. For immediate retention, only two variables were identified as important

discriminators, namely NART and stage of cancer and the three classes were best separated by

an average of these two variables (based on the first function). For delayed recall, the two

classes are again best separated by an average of NART and stage of cancer.

The eigenvalues associated with the discriminant functions in Table 3.4 were consistently low

and all less than unity, indicating that factors other than those included in the analysis may

have substantial influence on the responses in this study group.

Table 3.5 presents latent class means and standard errors for age, NART, education (years),


Table 3.5 Means for demographic variables, quality of life scores, and numbers for stage ofcancer [I/II&III] by classes for the Learning, Immediate Retention and Delayed Re-call outcomes.

Response Class 1 Class 2 Class 3Mean (se) Mean(se) Mean (se) p†

Learning Age 47.36 (0.99) 50.79 (1.05) 0.013NART 112.85 (0.95) 108.33 (1.15) 0.004Education 13.83 (0.43) 12.36 (0.40) 0.013FACT Fatigue 40.57 (1.04) 37.21 (1.12) 0.032FACT Breast 24.73 (0.67) 22.90 (0.71) 0.063FACT General 87.22 (1.47) 85.54 (1.58) 0.444Stage of Cancer 9:51 24:43 0.008n 60 70

Immediate Age 47.57 (1.19) 49.29 (0.98) 52.15 (1.55) 0.082Retention NART 112.56 (1.09) 110.34 (1.19) 106.45 (1.91) 0.032

Education 13.67 (0.45) 13.05 (0.45) 11.77 (1.91) 0.100FACT Fatigue 40.67 (1.39) 38.14 (1.08) 36.86 (1.86) 0.193FACT Breast 25.40 (0.81) 22.66 (0.71) 23.73 (1.14) 0.045FACT General 88.00 (2.04) 85.06 (1.47) 86.73 (2.58) 0.741Stage of Cancer 6:36 18:45 9:13 0.056n 43 65 22

Delayed Age 48.26 (0.91) 50.15 (1.02) 0.171Recall NART 112.44 (0.89) 108.40 (1.25) 0.009

Education 13.75 (0.42) 12.32 (0.41) 0.015FACT Fatigue 36.63 (1.09) 37.89 (1.11) 0.267FACT Breast 24.60 (0.65) 22.89 (0.73) 0.083FACT General 85.95 (1.66) 86.68 (1.41) 0.741Stage of Cancer 11:53 22:41 0.023n 65 65† p-value from F or t test or p-value from χ2 test for Stage of Cancer

FACT General, FACT Breast, FACT fatigue for learning, immediate and delayed recall;

numbers for stage of cancer, together with the p-values of differences between the classes.

Stepwise binary and multinomial logistic regressions produced similar results to the

discriminant analysis with odds ratios (OR), confidence intervals and significance values

presented in Table 3.6. Age, NART predicted IQ, FACT-breast subscale and modified stage of

cancer were predictors of class differences (Class 1 versus 2) for learning; FACT Breast

subscale for differences between Classes 1 and 2 for immediate retention and with age, NART

predicted IQ and modified stage of cancer for differences between classes 1 and 3 for this

measure. NART and modified stage again were predictors of differences between the two

classes of delayed recall.

All or a subset of the four variables, age, NART predicted IQ, FACT Breast subscale scores

3.3. Results 103

Table 3.6 Results of stepwise logistic/multinomial analysis for Learning, Immediate Retentionand Delayed Recall with age, NART, years of education, FACT General, Fact Breast,FACT fatigue, stage of cancer.

Response Class Variable Odds Ratio 95% CI pComparison

Learning 2 to 1 Age 1.075 1.019-1.134 0.008NART 0.940 0.897-0.985 0.010FACT Breast 0.920 0.856-0.990 0.026Stage of Cancer 0.240 0.091-0.633 0.004

Immediate 2 to 1 FACT Breast 0.906 0.838-0.980 0.014Retention 3 to 1 Age 1.095 1.017-1.180 0.017

NART 0.917 0.859-0.980 0.010Stage of Cancer 0.158 0.041-0.603 0.007

Delayed 2 to 1 NART 0.944 0.904-0.986 0.009Recall Stage of Cancer 0.349 0.147-0.830 0.017

and a measure of stage or severity of cancer were significant predictors of class membership

(Table 3.4 and Table 3.6) in both the logistic and discriminant analyses.

None of the categorical variables of menstrual status, marital status or surgery type

significantly differentiated between the classes of the three cognition measures.

All four of the identified variables were significant predictors for learning with class 1

comprising younger subjects (47.4 versus 50.8, OR= 0.93, p=0.008) with a higher NART

score (112.9 versus 108.3, OR=1.06, p=0.010), higher FACT Breast score (24.7 versus 22.9,

OR=1.09, p=0.260), and proportionally more patients with later stages of cancer (85% versus

64% in Stages II and III, OR=4.17 p=0.004).

Again, all four of the identified variables were significant predictors for the multinomial

regression on the three immediate retention classes. However in the comparison of class 2

with class 1 only FACT Breast was significant with class 2 having lower but not significantly

different scores than class 1 (22.7 versus 25.4, OR=0.91, p=0.14). In the comparison of class

3 with class 1, class 3 comprised of older patients (52.2 versus 47.6, OR=1.10, p=0.017) with

lower NART scores (106.5 versus 112.6, OR=0.917, p=0.010) and fewer subjects with later

stages of cancer (59% versus 86% in Stages II and III, OR=0.16 p=0.007).

For the two classes of delayed recall only the NART score and stage of cancer were significant

predictors, with NART scores being higher for class 1 (112.4 versus 108.4, OR=1.06,

p =0.009) and more subjects with later stages of cancer (85% versus 65% in Stages II and III,


OR=2.87 p=0.017).

3.4 Discussion

This study forms part of a larger research project into cognition associated with chemotherapy

among early breast cancer patients. The aim of the study was to identify classes of women

who demonstrated different patterns of response with respect to learning, immediate retention

and delayed recall aspects of cognition as measured by the AVLT instrument. To this end,

latent piecewise linear growth models were fitted to responses at three time points: prior to

chemotherapy, and one month and three months after chemotherapy. Two trajectory classes of

response were identified for learning and delayed recall, and three classes were identified for

immediate retention.

For all three cognition measures there was a difference between the classes with respect to the

level (intercept) of the score at the change point, one month after completion of treatment.

Classes, ordered by decreasing intercept, showed significant differences in numbers of words

of 10.2, 3.2, 3.2 at one month after completion of treatment between the first and second

classes for each of the outcomes respectively, and 1.9 words between the second and third

classes of immediate retention.

Significant decline was demonstrated for the first linear component (before treatment to one

month after treatment) for all classes with exception for the third (lowest) class for immediate

retention. However there was no significant change or recovery for the second linear

component (one month after treatment to six months after treatment) for any of the outcomes.

For learning, the first class was characterised by a less steep rate of decline, followed by a

faster non-significant rate of recovery compared to the second class. The first two classes of

immediate retention showed a similar pattern to that of learning, with the second class (lower

intercept) showing a significantly steeper decline after chemotherapy than the first class.

However, the third class differed by showed a non-significant decline for both the first and

second linear components. The two classes of delayed recall appeared to have similar initial

declines followed flat recoveries. So only delayed recall could be deemed to have parallel

trajectories for both classes, which may indicate the effects of memory, as typified by delayed

recall may follow a more homogeneous process.

3.4. Discussion 105

The K-means analysis showed strong agreement (between 90% and 100%) in the

identification of classes compared with the latent class piecewise mixture model approach,

using the resultant number of classes from the Latent Class approach.

Logistic regression and discriminant analysis demonstrated that a only four variables, namely,

age, NART predicted IQ, FACT Breast subscale scores and a measure of stage or severity of

cancer were significant predictors of class membership.

The relationship with age and NART scores and Auditory Verbal Learning scores are

consistent with the findings of Geffen et al [9], who reported a decreased recollection, in

general, with increased age and an increase with higher scores from the National Adult

Reading Test.

The differences between classes typically were as follows: compared to subjects in class 2, the

subjects in class 1 had a higher word count at one month after chemotherapy (mean 54.9

versus 44.7; 12.5 versus 9.3; 11.4 versus 8.1 for the three outcome measures respectively),

were younger with mean age of 47 to 48 (versus 49 to 51 years of age), had higher NART

scores (mean 112 to 113 versus 108 to 110) and higher FACT Breast scores (mean 25 versus

23) and were in later stages of cancer (83% to 85% versus 64% to 71% in Stages II and III).

For immediate retention, the third class was characterised by older subjects (mean 52 years),

with lower NART scores (mean 106) and contained proportionally fewer in the later stages of

cancer (59%).

Latent class growth model or growth mixture model provide a statistical methodology for

identifying distinct subgroups with differing longitudinal or trajectory profiles from data

which are generated from unobservable sub-populations. In so doing they maximise the

similarity of trajectories within each class and account for the heterogeneity between

individual trajectories with the class membership. This paper uses these methods to identify

subgroups of women with differing trajectory profiles on verbal learning cognition measures,

where trajectory differences relate to changepoint (intercept) level of cognition scores,

patterns of decline and recovery after receiving chemotherapy treatment for breast cancer. It is

hoped that this will facilitate greater understanding of this phenomenon and assist with

identification and management of susceptible patients.


Acknowledgements

This research was conducted as part of the Cognition in Breast Cancer Study undertaken by

the Wesley Research Institute and was supported by the Wesley Research Institute, the Cancer

Council of Queensland, the National Breast Cancer Foundation, and the Australian Research

Council Linkage Project. The authors would also like to sincerely thank all the women who

participated in the study.

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Chapter 4

Bayesian Longitudinal Models


presented in its entirety. This article was submitted to the journal Biostatistics in February

2009 and is currently awaiting a response.

Title: Bayesian Analysis Of Longitudinal Cognition Models: Verbal Memory

Performance In Women Undergoing Adjuvant Chemotherapy Treatment For Breast

Cancer


Brooke Andrewcd

aSchool of Mathematical Sciences, Queensland University of Technology, GPO Box 2434,Brisbane, QLD 4001, AustraliabTranslational Research Laboratory, Queensland Institute of Medical Research, Brisbane,QLD, AustraliacSchool of Psychology, University of Queensland, Brisbane, QLD, AustraliadWesley Research Institute, The Wesley Hospital, Brisbane, QLD, Australia

111

112 CHAPTER 4. BAYESIAN LONGITUDINAL MODELS

Bayesian piecewise linear random effects growth models were employed in this paper to

model the responses for the cognitive domain of verbal memory (learning, immediate

recognition and delayed recall) for four measurement occasions, namely prior to

chemotherapy treatment (post surgery), one month, six and eighteen months following

completion of chemotherapy. The two piecewise linear segments (Times 1 to 2, and Times 2

to 4) are used assess potential decline, recovery or no change in the cognitive functioning

profiles. These models were extended to latent class growth mixture models, whereby a

mixture of a small number of trajectory profiles would account for the heterogeneity among

subjects’ responses, and were also extended to incorporate mediating variables.


data analysis, interpretation, writing all drafts. Professor Kerrie Mengersen was responsible

for general advice and editorial comment. Katharine Veancombe and Brooke Andrew in their

role of neuropsychologists were responsible for neuropsychological testing, data entry and

integrity, advice on cognitive and self-report measures and editorial comment. Dr Geoffrey

Beadle as the principal clinical researcher in the Cognition and Breast Cancer Study retained

an overall advisory and editorial role.

113


Bayesian Analysis Of Longitudinal Cognition Models: Verbal Memory

Performance In Women Undergoing Adjuvant Chemotherapy

Treatment For Breast Cancer

Summary

Decline in cognitive functioning can be experienced by up to 50% of women while

undergoing adjunct chemotherapy and these effects may continue for several years for a

subset of patients. We analysed cognitive function in early stage breast cancer patients before,

on completion of, six months after and eighteen months after chemotherapy. Bayesian

piecewise linear random effects growth models were fitted to the four measurement occasions

to assess potential decline, recovery or no change in the cognitive function profile responses

for verbal learning, immediate retention and delayed recall (as aspects of verbal memory).

These models were extended to latent class growth mixture models, where a mixture of a

small number of trajectory profiles would account for the heterogeneity among subjects’

responses, and also to include mediating variables. The overall trajectory profile of all three

outcomes was well explained by a two piece linear process with a change-point at one month

after chemotherapy and investigation of covariates showed that more years of education and

higher cancer severity scores are associated with a higher level of the verbal memory

responses. The Bayesian latent class growth mixture models with adjustments for education

resulted in the three class models for learning and immediate retention, and the respective two

class model for delayed recall being the preferred models. Age and baseline estimated

intellectual functioning scores, together with the differing proportions between stages of

cancer reflected differences between classes.

Keywords: Bayesian, longitudinal, latent class, growth mixture, breast cancer, verbal memory,

cognition.


4.1 Introduction

Decline in cognitive function is a frequently reported side-effect for women undergoing

adjuvant chemotherapy treatment for breast cancer. The estimates of women suffering from

cognitive impairment after chemotherapy in the short term varies between studies and ranges

from 20% or 25% [30, 31] to 50% for women with moderate or severe impairment [5] and

similarly 50% for studies on older women (aged 65 years and above) by Hurria and others

[29]. The level of cognitive dysfunction has been shown to improve over time [31, 59] with a

subset of women suffering long term effects for up to 10 years [1, 51]. However the nature of

this cognitive impairment has been described as subtle by Brezden and others [5] and Wefel

and others [63].

In order to investigate this process more fully a study was undertaken to assess the impact of

adjuvant chemotherapy on cognitive functioning in early stage breast cancer patients drawn

from hospitals throughout south-east Queensland, Australia. The study follows a longitudinal

prospective design with assessments conducted before chemotherapy, and at one, six and

eighteen months after completion of chemotherapy. A battery of neuropsychology tests

measuring a range of cognitive domains was conducted at all four measurement occasions,

together with measures of anxiety, depression and self reported quality of life scales as

measured by the Functional Assessment of Cancer Therapy. Other demographic factors

comprising age, education, estimated intellectual functioning, medical and hormonal history

were also recorded.

This paper focuses on the analysis of the Auditory Verbal Learning Test, in particular the

potential patterns of decline, recovery or no change in verbal memory function before and

after chemotherapy treatment, as well as investigating a range of possible mediating variables

on these responses. In order to model this process, Bayesian latent growth curve models were

fitted to the learning, immediate retention and delayed recall tests at the four measurement

occasions. Latent growth curve models are widely used as multilevel or structural equation

models for longitudinal or repeated measures data with measurement over time being nested

in the same subject [3, 13, 40, 48, 54, 56]. These models enable each subject to have a

potentially unique trajectory and model trajectory parameters as random effects.


Bayesian latent curve growth models can be viewed as applications of Bayesian hierarchical

regression [13, 24, 25]. The early work on Bayesian hierarchical growth curve models can be

attributed to Fearn and Geisser in the 1970’s [16, 17, 22], with further applications of

Bayesian growth models extended to include autoregressive correlations [34, 35], nested

models [27, 52], use of ordinal [61] and binomial outcomes [11] and alternative

parameterizations [66].

Typically growth models follow linear or polynomial trajectory patterns [48, 54], but

piecewise linear growth models can be used as an alternative to break up a nonlinear or

curvilinear growth trajectory into separate linear components. This is useful in the comparison

of growth rates in different periods [48], with the periods being marked by a transitional

change [64] and the discontinuity occurring at a known or unknown time point. Multiple

discontinuities are also possible. A piecewise model is often preferable to a more general

nonlinear continuous model (of higher polynomial form) if the number of time periods is

small.

Piecewise linear trajectory models have been used in modelling developmental processes

primarily with fixed transition points in the frequentist multilevel [48, 54, 56] and structural

equation modelling frameworks [3, 14, 44]. The areas of application range from educational

assessment [53], alcohol use in adolescents [6, 37, 38], wage patterns for high school dropouts

[54] and effectiveness of intervention processes [33, 62]. These models are also known as

broken stick, linear spline, turning point or change point models.

Bayesian applications of these processes are often referred to as change point models where

the point(s) of change may also be estimated, and cover a range of applications, including

mining accidents [7], cognitive function in dementia sufferers [28], markers for ovarian cancer

[55] and daily menopausal symptoms relief with acupuncture [32]. Mixtures of cubic splines

[49] and combinations of other polynomial functions [41] are further examples of the use of

more complex piecewise trajectories in Bayesian growth models.

In order to model decline and recovery of verbal memory function in chemotherapy patients, a

two piece linear process was used as the underlying growth trajectory profile for the

longitudinal response. This allowed for the identification of two temporal responses: for the

treatment phase (baseline to one month post-chemotherapy) and the recovery phase from one

4.2. Methods 117

month to eighteen months post-chemotherapy.

The aim of this paper was two-fold. The first aim was to identify covariates of mediating

variables which influence the patterns of response over time, using Bayesian piecewise latent

growth models. The second aim was to identify sub-classes of women who demonstrated

different patterns of response, using Bayesian latent class growth mixture models. The

responses of interest were learning, immediate retention and delayed recall aspects of

cognition as measured by the Auditory Verbal Learning Test (AVLT) instrument [19]. The

possible mediating variables included age, IQ and education, which have been found to

influence overall cognitive performance [20, 31], and menopausal status, quality of life, and

measures of mood which have been implicated by the breast cancer literature [2, 8, 31, 50].




4.2 Methods

4.2.1 Study design and participants

This study is part of the Cognition in Breast Cancer (CBC) study undertaken with participants

who were recruited from community hospitals in south-east Queensland with histologically

proven breast cancer treated initially by definitive surgery. Other eligibility criteria included:

age between 18 and 70 years, proficiency in English since early childhood, geographically

accessible for assessment, and having a Karnofsky performance status index of equal or

greater than 80% (indicating normal activity with minor disruption). Participants were also

required to have no recent history of cancer, no previous history of cytotoxic drug treatment,

neurological or psychiatric symptoms or current use of medications which may lead to deviant

neuropsychological test results. Adjuvant chemotherapy was administered after surgery in all

cases and patients were eligible for participation if they were also receiving adjuvant

endocrine treatment or post-operative radiation treatment. All participants provided written,

informed consent. Approval for this study was provided by the Human Research Ethics

Committees of all the participating hospitals as well as the Queensland Institute of Medical


Research. Only participants undergoing chemotherapy treatment and who completed all four

neuropsychological assessment were included in these analyses.

Demographic data collected included age, marital status, family cancer history, menopausal

status, use of hormone replacement therapy, current and previous medications. Planned

treatments including drug regimens used in chemotherapy, endocrine and radiation treatments

as well as the site and extent of the cancer were also recorded. Self-report measures included

depression/anxiety using the Hospital Anxiety and Depression Scale (HADS) [67], and

quality of life as determined by the Functional Assessment of Cancer Therapy General

(FACT-G), Breast subscale (FACT-B), and Fatigue (FACT-F) scores [4, 10]. Higher scores on

the HADS anxiety and depression scales indicated higher levels of depression or anxiety. For

the quality of life variables, the FACT-G scale (27 items) combined the responses for the

domains of physical, emotional, social and functional well-being in a score for general

wellbeing. The 9 item FACT-B subscale comprised of questions specifically on coping with

breast cancer, and the 13 item FACT fatigue subscale measured the disruptiveness and

intensity of fatigue. Higher scores for the FACT scales indicated increased well being or

higher levels of energy (less fatigue). Pre-morbid intellectual functioning was estimated using

the predicted IQ from the National Adult Reading Test (NART) [47, 65].

Participants undertook an individually administered, comprehensive battery of neurological

tests comprising assessments on numerous domains, namely attention, visual and verbal

memory, speed of information processing and executive function. The cognitive functioning

measures considered in this paper are the scales of verbal learning and memory measured by

the Auditory Verbal Learning Test (AVLT) as prescribed in Geffen and Geffen [19] and

utilised in other papers by the same authors [20, 21]. The primary response variables for this

paper were the learning score which was derived from the sum of the words recalled in Trials

1-5 (Learning AVLT Trials 1-5), the immediate retention score after a distractor list (AVLT

Trial 7), and a delayed recall score comprising the total number of words recalled after a 30

minute delay (AVLT Trial 8). Higher scores on these measures imply more words learnt or

recalled and hence are indicative of better verbal memory ability. Age, gender, IQ and

education level have been shown to influence word counts in these tests [20].

4.2. Methods 119

4.2.2 Bayesian piecewise linear latent growth model

A two-piece linear growth model was used in this study to model potential decline and

recovery of the verbal memory process. The first piece L1 covers the period from before

chemotherapy, i.e. baseline or initial time (time 1) to one month after completion of

chemotherapy (time 2). The second piece L2 covers the period post chemotherapy i.e. from

one month after completion of chemotherapy (time 2), six month post-chemotherapy (time 3)

and eighteen months post-chemotherapy (time 4). Values for L1 were set as (0, 1, 1, 1) and L2

as (0, 0, 1, 3) for times 1 to 4 respectively.

The piecewise linear latent growth model was written as a random effects model which is

equivalent to a hierarchical model in which the variability at each level is specified separately.

For longitudinal data the observations over time on individual subjects are considered as level

1 measurements and subjects or participants as level 2 measurements.


=1,2,3,4). Then

yti ∼ Normal(λti, σ2) (4.1)

where λti = η0i + η1iL1t + η2iL2t

and η0i ∼ Normal(β0, σ20)

η1i ∼ Normal(β1, σ21)

η2i ∼ Normal(β2, σ22)

where η0i represents the expected intercept at time 1; η1i represents the expected first linear

change; η2i represents the expected second linear change; The prior distributions were

specified as Normal(0,10E6) for β0, β1, β2, Uniform[0,100] for σ0, σ1, σ2 and Inverse

Gamma (0.01,0.01) for σ2.

Covariates were considered to be time invariant and were added as fixed effects to either the

level (β0) or as interactions with either or both of the linear slopes L1, L2. If wmi is the mth

time invariant covariate interacting with all the three growth parameters then the model could


be rewritten as

yti ∼ Normal(λtim, σ2) (4.2)

where λtim = η0i + η1iL1t + η2iL2t

+ B0mwmi + B1mwmiL1t + B2mwmiL2t

with prior distributions for B0m,B1m,B2m distributed as Normal(0,10E4).

The piecewise linear growth models were fitted using WinBUGS 1.4 [57] with the

R2WinBUGS package [60] in R. Two Markov chains were used with 100,000 iterations and

with the first 10,000 iterations discarded. Convergence assessment was based on the

Gelman-Rubin R̂ statistic (output from R2WinBUGS), with R̂ < 1.1 for all parameters

indicating adequate mixing [24, 25] and Monte Carlo errors less than 5% of the standard

deviation [58].

The Deviance Information Criterion (DIC) is the Bayesian equivalent to the Akaike

Information Criterion (AIC) or Bayesian Information Criterion (BIC) of model fit, and is

determined as the posterior mean deviance with an added penalty component for model

complexity. There are different ways of defining model complexity, one being the measure of

the effective number of parameters of a Bayesian model [24, 58] or as half the posterior

variance of the deviance used by R2Winbugs [24]. The measure of model complexity (pD) is

estimated in the R2WinBUGS package as half the average within-chain variances of the

deviances [60].

A range of Bayesian growth models with random intercepts-slopes were fitted: unconditional

(Model A), years of education adjusted intercept (Model B), education and stage of cancer

(0=Stage I, 1=Stage II and III) adjusted intercept (Model C), education, stage and fatigue

score at time 1 adjusted intercept (Model D), and Model D with HADS depression score at

time 3 adjusted intercept and interaction with recovery slope (Model E). The covariates were

determined from correlations and partial correlations with time 1 AVLT scores and slope

estimates from differences between times 1 and 2, and times 2 and 4. All covariates were

centred around values close to their means, with years of education scores centred at 13 years,

fatigue centred at 38 and HADS depression at time 3 centred at 2.8.

4.2. Methods 121

4.2.3 Bayesian latent class growth mixture models

These models assume that the trajectories from n subjects are driven by an underlying

subject-level latent growth process. The mean structure of the process depends on the subject

belonging to one of K latent classes (K � n).

If there are K groupings the LCGA model can be written in a similar manner to Equation 4.1

with superscript k indicating class or group k

λkti = βk

0 + βk1L1t + βk

2L2t (4.3)

Similarly with the addition of covariates the equation is the same as Equation 4.2 with the

class indicated as superscript k, as presented in the following equation for the mth covariate

wim.

λkti = βk

0 + βk1L1i + βk

2L2t (4.4)

+Bkw0m

wim + Bkw1m

wimL1t + Bkw2m

wimL2t

Estimates of probability of group membership are also obtained. If classes are well defined,

then each subject will have a high probability of belonging to a single class. The Bayesian

latent class growth mixture model proposed is:

yti ∼ Normal(λkti, σ

2i )

λkti | Ci = k ∼ Nm(β∗k, σ

2)

Ci ∼ Multinomial(π1, . . . , πk)

πk ∼ Dirichlet(1, 1, . . . , 1)k)

1/σ2i = 1/σ2

k | Ci = k ∼ Gamma(0.1, 0.1)

Here, Ci gives the latent class membership for subject i with Ci = k if subject i belongs to

class k (k = 1, . . . ,K). The subject level variances σ2i were set to be equal over time but able

to vary across groups. The λkti is determined by class membership based on Equation 4.4. In

these models the covariates are included in the class trajectories in a similar way to the models


of Nagin [46], as opposed to being predictors of class membership [12, 15, 36, 45].

Only two and three class models were considered due to the possibility of small numbers for

group membership given the relatively small total sample size n=120. The regression

parameters βk1 and βk

2 used non-informative prior distributions, namely N(0, 1000) with βk0

being ordered, β10 < β

20 < β

30, so β2

0 = β10 + θ1, and β3

0 = β10 + θ1 + θ2 with θ1, θ2 restricted to

positive values from N(0, 1000) for k = 2 or k = 3 .

Model selection was undertaken with the DIC produced by the R2WinBUGS package, in

conjunction with desire that credible interval of additional regression parameters avoid

covering zero. Although there has been much debate on the appropriateness of the DIC with

mixture models, with particular concern about the pD estimate of effective numbers of

parameters estimated [9, 18, 42], there does not appear to be any clear resolution on this

matter. Average posterior probabilities of class membership were also considered as an

indication of model fit [15, 45, 46]. Again the convergence was asserted if R̂ < 1.1.

A range of two class Bayesian growth models with covariates included were fitted to the

models described earlier, Models B through to Model D. However a restricted set of models

was fitted to three class models due to problems with convergence and estimation of covariate

regression parameters. For the two class models, and the majority of the three class models,

two Markov chains were used with 20,000 iterations and with the first 2,000 iterations

discarded.

4.3 Results

The 120 participants were 25 to 68 years of age (mean 49.3, sd 7.8), with a mean of 13.1 (sd

3.4) years of education and a baseline NART predicted IQ ranging from 90 to 126 (mean

110.6, sd 8.6). At baseline, the majority of participants were married or living with a partner

(85.0%), and were pre-menopausal (52.5%); 43.3% had undergone a mastectomy and 58.3%

had breast conserving surgery. Stage is an measure of the severity of the cancer, and is

determined using the size of the tumour and axillary lymph node involvement. As being in

early stages of breast cancer was a requirement of the study very few participants were in

Stage III of the disease (6.7%) with the majority in Stage II (66.7%) and fewer in Stage I

(26.7%). Table 4.1 presents details of the demographic data for these participants.

4.3. Results 123

For the purposes of further analyses the measure of severity or Stage was categorised into two

levels with stages II and III combined. Participants were also categorised as being estrogen

producing negative or positive at each measurement occasion. Estrogen producing positive

was defined by having experienced menstruation within the past 12 months, and negative if

otherwise. The numbers of participants retaining estrogen producing ability (positive) after

chemotherapy treatment reduced from 69.2% to 50.4%.

The means over the four measurement occasions for the three outcome measures of interest,

learning, immediate retention and delayed recall, appeared to follow a similar pattern with

highest scores before chemotherapy, lowest at one month after chemotherapy, and increasing

improvement over the third and four measurement occasions. As described in Section 4.2.1

higher scores are indicative of better verbal learning ability. Table 4.2 presents the means,

standard deviations and score minima and maxima for the outcome variables for the four

occasions, and Figure 4.1 graphically depicts the verbal learning trajectory patterns.

4648

5052

5456

Learning

Before 1 mth 6 mths 18 mthsChemo After Chemotherapy

Mea

n W

ord

Cou

nt

9.0

9.5

10.0

10.5

11.0

11.5

12.0

Immediate Retention


9.0

9.5

10.0

10.5

11.0

11.5

12.0

Delayed Recall


Figure 4.1 Plots of mean scores for Learning, Immediate Retention and Delayed Recall frombefore chemotherapy to 18 months post-chemotherapy.

4.3.1 Results for Bayesian piecewise linear growth models

The five piecewise linear growth models A-E described in Section 4.2.2 were fitted to the data

and all models passed the convergence criteria also described in Section 4.2.2. Assessments of


Table 4.1 Sociodemographic and clinical data for participants with complete data for 4 mea-surement occasions n=120.

Variable Mean ± SD Min-Max

Age in Years 49.35 ± 7.81 25.2 − 67.9Education as FTE in Years 13.11 ± 3.42 6 − 20NART Predicted IQ baseline 110.64 ± 8.60 90 − 126Fatigue (FACT subscale) baseline 38.42 ± 8.92 16 − 52Depression HADS T1 3.12 ± 2.42 0 − 13

T2 3.50 ± 2.81 0 − 14T3 2.77 ± 2.85 0 − 17T4 2.35 ± 2.27 0 − 12

Variable Frequency Percent

Marital status Single/never married 7 5.8Married/living with partner 102 85.0Separated, Divorced, Widowed 11 9.2


Definitive surgery Breast conserving 70 58.3Mastectomy 52 43.3


Estrogen Producing ability T1 Negative 35 29.2T1 Positive 83 69.2T1 Unknown 2 1.7

T2 Negative 49 41.2T2 Positive 60 50.4T2 Unknown 10 8.4

4.3. Results 125

Table 4.2 Summary Statistics for Learning, Immediate Retention and Delayed Recall for fourmeasurement occasions n=120.

Variable Occasion Mean SD Min MaxLearning T1 53.11 6.97 33 68

T2 49.82 7.96 25 66T3 50.54 8.46 32 70T4 52.98 8.52 33 69



model fit using the Deviance Information Criterion (DIC) are presented in Table 4.3 with

smaller values indicating better fit.

Table 4.3 Assessment of Bayesian Model Fit with Deviance Information Criterion

Model Learning Immediate DelayedRetention Recall

Unconditional A 3019.1 1927.8 2008.8Education B 3018.9 1929.3 2007.5Education, Stage C 3017.1 1927.8 2003.5Education, Stage, Fatigue T1 D 3017.2 1926.9 1999.9D + Depression T3 (I, L2) E 3010.6 1929.4 2002.1

As can be seen in Table 4.3, the model with the lowest DIC value varies across the outcome

variables. Increasing model complexity did not always lead to better fitting models. However

Model C is an improvement on the models A and B for all three outcomes, with Model E

being preferred for Learning and Model D for immediate retention and delayed recall.

The posterior mean parameter estimates with posterior parameter standard deviations for

Model C (Education, Stage adjusted intercept) are presented in Table 4.4, together with the

best fitting Models D and E in Table 4.5.

Figure 4.2 presents a range of trajectories for each of the outcomes variables for Model C for


Table 4.4 Posterior parameter estimates for Bayesian Piecewise Linear Growth Model C edu-cation and stage adjusted intercept; posterior standard deviation in brackets

Parameter Learning Immediate DelayedRetention Recall

β0 I 48.46 (1.10) 9.95 (0.37) 9.76 (0.36)β1 L1 -3.43 (0.61) -1.23 (0.20) -1.66 (0.22)β2 L2 1.08 (0.21) 0.25 (0.07) 0.33 (0.08)

BEduc I 0.63 (0.17) 0.22 (0.05) 0.21 (0.05)BS tage I 4.82 (1.20) 1.52 (0.40) 1.68 (0.40)

σβ0 5.14 (0.51) 1.62 (0.15) 1.50 (0.17)σβ1 1.85 (0.93) 0.49 (0.27) 0.73 (0.30)σβ2 0.67 (0.31) 0.20 (0.10) 0.28 (0.11)σ 4.90 (0.21) 1.59 (0.07) 1.71 (0.07)

Table 4.5 Posterior mean parameter estimates for Bayesian Piecewise Linear Growth for ModelE for Learning and Model D for Immediate Retention and Delayed Recall; posteriorstandard deviation in brackets

Parameter Learning Immediate DelayedRetention Recall

E D D

β0 I 48.60 (1.45) 9.99 (0.36) 9.84 (0.36)β1 L1 -3.41 (0.62) -1.23 (0.20) -1.66 (0.22)β2 L2 1.09 (0.21) 0.25 (0.07) 0.33 (0.08)

BEduc I 0.64 (0.17) 0.23 (0.05) 0.21 (0.05)BS tage I 4.51 (1.26) 1.42 (0.40) 1.54 (0.39)

BFatigueT1 I 0.11 (0.07)† 0.03 (0.02)† 0.04 (0.02)BDepressionT3 I 0.30 (0.24)†BDepressionT3 L2 -0.19 (0.07)

σβ0 5.11 (0.51) 1.60 (0.15) 1.44 (0.18)σβ1 1.95 (0.94) 0.50 (0.28) 0.80 (0.30)σβ2 0.60 (0.29) 0.20 (0.10) 0.30 (0.11)σ 4.87 (0.20) 1.58 (0.07) 1.69 (0.07)† 95% credible intervals cover zero.

4.3. Results 127

combinations of education (at 10, 13, 16 years) and Stage for I and II/III. Solid lines indicate

Stage I and broken lines for Stage II/III.35

4045

5055

6065

Learning


68

1012

14

Immediate Retention


Educ=10,Stage=IEduc=10,Stage=II/III

Educ=13, Stage=IEduc=13, Stage=II/III

Educ=16,Stage=IEduc=16,Stage=II/III

68

1012

14

Delayed Recall


Figure 4.2 Plots of growth trajectories for Learning, Immediate Retention and Delayed Recallfor combinations of years of education (10, 13, 16), and Stage (I, II/III) for ModelC.


Two and three class growth mixture models were fitted to each of the Models B and C, with

non-informative priors as detailed in Section 4.2.3.

All two class models converged with all parameter estimates fulfilling the convergence

criterion of R̂ < 1.1. However a small number of individual class allocations had R̂ = 1.3. The

three class models converged for learning and immediate retention but not for delayed recall.

This failure appeared to be caused by one of the classes becoming redundant, which can be

indicative of attempting to fit too many classes or estimating too many parameters. While it

may be possible to overcome this problem by fixing at least one observation in at least two

classes [39], it is not obvious how this is best achieved in the context of piecewise linear

regression with noninformative prior information about class trajectory profiles. Thus the

response was excluded from further consideration of the three class models. Table 4.6

presents the DIC, pD, numbers and probabilities of class membership for two and three class


models with class specific variances.

Table 4.6 DIC, Posterior estimates for probabilities of class membership and numbers in classesModels B, C

Model Parameter Learning Immediate DelayedRetention Recall

Two classesB DIC 3091.3 1986.8 2058.8

pD 54.2 44.2 54.9πk 0.57,0.43 0.64,0.36 0.58,0.42nk 68,52 77,43 69,51

C DIC 3081.7 1994.6 2063.7pD 58.7 59.7 73.2π 0.52,0.48 0.60,0.40 0.57,0.43n 62,58 72,48 68,52

Three classesB DIC 3057.4 1961.6

pD 117.7 121.0πk 0.22,0.40,0.38 0.23,0.47,0.30nk 26,49,45 27,57,36

C DIC 3078.6 2034.2pD 128.2 184.4πk 0.14,0.42,0.44 0.24,0.49,0.27nk 16,51,53 28,59,32

Based on the goodness of fit evaluations, Model B with a three class mixture was deemed to

be best for learning and immediate retention, and the analogous model with a two class

mixture was deemed to be preferred for delayed recall. The posterior class estimates of the

trajectory parameters together with average mean posterior probabilities for class membership

for both the two and three class models with adjustments for years of education are presented

in Table 4.7. Figure 4.3 depicts the class trajectories for learning, immediate retention and

delayed recall with years of education set to 13.

There was reasonable agreement between the classes across the outcome variables with the

overall probabilities of agreement over class membership for the two class mixtures for Model

B being 0.817 (se 0.035), 0.808 (se 0.036), 0.908 (se 0.026) for learning/immediate retention,

4.4. Discussion 129

learning/delayed recall and immediate retention/delayed recall comparisons respectively, and

intra-class agreements ranging from 67.9% to 97.6%. The overall probability of agreement for

the three class mixture for Model B was lower than for the equivalent two class models [0.708

(se 0.032), 0.708 (se 0.032)] for learning and immediate retention) with intra-class agreements

ranging from 62.5% and 88.6%.

3540

4550

5560

65

Learning

Before 1 mth 6 mths 18 mths

n=26 (22%)

n=49 (40%)

n=45 (38%)

Chemo After Chemotherapy

Mea

n W

ord

Cou

nt

68

1012

14

Immediate Retention

n=27 (23%)

n=57 (47%)

n=36 (30%)


68

1012

14

Delayed Recall


n=69 (58%)

n=51 (42%)


Figure 4.3 Plots of two and three class trajectories for learning, Immediate Retention and De-layed Recall using Model B for years of education=13

Class means and standard deviations for mediating variables included in Models B to E, and

the probability of class mean differences are presented in Tables 4.8 and 4.9 for two and three

class mixtures respectively.

4.4 Discussion

From the results of the Bayesian latent growth models it is evident that the trajectory profile of

all three outcomes of verbal memory (learning, immediate retention and delayed recall) is

well explained by a two piece linear process with a changepoint at time two, (one month after

chemotherapy) comprising a decline in measurements before chemotherapy to one month post

chemotherapy (posterior slope estimates -3.43, -1.23, -1.66 respectively) and a recovery phase

from one month to eighteen months post-chemotherapy (posterior recovery slope estimates

1.08, 0.25, 0.33). Investigation of covariates showed that increasing years of education and


Table 4.7 Means and SD of parameters of Bayesian Latent Class Piecewise Linear GrowthModels - Education adjusted intercept Model B for two and three classes

Class Parameter Learning Immediate DelayedRetention Recall

1 β0 48.636 (0.865) 10.037 (0.270) 9.883 (0.329)β1 -4.213 (0.986) -1.445 (0.310) -1.917 (0.356)β2 0.784 (0.352) 0.257 (0.108) 0.281 (0.125)beduc 0.493 (0.121) 0.202 (0.037) 0.148 (0.046)σ 6.138 (0.282) 2.004 (0.089) 2.171 (0.103)Av.PP 0.967 (0.059) 0.940 (0.124) 0.927 (0.117)

2 β0 56.422 (0.854) 12.935 (0.325) 12.516 (0.324)β1 -2.387 (0.974) -0.826 (0.334) -1.311 (0.333)β2 1.461 (0.350) 0.233 (0.116) 0.397 (0.116)beduc 0.560 (0.166) 0.110 (0.053) 0.159 (0.061)σ 5.212 (0.291) 1.546 (0.119) 1.674 (0.133)Av.PP 0.935 (0.118) 0.930 (0.145) 0.914 (0.141)

1 β0 45.3 (1.758) 8.727 (0.509)β1 -5.832 (1.769) -1.107 (0.522)β2 1.067 (0.675) 0.257 (0.108)beduc 0.494 (0.228) 0.263 (0.064)σ 6.152 (0.578) 1.832 (0.166)Av.PP 0.891 (0.141) 0.853 (0.158)

2 β0 50.93 (0.927) 10.858 (0.321)β1 -3.228 (1.015) -1.555 (0.325)β2 0.722 (0.384) 0.233 (0.116)beduc 0.571 (0.152) 0.193 (0.052)σ 4.754 (0.419) 1.673 (0.115)Av.PP 0.859 (0.142) 0.872 (0.131)

3 β0 56.93 (0.912) 13.124 (0.324)β1 -2.279 (1.022) -0.766 (0.348)β2 1.479 (0.363) 0.257 (0.108)beduc 0.565 (0.154) 0.115 (0.049)σ 5.047 (0.304) 1.462 (0.117)Av.PP 0.931 (0.118) 0.947 (0.077)

4.4. Discussion 131

Table 4.8 Class means for demographic variables, quality of life scores, and numbers for stageof cancer [I/II&III].

Response Class 1 Class 2 ProbMean (se) Mean(se) mean diff

Two Class ModelsLearning Age 50.57 (0.98) 47.79 (1.00) 0.052

NART 109.45 (1.13) 112.16 (1.03) 0.087Education 13.06 (0.47) 13.17 (0.39) 0.862FACT Fatigue T1 36.87 (1.17) 40.40 (1.05) 0.031Depression T3 3.09 (0.37) 2.36 (0.35) 0.164Stage of Cancer 25:42 7:46 0.003n 67 53

Immediate Age 50.05 (0.90) 47.99 (1.15) 0.171Retention NART 109.85 (1.03) 112.18 (1.14) 0.161

Education 13.03 (0.41) 13.27 (0.45) 0.713FACT Fatigue T1 37.70 (1.01) 39.83 (1.37) 0.215Depression T3 2.81 (0.34) 2.68 (0.39) 0.818Stage of Cancer 28:51 4:37 0.003n 79 41

Delayed Age 50.54 (0.98) 47.68 (0.99) 0.047Recall NART 109.34 (1.12) 112.47 (1.01) 0.049

Education 12.89 (0.45) 13.42 (0.41) 0.400FACT Fatigue T1 37.19 (1.07) 40.16 (1.22) 0.072Depression T3 3.06 (0.40) 2.36 (0.27) 0.187Stage of Cancer 26:44 6:44 0.002n 70 50

Table 4.9 Class means for demographic variables, quality of life scores, and numbers for stageof cancer [I/II&III].

Response Class 1 Class 2 Class 3 ProbMean (se) Mean(se) Mean (se) mead diff

Three Class ModelsLearning Age 52.18 (1.50) 49.40 (1.12) 47.77 (1.13) 0.082

NART 110.45 (1.93) 109.58 (1.25) 111.96 (1.16) 0.402Education 13.176 (0.77) 12.96 (0.51) 13.24 (0.44) 0.918FACT Fatigue T1 36.00 (2.02) 37.39 (1.28) 40.89 (1.31) 0.051Depression T3 3.38 (0.56) 2.78 (0.43) 2.42 (0.40) 0.419Stage of Cancer 12:12 15:36 5:40 0.002n 24 51 45

Immediate Age 51.19 (1.57) 49.47 (1.01) 47.82 (1.30) 0.257Retention NART 111.64 (1.79) 109.27 (1.15) 112.29 (1.28) 0.208

Education 13.28 (0.73) 12.90 (0.46) 13.34 (0.51) 0.801FACT Fatigue T1 37.28 (2.20) 38.13 (0.99) 39.74 (1.58) 0.542Depression T3 3.20 (0.58) 2.65 (0.39) 2.66 (0.43) 0.687Stage of Cancer 12:13 16:44 4:31 0.007n 25 60 35


increased levels of cancer severity are associated with higher levels (intercepts) of all three

verbal memory responses (Table 4.4). The effect of stage on the level (intercept) of the

responses may be a function of the distribution of subjects into the severity categories of stage

of cancer, with 27% in Stage I (see Table 4.1).

The observed need for adjustment for years of education is not entirely consistent with

findings of previous studies where age was the predominant demographic factor affecting

learning and memory scores [20, 21, 31]. In these studies, the years of education did not differ

across the age groups, but the earlier studies comprised female participants with lower mean

years of education which ranged from 10.0 to 12.3 for the ten year age bands from 40 to 69

[20, 21]. In contrast, in the current study there was a significant reduction in years of

education with increased age (r=-0.297, p=0.001), with mean years of education highest at

15.3 for ages 30-39 and reducing to 10.2 for ages 60-69. Our results may be interpreted that

the increasing years of education as a partial surrogate for younger ages may offset the effect

of increased disease severity.

The more complex models, D and E presented in Table 4.5 indicated that only delayed recall

included a significant increase in the level (intercept) with the baseline FACT Fatigue score,

indicating that less fatigue increased overall memory ability. The Learning trajectory included

a significant reduction in the recovery response with an increased HADS depression score

measured at 6 month post-chemotherapy.

The Bayesian latent class growth mixture models with adjustments for education resulted in

the three class models for learning and immediate retention, and the respective two class

model for delayed recall being the preferred models based on the Deviance Information

Criterion (DIC) assessment. The resultant classes differed predominantly by the level of the

response as visible in Figure 4.3. However, for both the two and three class models, the

decline in the first linear component was less steep for classes with higher initial scores, and

learning and delayed recall exhibited faster rate of recovery for classes with higher initial

scores. The recovery rate remained constant over all classes for immediate retention.

Age and baseline NART scores, together with the differing proportions between stages of

cancer reflected differences between the two classes of delayed recall. Older ages, lower

NART scores and a relatively larger proportion of subjects with less severe cancer (Stage I)

BIBLIOGRAPHY 133

were indicative of the class with overall lower verbal memory performance levels (Class 1).

However the differing proportions in stages of cancer was the only mediating variable which

varied between the three classes for learning and immediate retention.

The latent classes identified by the Bayesian latent class growth mixture analysis, have

revealed multiple trajectory processes being followed by participants in the Cognition in

Breast Cancer study. The differences in trajectory classes were indicative of the relatively

subtle nature of the differences experienced, despite the commonality of a definite cognitive

decline followed by a recovery phase. It is hoped that this will facilitate greater understanding

of this phenomenon and assist with identification and management of susceptible patients.

Funding

This research was supported by The Wesley Research Institute (200320); the Cancer Council

Queensland (406900); the National Breast Cancer Foundation (406900) and the Australian

Research Council Linkage Project (LPO669670). Conflicts of Interest: None declared.

Acknowledgements




Council Linkage Project. The authors would like to thank Drs Toni Jones, Donna Spooner,

and Miss Elena Moody for their input in the design and implementation of the study. We

would also like to thank all the oncologists, surgeons, and research nurses who helped in the

recruitment process, and the research assistants involved in recruitment and data collection.

Finally, the authors would also like to sincerely thank all the women who participated in the

study at such a distressing period in their life.

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Chapter 5

Bayesian Estimation of Class

Predictors for Latent Class Growth

Models


presented in its entirety. This article has been submitted to the journal Memory in December

2009. Please note that the order of the references in this chapter differs from the submitted

paper.

Title: Impact of Chemotherapy on Verbal Memory in Breast Cancer patients. Who is at

Risk?


Brooke Andrewcd


The purpose of this study was to identify subgroups of women with different verbal memory

trajectory (growth) patterns, and identify predictors (baseline) for the subgroups or classes of

141

142CHAPTER 5. BAYESIAN ESTIMATION OF CLASS PREDICTORS FOR LATENT

CLASS GROWTH MODELS

breast cancer patients undergoing chemotherapy. The current study involved participants who

were recruited from community hospitals in south-east Queensland with histologically proven

breast cancer treated initially by definitive surgery who underwent chemotherapy. Participants

were assessed at four occasions: before chemotherapy post-surgery, one month, six months

and eighteen months post-chemotherapy with relevant medical, neuropsychological and

quality of life factors being undertaken. Learning, immediate retention and delayed recall

aspects of verbal memory were measured by the Auditory Verbal Learning Test. Two

trajectory classes Low and High were obtained for the three verbal memory outcomes, with

age, years of education and stage of cancer being identified as core predictors of learning,

immediate retention and delayed recall, The addition of baseline anxiety or depression was a

substantial predictor of immediate retention and depression for delayed recall. Increased age,

fewer years of education, having Stage I cancer and higher baseline anxiety/or depression

score were implicated as predictors for some Low verbal memory classes.








143


CLASS GROWTH MODELS

Impact of Chemotherapy on Verbal Memory in Breast Cancer patients.

Who is at Risk?

Abstract

The purpose of this study was to identify subgroups of women with different verbal memory

trajectory (growth) patterns, and identify predictors (baseline) for the subgroups or classes of

breast cancer patients undergoing chemotherapy. The current study involved participants who

were recruited from community hospitals in south-east Queensland with histologically proven

breast cancer treated initially by definitive surgery who underwent chemotherapy. Participants

were assessed at four occasions: before chemotherapy post-surgery, one month, six months

and eighteen months post-chemotherapy with relevant medical, neuropsychological and

quality of life factors being undertaken. Learning, immediate retention and delayed recall

aspects of verbal memory were measured by the Auditory Verbal Learning Test. Two

trajectory classes Low and High were obtained for the three verbal memory outcomes, with

age, years of education and stage of cancer being identified as core predictors of learning,

immediate retention and delayed recall, The addition of baseline anxiety or depression was a

substantial predictor of immediate retention and depression for delayed recall. Increased age,

fewer years of education, having Stage I cancer and higher baseline anxiety/or depression

score were implicated as predictors for some Low verbal memory classes.

Keyword: Bayesian ; longitudinal; latent class growth ; breast cancer ; verbal memory ;

cognition ; predictors


5.1 Introduction

Success in the treatment of breast cancer, especially when diagnosed in early stages of the

disease has resulted in an increased relative 5 year survival rate in the last quarter century

from 74% in 1982 to nearly 90% in 2006 [1] in Queensland, Australia. During this period the

annual incidence increased from 86.3 to 116.4 rate per 100,000 population. For the whole of

Australia in a comparable period the incidence rate increased from 69.2 to 118.5, with a 2.3

fold increase in new cases from 5304 (1982) to 12,170 (2005) and improved 5 year survival

rate (72% to 88%) [2]. This survival pattern is replicated in the other western countries with

improvements in survival rates being attributed to improved availability of diagnostic

screening, increased awareness and more effective treatment regimes [3].

With this increase in breast cancer survival, survivorship issues such as quality of life

becomes an area of primary attention. Decline in cognitive functioning is a frequently

reported side-effect for women undergoing adjuvant chemotherapy treatment for breast

cancer. The proportion of women suffering from cognitive impairment after chemotherapy in

the short term varies between studies, with reported estimates of 20% or 25% [4, 5] to 50%

for women with moderate or severe impairment [6] and similarly 50% for studies on older

women (aged 65 years and above) [7]. The level of cognitive dysfunction has been shown to

improve over time [4, 8], although a subset of women are still below baseline levels 12

months post-chemotherapy [9, 10] and there is evidence of long term effects for up to 10 years

post chemotherapy completion [11, 12].

The cognitive domains of attention, concentration, verbal and visual memory, processing

speed [6, 9, 11, 12, 13, 14] and executive function [15] have been specifically indicated as

areas of functional deficit. In particular, the domain of verbal memory was consistently

identified in several studies [11, 13, 14, 16, 17, 18, 19] as suffering compromise after

chemotherapy treatment. While age, IQ and education have been reported to affect verbal

memory performance [4, 20, 21], stage of cancer, menopausal status, baseline measures of

mood (depression and anxiety) and fatigue have also being implicated in the extant literature,

albeit inconsistently [4, 11, 14, 15, 18, 22, 23, 24, 25, 26].

In order to investigate this process more fully a study was undertaken to assess the impact of


CLASS GROWTH MODELS

adjuvant chemotherapy on cognitive functioning in early stage breast cancer patients drawn

from hospitals throughout south-east Queensland, Australia. The study follows a prospective

longitudinal design with four assessments conducted before chemotherapy, and at one, six and

eighteen months after completion of chemotherapy. A comprehensive battery of

neuropsychological tests measuring a range of cognitive domains was conducted at all four

measurement occasions, together with self-report measures of anxiety, depression and quality

of life. Other demographic factors comprising age, education, estimated intellectual

functioning, medical and hormonal history were also recorded.

This paper focuses on the analysis of the verbal memory domain measured by the Auditory

Verbal Learning Test [27], the identification of the possible classes of subjects who

demonstrate distinct trajectory patterns of decline, recovery or no change in verbal memory

function before and after chemotherapy treatment, and the role of predictor or mediating

variables in the determination of the probability of class membership. The impact of

time-varying factors of fatigue, depression, anxiety and estrogen producing status is also

considered.

In order to model this process, Bayesian latent class growth models were fitted to the learning,

immediate retention and delayed recall tests at the four measurement occasions. A

two-piecewise linear trajectory profile was used as the underlying process to assess the two

phases, from baseline to after chemotherapy, and post-chemotherapy. The predictors of class

membership were restricted to baseline measures and included age, premorbid intelligence

estimate (NART), years of education, stage of cancer, type of surgery and baseline scores for

estrogen producing ability, fatigue, anxiety and depression.

Latent class growth models or latent growth mixture models are extensions of latent growth

models from the multilevel [28, 29, 30, 31] and structural equation modeling literature

[32, 33]; instead of all subjects having distinct growth trajectories, a finite (usually small)

number of classes or groups of trajectories are used to model the process. The latent class

growth models (LCGM) were developed by Nagin, Land and Tremblay [34, 35, 36], and are a

subset of the growth mixture models (GMM) presented by Muthen [37, 38, 39]. Bayesian

latent class growth models can be viewed as applications of Bayesian hierarchical regression

[31, 40, 41] combined with Bayesian finite mixture models [31].


The piecewise linear model, also known as a change point model, was adopted in preference

to a quadratic model in light of the small number of time points. This model can provide a

flexible representation of nonlinear response as well as explicitly characterize and compare

growth rates in clearly identified periods [28, 33, 42, 43, 44]. Piecewise growth mixtures have

been used in a frequentist framework to model smoking patterns [45] and cognitive

development [46], and in a Bayesian framework to describe cognitive function in dementia

sufferers [47], markers for ovarian cancer [48] and daily menopausal symptom relief with

acupuncture [49].

The inclusion of covariates or predictors in the probability of group membership takes a form

similar to a logistic or multinomial logistic model and is described in detail by Nagin [36] and

Muthen [39]. The inclusion of covariates in predicting the probability of group membership

for LCGM and GMM has been used in the areas of education [50, 51], cancer biomarker

research [52], smoking likelihood [53], alcohol use [54, 55], PTSD in Gulf War veterans [56]

and adolescent delinquency [57, 58].

Published literature on Bayesian latent class growth models or growth mixture models is

limited, and includes Bayesian general growth mixture models of binary daily adverse event

occurrences and continuous daily affect scores following myocardial infarctions [59], the

identification of multivariate responders and non-responders for the treatment of cystitis [60],

enabling the variance structure to include group differences with application to calcium

absorption/malabsorption [61].

The aims of this paper were three-fold. The first aim was to identify sub-classes of women

who demonstrated different patterns of response, using Bayesian latent class growth mixture

models. The second aim was to identify covariates or predictors of these classes, and the last

aim was to investigate the impact of time-varying trajectory covariates on the subgroup

structure.



discussion in Section 5.4.


CLASS GROWTH MODELS

5.2 Methods

5.2.1 Participants

This study is part of the prospective longitudinal Cognition in Breast Cancer (CBC) study

undertaken with 178 participants who were recruited from community hospitals in south-east

Queensland from May 2004 to April 2006. All participants had histologically proven early

stage breast cancer treated initially by definitive surgery. Although the larger study embraced

participants who had undergone a range of treatments, including adjuvant chemotherapy,

radiation or endocrine treatment, only participants who experienced chemotherapy treatment

(n=155) and who presented at all four measurement occasions (120) are considered in this

paper.

Further eligibility criteria for the study included being aged between 18 and 70 years;

proficiency in English since early childhood; geographically accessible for assessment; a

Karnofsky performance status index of equal or greater than 80%; no recent history of cancer;

no previous history of cytotoxic drug treatment; no neurological or psychiatric symptoms; and

no current use of medications which may lead to deviant neuropsychological test results.

All participants provided written, informed consent. Approval for this study was provided by

the Human Research Ethics Committees of all the participating hospitals as well as the

Queensland Institute of Medical Research.

Demographic data collected included age, years of education, menopausal status, marital

status, family cancer history, use of hormone replacement therapy, and current and previous

medications. Medical information on chemotherapy treatment, number of treatment courses

and the use of other treatments were also recorded.

Participants undertook an individually administered, comprehensive battery of

neuropsychological tests comprising assessments on numerous domains, namely attention,

visual and verbal memory, speed of information processing and executive function. The first

assessment (denoted here as T1 or baseline) was conducted after surgery but prior to the

commencement of chemotherapy, with the second, third and fourth (denoted as T2, T3, T4)

conducted one month, six and eighteen months post-chemotherapy.

5.2. Methods 149

5.2.2 Instruments

The Hospital Anxiety and Depression Scale (HADS) [62] was used to assess depression and

anxiety levels. The self-report questionnaire consists of 7 items each for anxiety and

depression which are summed to obtain scores ranging from zero to 21. Higher scores on the

HADS anxiety and depression scales indicate higher levels of depression or anxiety. Scores

less than 8 are regarded to be in the normal range, with scores 11 or greater as a probable

indication of the mood disorder [63].

Quality of life was measured with the Functional Assessment of Cancer Therapy Scales of

General (FACT-G), Breast (FACT-B), and Fatigue (FACT-F) scores [64, 65]. For the quality

of life variables, the FACT-G scale (27 items) combined responses over the physical,

emotional, social and functional domains into one general wellbeing score. The 9 item

FACT-B subscale comprised questions specifically on coping with breast cancer, and the 13

item FACT fatigue subscale measured the disruptiveness and intensity of fatigue. Higher

scores for the FACT scales indicated increased well being or better quality of life, so for the

fatigue scale higher scores indicated higher levels of energy (less fatigue).

Pre-morbid intellectual functioning (IQ) was predicted by the the National Adult Reading Test

or NART [66, 67].

5.2.3 Verbal Memory

The domain of verbal memory was assessed with the Auditory Verbal Learning Test (AVLT)

as prescribed in Geffen and Geffen[27] and utilised in other papers by the same authors

[20, 68]. A list of 15 words (List A) are read aloud with a one second interval between words,

for each five consecutive trials followed by a free-recall test (Trials 1 to 5). An interference

word list (B) is then presented followed with a free-recall test (Trial 6). Trial 7 and 8 require

the participant to free recall as many words as possible from the original list (A), with Trial 8

occurring after at least a 20 minute delay. The domain of verbal memory for this paper

consists of three parts: verbal learning measured by the sum of the words correctly recalled on

Trial 1 to 5, immediate retention from Trial 7, and delayed recall from Trial 8. Higher scores

are indicative of better performance. Alternative forms were utilised to decrease practise

effects.


CLASS GROWTH MODELS

Age, gender and IQ have been shown to influence the number of words recalled in the AVLT

[20]. There is some debate as to the relationship between years of education and verbal

memory ranging from no significant contribution [69], no significant contribution after

adjusting for intellectual functioning [70], to being included as a factor in the construction of

published norms [71]. Extent of education, or years of fulltime study is closely related to

intellectual functioning, and in our study was strongly correlated (0.79, n = 120). As a

stronger correlation existed between education and the verbal memory outcomes, education is

used in the models instead of the NART measure of intellectual function.

The verbal memory outcomes are displayed with age and gender adjusted Geffen published

norms [20].

5.2.4 Medical indicators

Stage of cancer is a measure of the severity of the disease, and is a function of the size of the

tumour and axillary lymph node involvement. The increased stage indicates increased disease

severity. For analysis purposes Stages II and III were combined due to the low numbers of

participants (n=8; 6.7%) in Stage III which was a consequence of the inclusion criteria of the

study.

Participants were categorised as being estrogen producing negative or positive at each

measurement occasion. Estrogen producing positive was defined by having experienced

menstruation within the past 12 months, and negative otherwise. Missing estrogen producing

ability scores were imputed from the participant’s remaining profile, with 2, 11, 6 and 3

participants having their scores imputed at times 1 to 4 respectively.

Surgery type was used as an indicator of the type of surgery undergone with having had a

mastectomy coded as 1 and 0 otherwise.

5.2.5 Statistical Methods

A two-piecewise linear trajectory profile was used to assess the change in outcomes over the

assessment times, where the four assessment times are defined as baseline (T1) or prior to

chemotherapy treatment but after surgery, one month (T2), six months (T3) and eighteen

months (T4) post-chemotherapy treatment completion. The two-piece linear model consists of

5.2. Methods 151

two linear components, L1 and L2 where the first linear component (L1) models the linear

pattern from baseline (T1) to one month post-chemotherapy (T2) and the second (L2) the

linear pattern from T2 to T4, with the change-point, or node at T2. The two linear components

L1 and L2 take the values L1=0,1,1,1 and L2=0,0,1,3 for the pre- and post-chemotherapy

phases for times T1 to T4 respectively.

The latent class growth models assume that the trajectories from n subjects are driven by an

underlying subject-level latent growth process, in which the mean structure of the process

depends on the subject belonging to one of K latent classes (K � n). Only two and three class

models were considered due to the possibility of small numbers for group membership given

the total sample size n=120.

A range of models were fitted: two and three class unconditional models, that is without

covariates; models using single predictors for class membership; models with combinations of

two and three predictors of class membership; a core model with age, education and stage of

cancer (AES); and the core model with an additional fourth predictor. Finally, class specific

time covariates (anxiety, depression, fatigue, estrogen producing ability) were added

individually to the core (AES) model. The following variables were centred in all models: age

(at 49), NART (at 110), education (at 13), and fatigue (at 38).

For the unconditional models, with only an intercept predictor, a Dirichlet prior distribution

was used for the probability of class membership and for models with predictors a N(0,100)

prior was used for the logistic (or multinomial) regression parameters. The class dependent

intercept parameters were ordered to ensure the first class was always the lowest.

Non-informative prior distributions were employed, namely N(0,1000) for the lowest class

intercept and similar but zero bounded priors were employed for subsequent intercept

increases for additional classes. The class specific regression parameters for the two linear

segments, L1 and L2 used non-informative prior distributions, namely N(0,1000). The class

specific residual standard deviations took uniform U(0.01,10) prior distributions. Again the

prior distributions for the regression coefficients for the time-varying covariates employed

non-informative N(0,100) prior distributions.

Model comparison was undertaken using the Deviance Information Criterion (DIC). The DIC

is determined as the posterior mean deviance with an added penalty component for model


CLASS GROWTH MODELS

complexity (pD) which may be estimated, based on multiple MCMC chains [74], as half the

average within-chain variance of the deviances [72]. Model selection was based on the

consideration of both the DIC and model parsimony, with the latter being indicated by all

regression parameters having 95% credible intervals that exclude zero.

Bayesian analyses used the Winbugs 1.4 package [73] with the R2Winbugs package [72] in

the R statistical software program. Two Markov chains were used with 10,000 iterations and

with the first 1000 iterations discarded for the unconditional and the probability of

membership only models, and 20,000 with 2000 burnin for models including time-varying

trajectory covariates. Convergence assessment was based on the Gelman-Rubin R̂ statistic

(output from R2WinBUGS), with R̂ < 1.1 for all parameters indicating adequate mixing

[40, 41] and Monte Carlo errors less than 5% of the standard deviation [74].

Sensitivity analyses were also undertaken with the core AES model using alternative logistic

parameter prior distributions of N(0, 9/4) as specified by Garrett and Zeger [75] and used in

the papers of Elliott and Leiby [59, 60].

5.3 Results

The 120 participants with complete data were between 25 and 68 years of age (mean 49.3, sd

7.8), with a mean of 13.1 (sd 3.4) years of education and a baseline NART predicted

intellectual functioning ranging from 90 to 126 (mean 110.6, sd 8.6). At baseline, the majority

of participants were married or living with a partner (85.0%), and were pre-menopausal

(52.5%); 43.3% had undergone a mastectomy and 58.3% had breast conserving surgery. As

being in early stages of breast cancer was a requirement of the study very few participants

were in Stage III of the disease (6.7%) with the majority in Stage II (66.7%) and fewer in

Stage I (26.7%). The numbers of participants retaining estrogen producing ability (positive)

after chemotherapy treatment reduced from 69.2% at baseline (T1) to 50.4% at time T2, with

further reductions to 48.3% and 29.4% at times T3 and T4 respectively. Table 5.1 presents

details of the sociodemographic characteristics of the participants.

For comparative purposes Table 5.2 presents the means of the verbal memory variables for the

four measurement occasions in the present study along with published norms [68]. The

Geffen norms were used as opposed to the norms published by Strauss [21] as the intellectual

5.3. Results 153

Table 5.1 Sociodemographic and clinical data for participants with complete data for 4 mea-surement occasions n=120.

Variable Mean ± SD Min-MaxAge in Years 49.35 ± 7.81 25.2 − 67.9Education as FTE in Years 13.06 ± 3.27 6 − 20NART Predicted IQ baseline 110.65 ± 8.60 90 − 126Fatigue (FACT subscale) baseline 38.42 ± 8.92 16 − 52Anxiety HADS baseline 6.77 ± 3.79 0 − 21Depression HADS baseline 3.12 ± 2.42 0 − 13Variable Frequency PercentMarital status Single/never married 7 5.8

Married/living with partner 102 85.0Separated, Divorced, Widowed 11 9.2


Definitive surgery Breast conserving 70 58.3Mastectomy 52 43.3


functioning levels in the present study were comparable to the higher intellectual functioning

means in Geffen. From observation, the 30-39 years age class (n=8) at baseline had verbal

memory scores lower than the published norms, but increased beyond these norms by T4,

whereas the older two age classes 50-59 years and 60-69 years were higher than the norms at

baseline but after a considerable drop at T2 one month post-chemotherapy increased but had

on average not returned to baseline scores by time 4 (18 months post-chemotherapy).

The means over the four measurement occasions for the three outcome measures of interest,

learning, immediate retention and delayed recall, appeared to follow a similar pattern with

highest scores before chemotherapy, lowest at one month after chemotherapy, and increasing

improvement over the third and fourth measurement occasions; this is consistent with the

observed age-specific norms in Table 5.2. Higher scores are indicative of better verbal

learning ability. Table 5.3 presents the overall means, standard deviations and score minima

and maxima for the outcome variables for the four occasions.

Also presented in Table 5.3 are details of the responses for the time-varying covariates of

fatigue, anxiety, depression and estrogen producing ability, with means and standard

deviations for the continuous variables and numbers and odds for the binary variable. The


CLASS GROWTH MODELS

Table 5.2 Comparison Of Verbal Memory Scores By Ten Year Age Classes With PublishedNorms.

Age Groups30-39 40-49 50-59 60-69

n 8 59 39 13Learning Study Sample T1 53.1 52.7 51.1 51.1

T2 51.1 50.5 44.9 44.9T3 54.3 50.3 46.5 46.5T4 59.5 52.6 49.6 49.6

Published Norms 55.9 52.1 47.6 49.0

Immediate Study Sample T1 11.6 11.5 11.1 11.0Retention T2 10.5 10.3 10.6 8.1

T3 11.5 10.4 10.1 9.5T4 12.6 11.1 10.7 9.4


Delayed Study Sample T1 11.5 11.5 11.2 11.0Recall T2 10.4 9.7 10.3 7.9

T3 10.4 9.8 10.0 9.2T4 11.9 11.1 10.4 9.5


Age Study Sample T1 35.7 45.4 54.3 62.737.7 44.8 57.5 62.7

Education Study Sample T1 14.4 13.6 13.0 10.210.9 11.7 12.3 10.0

Intellectual Study Sample T1 108.8 111.8 111.1 106.2function 111.9 113.3 116.6 113.9

mean scores for fatigue are lowest after chemotherapy (T2) and improve to a higher than

baseline level at time 4. Anxiety means are highest at baseline, then follow a continual

decline. While mean anxiety levels reduce over time the proportion of participants with

anxiety levels higher than 11 were more constant with 14%, 7.5%, 8% and 12% over the four

measurement occasions. Depression scores increased from baseline and were maximal just

after chemotherapy completion (T2) and then declined over time. Very few participants had

scores greater than 10, with only 3 participants (2.5%) at time 2 and 1 at time 4. The odds for

estrogen producing ability are highest at baseline, 2.43 then reduce over time to 0.46 at T4 (18

month post-chemotherapy).

The two MCMC chains used in the estimation of the two class single covariate models

achieved convergence with R̂ < 1.1. However the three class single covariate model resulted

in a class with zero membership. Hence only two class models were considered for further

investigation. All two class models satisfied the convergence criteria.

The posterior estimates for the logistic parameters of the probability of class membership for

the two class models are presented in Table 5.4. When the 95% credible interval encloses

5.3. Results 155

Table 5.3 Summary Statistics for Learning, Immediate Retention and Delayed Recall, Fatigue,Depression, Anxiety and Estrogen Producing ability over four measurement occa-sions n=120.

Occasions1 2 3 4

Mean SD Mean SD Mean SD Mean SDLearning 53.11 6.97 49.82 7.95 50.54 8.46 52.98 8.52Immediate Retention 11.32 2.35 10.15 2.47 10.26 2.68 10.87 2.60Delayed 11.33 2.37 9.77 2.52 9.87 2.82 10.71 2.80Fatigue 38.42 8.92 33.82 10.70 38.22 10.34 40.42 9.17Anxiety 6.77 3.79 5.82 3.45 5.79 3.44 5.76 3.52Depression 3.13 2.42 3.50 2.81 2.77 2.85 2.35 2.27

Estrogen Producing +/- n 85/35 71/49 58/62 38/82odds 2.43 1.45 0.94 0.46

95%CI 1.61,3.67 0.99,2.12 0.64,1.36 0.21,0.69

zero, the predictor is considered not to contribute substantively to the class membership

probability. Age, NART, education, and stage were consistently substantive predictors of class

membership probability when considered individually, for all three verbal memory outcomes.

However fatigue was indicated as a substantive predictor only for learning and immediate

retention, anxiety with immediate retention and depression with both immediate retention and

delayed recall. The other binary covariates, estrogen producing ability and surgery type were

not indicated as substantive predictors for any of the three verbal memory outcomes.

The full models with the inclusion of the trajectory parameters for the unconditional (no

covariate) model and the age-education-stage covariate models are presented in Tables 5.5 and

5.6 respectively. The addition of the covariates in determining the probability of class

membership had minimal impact on the trajectory parameters, with the only a slight shift of

the intercept and virtually no change to the two linear slope parameters. All verbal memory

outcomes showed a substantive decline for the first linear slope (L1), from baseline to after

chemotherapy completion, with the lower class having a more rapid decline than the higher

class; -4.32 versus -2.37 for learning, -1.47 versus -0.92 for immediate retention and -1.86

versus -1.40 for delayed recall. All three outcomes exhibited a substantive positive recovery

with the second linear slopes (L2) for the low and high classes of 0.86 and 1.34, 0.21 and

0.29, and 0.26 and 0.41 for learning, immediate retention and delayed recall respectively. The

lower class always showed a slower rate of recovery post-chemotherapy than the higher class,

although not substantively different.


CLASS GROWTH MODELS

Table 5.4 Posterior Estimates For Logistic Regression Parameters For Predictors Used SinglyWith Probability Of Class Membership (for the Low class of the two class model).

Outcome Predictor Posterior Posterior 95% PosteriorMean SD Cred Interval DIC PD Class numbers

Learning Age † 0.069 0.029 0.015 0.088 3107.7 52.3 64.5/55.4NART † -0.073 0.025 -0.124 -0.025 3105.8 51.5 61.3/58.7Education † -0.167 0.067 -0.303 -0.039 3106.7 52.5 61.6/58.4Stage † -1.610 0.570 -2.780 -0.560 3106.9 51.7 64.9/55.1Fatigue † -0.052 0.025 -0.102 -0.006 3106.8 52.0 64.4/55.6Anxiety 0.045 0.054 -0.058 0.150 3103.1 49.3 63.5/56.5Depression † 0.190 0.096 0.012 0.390 3105.4 50.9 64.5/55.5EP ability -0.200 0.426 -1.050 0.620 3105.4 51.0 63.7/56.3Surgery Type -0.035 0.401 -0.820 0.760 3103.6 50.1 63.2/56.8

Immediate Age † 0.061 0.029 0.008 0.120 2012.0 50.1 67.2/52.8Retention NART † -0.075 0.027 -0.131 -0.023 2014.8 51.9 62.9/57.1

Education † -0.221 0.078 -0.390 -0.075 2019.8 55.8 61.5/58.5Stage † -1.690 0.622 -3.000 -0.580 2011.6 49 68.8/51.2Fatigue † -0.047 0.024 -0.096 -0.002 2007.5 46.4 67.5/52.5Anxiety † 0.260 0.112 0.048 0.490 2004.7 44.5 66.9/53.1Depression † 0.240 0.098 0.054 0.440 2007.3 46.1 64.3/55.7EP ability -0.370 0.441 -1.200 0.490 2010.7 48.6 66/54Surgery Type -0.190 0.424 -1.000 0.660 2007.5 45.8 67.2/52.8

Delayed Age † 0.060 0.028 0.008 0.120 2057.6 48.0 68/52Recall NART † -0.076 0.026 -0.130 -0.026 2056.7 46.8 66.2/53.8

Education † -0.170 0.069 -0.310 -0.043 2055.9 46.1 66.1/53.9Stage † -1.880 0.702 -3.390 -0.670 2061.5 48.6 71.4/48.6Fatigue -0.047 0.026 -0.099 0.003 2057.0 46.5 70.4/49.6Anxiety 0.061 0.573 -0.049 0.180 2055.6 45.7 68.6/51.4Depression † 0.283 0.117 0.069 0.520 2054.4 44.5 72.4/47.6EP ability -0.530 0.434 -1.398 0.310 2058.4 49.1 67.5/52.5Surgery Type -0.012 0.422 -0.834 0.820 2060.1 49.8 68.5/51.5† indicates 95% credible interval not covering zero

5.3. Results 157

Table 5.5 Posterior Estimates For The Latent Class Growth Full Unconditional Model.

Class 1 Low Class 2 HighPosterior Posterior 95% Posterior Posterior 95%

Mean SD Cred Interval Mean SD Cred IntervalLearning Intercept 47.970 0.922 46.190 49.790 56.520 0.860 54.830 58.220

L1 -4.300 1.049 -6.390 -2.250 -2.430 0.973 -4.340 -0.540L2 0.850 0.367 0.130 1.570 1.340 0.353 0.650 2.030Resid sd 6.150 0.295 5.600 6.760 5.490 0.293 4.940 6.090Number 63.310 3.669 56 71 56.690 3.669 49 64Prob 0.530 0.054 0.420 0.630 0.470 0.054 0.370 0.580DIC 3101.4pD 48.0

Immediate Intercept 9.780 0.306 9.200 10.370 12.720 0.309 12.000 13.350Retention L1 -1.450 0.340 -2.100 -0.800 -0.920 0.321 -1.500 -0.290

L2 0.240 0.120 0.001 0.470 0.260 0.113 0.043 0.490Resid sd 2.050 0.099 1.900 2.250 1.660 0.111 1.400 1.880Number 67.370 5.270 57 77 52.630 5.270 43 63Prob 0.560 0.062 0.440 0.680 0.440 0.062 0.320 0.560DIC 2007.6pD 45.9

Delayed Intercept 9.780 0.301 9.200 10.360 12.730 0.303 12.000 13.330Recall L1 -1.460 0.339 -2.100 -0.790 -0.920 0.318 -1.500 -0.290

L2 0.240 0.119 0.005 0.470 0.260 0.112 0.038 0.480Resid sd 2.050 0.099 1.900 2.250 1.660 0.111 1.400 1.880Number 67.640 5.242 57 77 52.370 5.242 43 63Prob 0.560 0.062 0.440 0.680 0.440 0.062 0.320 0.560DIC 2008.2 pd 46.5pD 46.5


CLASS GROWTH MODELS

Table 5.6 Posterior Estimates for the LCGM full trajectories with the predictor Age-Education-Stage model

Class 1 Low Class 2 HighPost. Post. 95% Post. Post. 95%

Mean SD Cred Interval Mean SD Cred IntervalLearning Intercept 48.160 0.923 46.000 49.980 56.620 0.875 55.000 58.310

L1 -4.322 1.036 -6.400 -2.318 -2.370 0.993 -4.300 -0.398L2 0.863 0.364 0.150 1.582 1.342 0.354 0.640 2.042Resid sd 6.195 0.296 5.600 6.795 5.470 0.301 4.900 6.082Numbers 65.288 4.049 57 73 54.712 4.049 47 63Logistic intercept 1.797 0.661 0.650 3.223logistic- age 0.075 0.035 0.010 0.144logistic -educ -0.167 0.079 -0.320 -0.019logistic -stage -2.082 0.676 -3.500 -0.875DIC 3121.3 pD 63.3

Immediate Intercept 9.790 0.307 9.200 10.380 12.616 0.293 12.000 13.200Retention L1 -1.469 0.340 -2.100 -0.793 -0.923 0.321 -1.500 -0.283

L2 0.213 0.120 -0.019 0.448 0.287 0.111 0.070 0.507Resid sd 2.037 0.099 1.800 2.239 1.686 0.110 1.500 1.899Numbers 65.537 5.348 55 76 54.463 5.348 44 65Logistic intercept 1.757 0.646 0.620 3.177logistic- age 0.055 0.033 -0.010 0.121logistic -educ -0.208 0.083 -0.380 -0.052logistic -stage -2.014 0.654 -3.400 -0.828DIC 2016.2 pD 53.3

Delayed Intercept 9.799 0.312 9.200 10.420 12.602 0.326 12.000 13.250Recall L1 -1.859 0.362 -2.600 -1.147 -1.405 0.328 -2.000 -0.765

L2 0.263 0.127 0.011 0.509 0.415 0.114 0.190 0.640Resid sd 2.210 0.104 2.000 2.421 1.696 0.116 1.500 1.926Numbers 68.179 5.015 58 77 51.821 5.015 43 62Logistic intercept 2.038 0.744 0.800 3.630logistic- age 0.058 0.033 -0.006 0.123logistic -educ -0.198 0.082 -0.360 -0.044logistic -stage -2.213 0.746 -3.800 -0.897DIC 2064.9 pD 51.4

5.3. Results 159

All three covariates, age, education and stage were substantive predictors for learning but the

impact of age as a substantive predictor was reduced in the core set (AES) for immediate

retention and delayed recall. However age was retained in the core set of predictors due to

recommendations by Vardy and colleagues [76]. The shift in class membership with the

addition of covariates resulted in the posterior median changing by a maximum of 2; the low

class for learning increased from 63 to 65, the immediate retention decreased from 66 to 68

and delayed recall had no change in class numbers of 68. The class composition altered by

three participants changing their class allocations: three from high to low for learning; two

from low to high and one from high to low for both for immediate retention and delayed recall.

Figure 5.1 presents graphically the trajectory profiles for the low and high classes from the

core AES models as shown in Table 5.6. The results of the covariates fatigue, anxiety,

3540

4550

5560

65

Learning


Lown=65 (54%)

Highn=55 (46%)


Mea

n W

ord

Cou

nt

68

1012

14

Immediate Retention

Lown=66 (55%)

Highn=54 (45%)


68

1012

14

Delayed Recall


Lown=68 (57%)

Highn=52 (43%)


Figure 5.1 Two Class Trajectory Plots for Learning, Immediate Retention and Delayed Recallfrom Latent Class Growth Models with core (AES) Predictors.

depression, estrogen producing and surgery type when added to the AES model are presented

in Table 5.7. None of these additional covariates substantively contributed to class prediction

for learning, but each of anxiety and depression were substantive additional predictors for

immediate retention, and depression for delayed recall. There was a minimal shift in class

numbers for the addition of anxiety, with the lower class changing from 66 to 65, however, the

addition of depression reduced numbers in the low class from 66 to 63, a reduction of 5 from

the unconditional model. For delayed recall, the addition of depression to the AES model

increased the low class numbers by 4 (68 to 72). Changes in composition relative to the


CLASS GROWTH MODELS

unconditional model for the AES models and the substantive four predictor models are

presented in Table 5.8.

Table 5.7 Posterior Estimates for Logistic regression parameters for Predictors for the coremodel (Age, Education and Stage) plus additional single covariates for the probabil-ity of class membership.

Outcome Predictor Post. Post. 95% Post. Post.Mean SD Cred Interval DIC pD Deviance Class no.

Learning + fatigue -0.050 0.029 -0.110 0.006 3120.1 61.3 3058.8 65/55+ anxiety 0.072 0.065 -0.053 0.202 3121.2 62.4 3058.8 65.5/54.5+ depression 0.165 0.107 -0.330 0.386 3124.3 65.3 3059.0 65.2/54.8+ EP ability 0.943 0.634 -0.280 2.223 3117.3 60.6 3056.7 64.5/55.5+ Surgery Type 0.250 0.496 -0.720 1.233 3123.4 64.3 3059.1 65.1/54.6

Immediate + fatigue -0.047 0.028 -0.100 0.007 2018.0 55.4 1962.5 65.4/54.6Retention + anxiety † 0.134 0.066 0.009 0.268 2012.0 50.8 1961.2 64.7/55.3

+ depression † 0.242 0.113 0.032 0.474 2016.0 53.3 1962.7 62.9/57.1+ EP ability 0.599 0.730 -0.085 2.028 2016.6 53.9 1962.7 67.4/52.6+ Surgery Type -0.130 0.503 -1.100 0.848 2017.0 54.0 1963.0 64.9/55.1

Delayed + fatigue -0.049 0.030 -0.110 0.009 2066.2 51.9 2014.2 69.7/50.2Recall + anxiety 0.109 0.067 -0.180 0.244 2064.1 50.1 2014.0 68.1/51.9

+ depression † 0.293 0.132 0.043 0.555 2063.2 50.0 2013.2 72.0/48.0+ EP ability 0.255 0.651 -1.000 0.547 2064.4 50.9 2013.5 68.9/51.1+ Surgery Type 0.024 0.506 -0.970 1.013 2066.9 53.2 2013.6 68.7/51.3†indicates 95% credible interval not covering zero

Table 5.8 Changes in class membership composition with the addition of covariates.

Outcome Base Covariates Low to High High to LowNumbers Numbers

Learning Unconditional Age-Education-Stage 0 3Immediate Unconditional Age-Education-Stage 2 1Retention Unconditional Age-Education-Stage-Anxiety 2 0

Unconditional Age-Education-Stage-Depression 5 0Delayed Unconditional Age-Education-Stage 2 1Recall Unconditional Age-Education-Stage-Depression 8 1

Figure 5.2 presents surface plots for the probability of being in the Low class based on the

AES model for learning and the AES-depression model for immediate retention and delayed

recall, with age set to 50. Similar profiles can be generated for other ages and configurations

of covariates.

The sensitivity analyses for the core AES models revealed a slight influence of priors on

posterior parameter estimates, smaller posterior standard deviations and a change in class

membership of one participant. The coverage of zero by the 95% credible interval did not

differ between the two prior distributions, resulting in no change in the substantive

5.3. Results 161

Age

30

40

50

60

70

Years of E

duc

10

15

20

Probability: Low

er Class

0.2

0.4

0.6

0.8

Learning Stage= I

Age

30

40

50

60

70

Years of E

duc

10

15

20

Probability: Low

er Class

0.2

0.4

0.6

0.8

Learning Stage= II/III

Years of Educ

10

15

20

Depression

0

2

4

6

8

10

Probability: Low

er Class

0.4

0.6

0.8

Immediate Retention Stage= I Age=50

Years of Educ

10

15

20

Depression

0

2

4

6

8

10

Probability: Low

er Class

0.2

0.4

0.6

0.8

Immediate Retention Stage= II/III Age=50

Years of Educ

10

15

20

Depression

0

2

4

6

8

10

Probability: Low

er Class

0.6

0.7

0.8

0.9

Delayed Recall Stage= I Age=50

Years of Educ

10

15

20

Depression

0

2

4

6

8

10

Probability: Low

er Class

0.2

0.4

0.6

0.8

Delayed Recall Stage= II/III Age=50

Figure 5.2 Probability surface plots of Low class membership for Learning, Immediate Reten-tion and Delayed Recall.


CLASS GROWTH MODELS

contribution of each logistic parameter.

Time-varying covariates of anxiety, depression, and estrogen producing ability were on the

whole not substantive predictors of the verbal memory trajectory processes. Individual

inclusion of the class specific time-varying covariates revealed only a small number of

substantive associations: depression for the High class of learning at time 4, and estrogen

producing ability with the Low class for learning and delayed recall at time 4. These results

are presented in Table 5.9.

Table 5.9 Posterior Estimates for substantive time-varying trajectory covariates added to theAES class membership predictor models.

Class 1 Low Class 2 HighPost. Post. 95% Post. Post. 95%

Mean SD Cred Interval Mean SD Cred IntervalLearningDepression T1 -0.080 0.301 -0.670 0.513 0.277 0.402 -0.500 1.078

T2 0.337 0.268 -0.190 0.860 -0.121 0.315 -0.730 0.502T3 -0.154 0.253 -0.660 0.346 0.171 0.333 -0.490 0.816T4 -0.042 0.358 -0.740 0.667 -0.814 0.359 -1.500 -0.100numbers 65.019 4.286 57 73 54.981 4.286 47 63DIC 3132.5 pD 73.5

Estrogen T1 2.583 1.742 -0.820 6.048 1.546 1.681 -1.800 4.817Producing T2 2.622 1.505 -0.320 5.567 1.063 1.421 -1.800 3.844

T3 1.448 1.465 -1.400 4.286 1.953 1.307 -0.650 4.495T4 4.466 1.840 0.830 8.060 1.945 1.582 -1.200 5.061numbers 62.181 3.478 55 69 57.819 3.478 51 65DIC 3126.4 pD 79.1

Delayed recallEstrogen T1 0.012 0.582 -1.100 1.148 1.175 0.603 -0.039 2.327Producing T2 0.390 0.495 -0.580 1.371 -0.502 0.483 -1.500 0.439

T3 -0.194 0.476 -1.100 0.736 -0.530 0.423 -1.400 0.297T4 1.245 0.628 0.001 2.464 0.164 0.508 -0.840 1.160numbers 71.864 3.334 65 78 48.136 3.334 42 55DIC 2070.5 pD 63.3

5.4 Discussion

The three aims of this paper were to: 1) identify sub-classes of women who demonstrated

different patterns of response using Bayesian latent class growth mixture models, 2) identify

covariates or predictors of these classes, and 3) investigate the impact of quality of life

time-varying trajectory covariates on the subgroup structure.

Two classes were consistently identified for each of the three outcomes of learning, immediate

retention and delayed recall. These classes were denoted as Low and High depending on the

5.4. Discussion 163

level of the estimated intercept. For all three verbal memory outcomes, the decline was

steeper from baseline to one month post-chemotherapy for the Low class than the High class

and the Low class exhibited a less rapid recovery from one month to eighteen months

post-chemotherapy.

The second aim was addressed by using a range of predictors of the probability of class

membership to identify patient characteristics which may impact on the trajectory profile class

membership. Increased age, lower NART estimates of intellectual function, fewer years of

education, being in Stage I, being more fatigued (lower score), and having higher baseline

indicators of anxiety or depression individually increased the probability of being in the Low

class for all three verbal memory outcomes.

For learning only the core covariates of age, education and stage were substantive predictors

for class membership, with being older, less educated and having Stage I of the disease

increasing the chance of being in the Low class. For immediate retention, the substantive

predictors were the core covariates with the addition of either baseline anxiety or depression.

Higher baseline anxiety and depression scores predicted membership of the Low class.

Similarly in addition to the core predictors, baseline depression was a substantial predictor for

delayed recall. The addition of baseline fatigue to the age, education and stage model was not

a substantial predictor for any of the three outcome measures.

The interpretation of disease severity (Stage) as a predictor of the trajectory class membership

is not straightforward. The participants categorized by Stage (I and II/III) were independent of

age, NART score, years of education, assessment time from surgery, baseline fatigue, anxiety

and depression, chemotherapy regime, marital status and estrogen/progesterone receptor

status (p>0.05). Predictably, Stage I participants were more likely to undergo breast

conserving surgery 71.8% compared to 51.1% for participants with StageII/III (p=0.43) rather

than having a mastectomy. However baseline scores for learning, immediate retention and

delayed recall were all significantly lower for Stage I than Stage II/III and this would be a

major factor in the class assignment. This relationship of low verbal memory scores and Stage

I cancer severity scores may be a function of relatively low numbers (32) for Stage I group

and warrants further research.

The influence of depression and anxiety is consistent with previous research with depression


CLASS GROWTH MODELS

being found to be significantly related to cognitive function in a number of studies

[18, 77, 78, 79] but anxiety in fewer studies [79, 80].

The third aim was addressed by investigating the effect of changes in estrogen producing

ability, depression, anxiety and fatigue over time by inclusion of these predictors as

time-varying covariates, in the respective models. Estrogen producing ability at time 4 (18

months post-chemotherapy) was a substantive positive predictor of both learning and delayed

recall ability for the Low class. Depression scores at time 4 were a substantive negative

predictor of learning for the High class, but did not impact on any other class/measurement

occasion combination.

Women in the Low class who retained estrogen producing ability at 18 months

post-chemotherapy showed an increased learning and delayed recall ability than otherwise

would have been the case. In a similar way, women in the High class with increased

depression mood scores scored lower for learning at 18 months measurement occasion. All

other time-varying covariates, namely anxiety and fatigue, had no substantive additional

impact on the verbal memory outcomes at any of the four measurement occasions or class

combinations.

A limitation of this study has been the relatively small sample size. This impacted on the

number of latent classes that could be considered and the power to identify substantive effects

of some covariates. Despite this, the results of this study contribute strongly to our

understanding of this complex problem. In summary, the Bayesian two-piece linear latent

class growth models identified distinct subgroups of women that differed in the overall

response level, rate of decline from baseline to after chemotherapy and the rate of recovery

post-chemotherapy. The substantive contribution of subsets of predictors, which include age,

education, stage of cancer, baseline indicators of anxiety and depression, to the probability of

class membership and the degree to which the numbers in class membership changes are

indicative of the importance of including predictors in these models. In contrast the inclusion

of time-varying covariates in the verbal memory context had minimal impact. These findings

may assist in the identification of breast cancer patients who may be at risk of higher levels of

verbal memory impairment with chemotherapy treatment.

5.5. Acknowledgements 165

5.5 Acknowledgements




Council Linkage Project. The authors would like to thank Drs Toni Jones, Donna Spooner,

and Miss Elena Moody for their input in the design and implementation of the study. We

would also like to thank all the oncologists, surgeons, and research nurses who helped in the

recruitment process, and the research assistants involved in recruitment and data collection.

Finally, the authors would also like to sincerely thank all the women who participated in the

study at such a distressing period in their life.

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Chapter 6

Bayesian Estimation Of Extent Of

Recovery


presented in its entirety. This article has been submitted to the Journal of the Royal Statistical

Society C: Applied Statistics in December 2009.

Bayesian Estimation Of Extent Of Recovery For Aspects Of Verbal Memory In Women

Undergoing Adjuvant Chemotherapy Treatment For Breast Cancer

Authors: Margaret Rolfea, Kerrie Mengersena, Katharine Vearncombecd,

Brooke Andrewcd,Geoffrey Beadleb,


175

176 CHAPTER 6. BAYESIAN ESTIMATION OF EXTENT OF RECOVERY

Bayesian latent basis random effects growth models and latent basis growth mixture models

were employed in this paper to estimate the degree of recovery of aspects of verbal memory

(learning, immediate recognition and delayed recall) measurement over four occasions,

namely prior to chemotherapy treatment but post surgery (initial level), one month, six and

eighteen months following completion of chemotherapy. The latent basis models fixed the

first two time loadings to 1 and 0 respectively and estimated the third and fourth from the data,

thus measuring the degree of recovery at 12 and 18 months post-chemotherapy completion.

These models were extended to latent class growth mixture models for two and three classes,

whereby a mixture of a small number of trajectory profiles would account for the

heterogeneity among subjects’ responses. The overall degree of recovery at 18 months

post-chemotherapy in the single class models had virtually returned to initial levels for

learning and to 59% and 56% of the initial level for immediate retention and delayed recall

respectively. Two and three class models are reported for the subgroup analysis with the Low

class of the three class scenario indicating 29.0%, 11.0% and 9.6% recovery for learning,

immediate retention and delayed recall respectively. The 95% credible intervals for the initial

level (β0 + β1) can assist in the identification of breast cancer patients at risk of long term

memory impairment, who can be targeted for intervention.








177


Estimation Of Extent Of Recovery For Aspects Of Verbal Memory In

Women Undergoing Adjuvant Chemotherapy Treatment For Breast

Cancer

Abstract

Decline in cognitive function can be experienced by up to 50% of women while undergoing

adjunct chemotherapy and a subset of patients may experience effects for several years.

Bayesian latent basis longitudinal random effects and latent class growth models were used to

assess the degree of verbal memory recovery for learning, immediate retention and delayed

recall in women undergoing chemotherapy for early stage breast cancer and who were

assessed before, one, six and eighteen months post-chemotherapy. The latent basis model,

with the initial time point fixed to 1 and the second set to zero, enabled the estimation of

degree of recovery at times 3 and 4. In the single class model, learning scores at eighteen

months post-chemotherapy had returned to initial levels with a 60% to 57% recovery for

immediate retention and delayed recall. Two and three class mixture models were fitted, with

classes differing for baseline and degree of recovery. In the three class model, the Low class

indicated 29.0%, 11.0% and 9.6% recovery for learning, immediate retention and delayed

recall respectively. The 95% credible intervals for the initial verbal memory scores may be a

useful indicator of breast cancer patients at risk of long term memory dysfunction.

Keywords: Bayesian; longitudinal; latent basis; latent class growth; breast cancer; verbal

memory.


6.1 Introduction

A frequently reported side effect for women undergoing chemotherapy treatment for breast

cancer is that of decline in cognitive function. Studies have shown that between 20% and 50%

of women suffer short term dysfunction [3, 15, 16, 17], and although most women recover

over time [17, 41], a small subset experience longer term dysfunction [1, 35]. Although

several cognitive domains (attention, concentration, verbal and visual memory, processing

speed) have reported to be affected [1, 3, 35, 36, 45, 46], the domain of verbal memory has

been consistently identified in several studies [1, 30, 34, 36, 42, 46, 47] as being compromised

after chemotherapy treatment.

In order to assess the degree of cognitive recovery after chemotherapy relative to initial

scores, this paper uses data obtained from the Cognition in Breast Cancer Study involving

early stage breast cancer patients drawn from hospitals throughout south-east Queensland,

Australia. The study followed a longitudinal prospective design with assessments conducted

before chemotherapy, and at one, six and eighteen months after completion of chemotherapy.

A battery of neuropsychological tests measuring a range of cognitive domains was

administered at all four measurement occasions, together with measures of anxiety,

depression and self reported quality of life scales.

This paper uses Bayesian latent basis growth models to assess non-linear response over time

and the degree of recovery at assessment times 3 and 4 for the three main outcomes of the

verbal memory cognitive domain (learning, immediate retention and delayed recall) measured

by the Auditory Verbal Learning Test [10]. The latent basis growth models can be viewed as

multilevel or structural equation models, where measurements over time are nested in the

same subject [2, 6, 20, 32, 37, 38]. With the inclusion of full random effects these models

enable each subject to have a potentially unique trajectory, where the trajectory parameters

(intercept and slope) can include random effects. In the Bayesian context latent basis growth

models are forms of Bayesian hierarchical regression [6, 13, 14].

The parameterisation of nonlinear growth trajectories by a latent basis model, where rather

than fixing the basis coefficients for the slope to some predetermined values as is modelled for

a linear response (0,1,2,3...), has the optimal shape estimated from the data [19, 20, 22], in a


similar manner to estimating factor loadings in the measurement part of a structural equation

model. The latent basis growth model ensures flexibility in modelling different trajectory

shapes by the way the basis coefficients are set, either being fixed or partially estimated. For

example a linear growth model over four measurement occasions can be specified with fixed

basis coefficients of (0, 1, 2, 3) or (0, 0.33, 0.666, 1) [20, 49] where the latter case shifts the

units of the slope to a proportion of the time range while retaining the linear trajectory. A

monotonic increasing non-linear growth model can be modelled with the first and last basis

coefficients being fixed to zero and one, but with the intermediate latent basis coefficients

being estimated from the data [20, 49], so as to obtain an optimal shape for the growth

trajectory. An alternative model would have the first two basis coefficient fixed to (0,1) and

have subsequent coefficients estimated [20, 22]. Nonlinear decline, and fluctuating change

trajectories are a few of the possible other options handled by latent basis coefficient models

[49].

Figure 6.1 presents some different trajectory profiles which latent basis models can represent,

with linear decline α=(1.00, 0.66, 0.33, 0.00), nonlinear decline with α=(1.00, 0.33, 0.17,

0.00), decline followed by a flat response α=(1.00, 0.00, 0.03, 0.03), and decline with

recovery α=(1.00, 0.00, 0.17, 0.5) for four measurement occasions.

1 2 3 4

45

67

89

10

Time

Out

com

e

1 2 3 4

45

67

89

10

Time

Out

com

e

1 2 3 4

45

67

89

10

Time

Out

com

e

1 2 3 4

45

67

89

10

Time

Out

com

e

Figure 6.1 Plots of some different trajectory profiles for latent basis models


Although latent basis models have been proposed for some time, there has been limited

implementation of these models. Recent research has utilised latent basis models for

nonlinear monotonically increasing responses, generally fitted in a non-Bayesian framework.

Applications have included modelling of cortisol responses over 8 measurement occasions

[31], the assessment of the five individual learning trials which when summed produce an

overall verbal memory learning score for the Rey Auditory Verbal Learning Test [48], and the

developmental trajectories of body mass index (BMI) measurements for girls from childhood

to adolescence over six measurement occasions [44].

Latent basis growth models can be discussed in terms of random coefficient, multilevel or

mixed effects models, with observed scores over time being one level and the subject-specific

intercepts and slopes another level [20, 32, 38]. In the Bayesian context, the random effects

for the intercept or level, slope and intercept/slope interaction can be estimated using a range

of different prior distributions and include the inverse-Wishart or Wishart, Uniform and

half-Cauchy distributions [12, 13, 14]. The inverse-Wishart distribution was initially

considered appropriate to estimate the variance/covariance parameters of a multivariate

normal distribution, but can be problematic with variances close to zero, whereas the

Uniform, or half-Cauchy or scaled-Wishart [12, 14] reduces this problem. This paper uses

Wishart and Uniform distributions to estimate the random effects for latent basis models.

Bayesian latent basis growth models were introduced by Zhang et al. [49] for the analysis of

readings scores of children at four measurement occasions over a six year period.

Growth mixture models have been developed to identify a finite or small number of distinct

latent classes or subgroups of trajectory profiles which account for unobserved heterogeneity

for the full sample. The combining of latent class models and random effects growth models

were initially developed by Muthen and Shedden [25], and expanded by Muthen [23, 24],

while in parallel Nagin and collegues developed latent class growth models where the latent

classes were assumed to account for all heterogeneity as opposed to the inclusion of random

effects [26, 27, 28, 29]. Although applications of growth mixture models are widely published

very few consider the latent basis formulation with estimated basis coefficients nor use

Bayesian methodology. Bayesian growth mixture models presented by Elliott et al. [7] used

linear trajectories for two variables and Leiby et al. [18] used multivariate factor analytic


models with quadratic trajectories to determine the underlying latent classes. Bayesian

estimation growth mixture or latent class growth models submitted for publication by the

authors of this paper utilised piecewise linear or linear spline functions to model the trajectory

response. Latent basis functions and growth mixture models have been combined to identify

four curvilinear trajectory classes or patterns of weight status change (BMI) across ages 5 to

15 years for a sample of 182 girls [44]. The latent basis coefficients were fixed for the first and

last of the six measurement occasions to zero and 10 respectively and the intermediate times

or ages estimated from the data.

The latent basis configuration used in the current paper fixes the first and second basis

coefficients to 1 for commencement and zero for one month post-chemotherapy with the latent

basis coefficients for six and eighteen months post-chemotherapy estimated from the data.

These estimates will be a measure of the degree of recovery relative to levels at initial, that is,

prior to chemotherapy treatment.

The aim of this paper is two-fold. The first aim is to identify the degree of recovery using

random effects latent basis growth models for all the study participants, thereby obtaining an

average degree of recovery at 6 and 18 months post-chemotherapy, based on posterior

estimates of the latent basis coefficients under a Bayesian model. The second aim is to

estimate the extent of recovery for identified sub-classes of women who demonstrated

different patterns of response, with latent basis latent class growth mixture models, again

using a Bayesian framework.




6.2 Methods

6.2.1 Study design and participants

This study is part of the Cognition in Breast Cancer (CBC) study undertaken with participants

who were recruited from community hospitals in south-east Queensland with histologically

proven breast cancer treated initially by definitive surgery. Although the larger study

6.2. Methods 183

embraced participants who had undergone a range of treatments, including adjuvant

chemotherapy, radiation or endocrine treatment, only participants who experienced

chemotherapy treatment (n=155) and who completed at all four measurement occasions

(n=120) are considered in this paper.

Eligible participants were required to be between 18 and 70 years old; proficient in English;

and have no previous history of cytotoxic drug treatment, neurological or psychiatric

symptoms, or current use of medications that might affect neuropsychological test

performance. All participants provided written, informed consent, and this study was

approved by the ethics committees of participating hospitals, the Queensland Institute of

Medical Research and the University of Queensland. Demographic data collected included

age, years of education, overall intellectual function, menopausal status, marital status, family

cancer history, use of hormone replacement therapy, and current and previous medications.

Medical information on severity, chemotherapy treatment, number of treatment courses and

the use of other treatments, together with self report measures of quality of life and mood

were recorded.

Participants undertook an individually administered, comprehensive battery of

neuropsychological tests which assessed the cognitive functioning domains of attention,

visual and verbal memory, speed of information processing and executive function. These

assessments were conducted after surgery but prior to the commencement of chemotherapy

(T1 or initial), one month (T2), six (T3) and eighteen months (T4) post-chemotherapy.

The neuropsychological measure considered in this paper is the Auditory Verbal Learning

Test (AVLT) as prescribed in Geffen [10] and utilised in other papers by the same authors

[9, 11]. The primary response variables considered from this test were the learning score

which was derived from the sum of the words recalled in Trials 1-5 (Learning AVLT Trials

1-5), the immediate retention score after a distracter list (AVLT Trial 7), and a delayed recall

score comprising the total number of words recalled after a 30 minute delay (AVLT Trial 8).

Higher scores on these measures imply more words learnt or recalled and hence are indicative

of better verbal memory ability. Age, gender, IQ and education level have been shown to

influence performance in this measure [9].


6.2.2 Bayesian random effects latent basis growth models

The latent basis growth model was written as a random effects model which is equivalent to a

hierarchical model in which the variability at each level is specified separately. For

longitudinal data, the observations over time on individual subjects are considered as level 1

measurements and subjects or participants as level 2 measurements.


=1,2,3,4). Then

yti ∼ Normal(λti, σ2ε ) (6.1)

where λti = η0i + η1iαtη0i

η1i

∼ Normal

β0

β1

,σ2

0 σ01

σ01 σ21

(6.2)

where η0i represents the expected intercept or level at time 2; η1i represents the linear slope;

and the latent basis coefficients α1 and α2 are fixed to 1 and 0 respectively, and α3 and α4 are

estimated degrees of recovery at T3 and T4, respectively. Thus the degree of recovery at these

two time points is expressed as a percentage of the change between 1st and 2nd time points.

The random effects η0i and η1i are considered to be random effects with means β0 and β1 and

variances σ20 and σ2

1, respectively, and covariance σ01 which can also be expressed as ρσ0σ1.

Non-informative prior distributions were specified as Normal(0,1000) for β0, β1 and

Uniform[0.01,100] for σε . After preliminary analysis of the data, mildly informative priors

for the latent basis coefficients α3 and α4 were specified as Normal(0,4) in order to be

non-influential (variance an order of magnitude larger than the posterior variance) but

sufficiently well defined for good parameter estimation.

A number of priors were considered for the random effects variance-covariance matrix:

1. σ2

0 σ01

σ01 σ21

∼ Wishart

1 0

0 1

, 2 and where ρ =

σ01

σ0σ1

2. with a Wishart with degrees of freedom set to 3 instead of 2

6.2. Methods 185

3. Uniform distributions for σ0, σ1 and ρ with Uniform[0.001,10], Uniform[0.0001,3],

Uniform [-1,1] respectively.

The use of 2 degrees of freedom is used to replicate the models of Zhang and colleagues [49].

However recommendations of using degrees of freedom one more than the number of random

effects estimated by the model also motivated consideration of a Wishart prior distribution

with 3 degrees of freedom [14]. This has the effect of setting a uniform distribution on the

correlation parameter.

The latent basis growth models with full random effects as per Equation 2 were fitted for the

three verbal memory outcomes with WinBUGS 1.4 [40] with the R2WinBUGS package [43]

in R. Three Markov chains were used with 20,000 iterations and with the first 2,000 iterations

discarded. Convergence assessment was based on the Gelman-Rubin R̂ statistic, as part of the

output from R2WinBUGS, so that R̂ < 1.1 for all parameters indicated adequate mixing

[13, 14] and Monte Carlo errors less than 5% of the standard deviation [39].

Model fit was assessed using the Deviance Information Criterion (DIC) [13, 39], which is

computed as the posterior mean deviance with an added penalty component for model

complexity. The measure of model complexity (pD) is estimated in R2WinBUGS as half the

average within-chain variance of the deviances [43].


These random effects growth models assume that the trajectories from n subjects are driven by

an underlying subject-level latent growth process. The mean structure of the process depends

on the subject belonging to one of K latent classes (K � n). A random effects model could

then be fit within each class by applying Equation 6.2 within each class, but this can induce a

large degree of instability into the model. An alternative is to consider a fixed effects model

within each class, in which case the components of the variance-covariance matrix in

Equation 6.2 are set to zero and the model reduces to the latent class growth model which is

written in a similar manner to Equation 6.1 with superscript k indicating class or group k as in


Equation 6.3.

yti ∼ Normal(λkti, (σ

kε)

2) (6.3)

where λkti = βk

0 + βk1α

kt

Only two and three class models were considered due to the possibility of small numbers for

group membership given the relatively small total sample size (n=120). The regression

parameters βk0 and βk

1 used non-informative prior normal distributions, namely N(0, 1000) with

βk0 was ordered as β1

0 < β20 < β

30, so β2

0 = β10 + θ1, and β3

0 = β10 + θ1 + θ2 with θ1, θ2 restricted to

positive values from N(0, 1000) for k = 2 or k = 3 . Prior distributions for the other parameters

remained the same as for the single class model, with Uniform[0.01,100] for (σkε)

2,

Normal(0,4) for the estimated latent basis coefficients α3 and α4. The Dirichlet distribution, as

the conjugate prior for the multinomial distribution, was used as the non-informative prior for

the probability of class membership πk, k = 1, . . . ,K where the K Dirichlet parameters were

set to 1 [13]. Model selection was undertaken with the DIC produced by the R2WinBUGS

package. Although there has been much debate on the appropriateness of the DIC with

mixture models, with particular concern about the pD estimate of effective numbers of

parameters estimated [4, 8, 21], there does not appear to be any clear resolution on this matter.

Average posterior probabilities of class membership were also considered [7, 24, 27].

Three Markov chains were used with 40,000 iterations and with the first 10,000 iterations

discarded and retaining every fifth simulation. Again the convergence was asserted if R̂ < 1.1.

The posterior probability of class membership is a measure of an individual’s likelihood of

belonging to each of the k trajectory groups or classes. The posterior probabilities of group

membership can determine the ability of the model to clearly differentiate between subjects.

An average posterior probability of group membership equal to 1 demonstrates the optimal or

ideal situation, with [27] specifying a rule of thumb of at least 0.7 for all groups as a

acceptable measure.

In the Bayesian analyses performed, each individual has a mean posterior probability of

membership for each of the k classes, which is the proportion of simulations whereby the

individual has been allocated to the kth class. The posterior median for the class membership

6.3. Results 187

for individual participants was used to assign participants to the most likely of the k classes.

An average of the mean posterior probabilities for individuals or participants allocated to each

of the k classes can be obtained and will indicate the ability of the model to differentiate

between classes.

Estimates of agreement were assessed for the resultant class membership over the three verbal

memory measures for the two and three class assignments separately, in order to assess the

consistency of participant class membership.

6.3 Results

The 120 study participants were aged between 25 to 68 years (mean 49.3, sd 7.8), with a mean

of 13.1 (sd 3.4, range 6 to 20) years of education and a initial NART predicted intellectual

functioning ranging from 90 to 126 (mean 110.6, sd 8.6). At the study commencement, the

majority of participants were married or living with a partner (85.0%), and were

pre-menopausal (52.5%); 43.3% had undergone a mastectomy and 58.3% had breast

conserving surgery. As being in early stages of breast cancer was a requirement of the study

very few participants were in Stage III of the disease (6.7%) with the majority in Stage II

(66.7%) and fewer in Stage I (26.7%).

The sample means over the four measurement occasions for the three outcome measures of

interest, namely learning, immediate retention and delayed recall, appeared to follow a similar

pattern with highest scores before chemotherapy, lowest at one month after chemotherapy, and

increasing improvement over the third and four measurement occasions. As described in

Section 6.2.1 higher scores are indicative of better verbal memory. Table 6.1 presents the

sample means, standard deviations and score minima and maxima for the outcome variables

for the four occasions, and Figure 6.2 graphically depicts the mean verbal memory trajectory

patterns.

6.3.1 Bayesian single class random effects latent basis growth models

The posterior means for the recovery parameters α3 and α4 differed minimally between the

three prior distributions used, Wishart (2 df), Wishart (3 df) and Uniform distribution, for the

variance/covariance of the random effect parameters. The parameters β0, β1, σε , σ0 and σ1


Table 6.1 Summary Statistics for Learning, Immediate Retention and Delayed Recall

Variable Occasion Mean SD Min MaxLearning T1 53.11 6.97 33 68

T2 49.82 7.96 25 66T3 50.54 8.46 32 70T4 52.98 8.52 33 69



3040

5060

70

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Before 1 mth 6 mths 18 mths Chemo PostChemotherapy

46

810

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Delayed Recall


Figure 6.2 Sample mean scores for Learning, Immediate Retention and Delayed Recall for fourtimes

6.3. Results 189

Table 6.2 Posterior parameter estimates for Bayesian latent basis using Wishart df=2, Wishartdf=3 and Uniform priors

Wishart df=2 Wishart df=3 Uniform

Posterior 95% Posterior 95% Posterior 95%Mean SD Cred Int Mean SD Cred Int Mean SD Cred Int

Learning α3 0.167 0.228 -0.34 0.57 0.175 0.216 -0.29 0.55 0.168 0.232 -0.36 0.57α4 0.961 0.258 0.55 1.56 0.942 0.250 0.54 1.51 1.009 0.265 0.59 1.64β0 49.954 0.759 48.48 51.45 49.919 0.758 48.42 51.39 49.965 0.768 48.46 51.47β1 3.083 0.723 1.61 4.44 3.170 0.699 1.76 4.56 3.007 0.731 1.51 4.39σε 5.082 0.194 4.72 5.48 5.085 0.192 4.73 5.48 5.075 0.194 4.71 5.47σ0 6.358 0.588 5.25 7.57 6.345 0.582 5.27 7.55 6.377 0.539 5.39 7.51σ1 1.002 0.442 0.40 2.09 0.914 0.414 0.36 1.92 0.792 0.631 0.04 2.36ρ -0.576 1.052 -3.35 0.96 -0.591 1.005 -3.29 0.65 -0.221 0.494 -0.94 0.90D 2920.9 19.3 2885.0 2961.0 2921.7 19.0 2887.0 2961.0 2919.5 19.6 2882.0 2959.0DIC 3030.4 3030.2 3029.4pD 109.4 108.3 110.5

Immediate α3 0.031 0.198 -0.40 0.38 0.021 0.203 -0.43 0.37 0.041 0.194 -0.39 0.38Retention α4 0.569 0.174 0.23 0.92 0.571 0.178 0.22 0.93 0.590 0.173 0.26 0.94

β0 10.202 0.240 9.73 10.67 10.210 0.237 9.74 10.67 10.189 0.234 9.73 10.65β1 1.099 0.221 0.67 1.54 1.083 0.217 0.67 1.51 1.111 0.203 0.72 1.52σε 1.615 0.064 1.50 1.75 1.617 0.063 1.50 1.75 1.625 0.063 1.51 1.75σ0 2.032 0.175 1.71 2.40 2.011 0.172 1.69 2.37 2.017 0.169 1.71 2.38σ1 0.647 0.182 0.36 1.07 0.601 0.169 0.34 0.99 0.378 0.274 0.01 1.01ρ -0.206 0.189 -0.67 0.06 -0.177 0.174 -0.60 0.07 -0.380 0.468 -0.97 0.86D 1820.3 21.4 1779.0 1863.0 1822.1 20.9 1782.0 1864.0 1826.5 20.6 1786.0 1867.0DIC 1937.3 1937.2 1938.5pD 116.5 114.9 111.1

Delayed α3 -0.001 0.165 -0.36 0.29 -0.005 0.166 -0.37 0.28 0.002 0.160 -0.34 0.29Recall α4 0.545 0.142 0.27 0.82 0.542 0.143 0.26 0.82 0.562 0.145 0.28 0.85

β0 9.840 0.252 9.35 10.33 9.843 0.253 9.34 10.34 9.835 0.248 9.35 10.32β1 1.488 0.243 1.01 1.96 1.487 0.244 1.01 1.96 1.481 0.233 1.06 1.94σε 1.767 0.069 1.64 1.91 1.770 0.069 1.64 1.91 1.775 0.069 1.65 1.92σ0 2.043 0.181 1.71 2.42 2.026 0.181 1.69 2.40 2.040 0.178 1.71 2.41σ1 0.668 0.198 0.36 1.13 0.630 0.188 0.34 1.07 0.469 0.293 0.02 1.10ρ -0.238 0.226 -0.81 0.06 -0.219 0.213 -0.76 0.06 -0.422 0.420 -0.96 0.70D 1907.0 21.1 1867.0 1950.0 1908.6 20.9 1869.0 1951.0 1911.0 20.89 1870.0 1952.0DIC 2019.9 2020.300 2021.0pD 112.8 111.700 109.8

had very similar posterior means, posterior standard deviations and 95% credible intervals

with the three prior distribution models. The posterior means for ρ were all negative but were

more variable in their posterior means and standard deviations or standard errors, although for

all estimates of ρ the zero point was covered by the 95% credible interval, indicative of no

correlation between the β0 and β1 (intercept and slope) estimates.

For all Bayesian random effects models convergence was achieved with R̂ < 1.1 for all

parameters indicating adequate mixing of the resultant MCMC chains. The DIC as an

estimate of adequate fit also differed minimally between the comparable models for learning,

immediate retention and delayed recall.

The results of the sensitivity analyses are presented in Table 6.2, where the main differences

were for the posterior means of σ0, σ1 and ρ which was to be expected. However minimal or

no differences were shown for the recovery parameters α3, α4, β0, β1 and DIC estimates.

Due to the similarity of the posterior estimates of the three prior distribution models only the

results for the Uniform prior model are discussed. The posterior means with 95% credible


interval with the Uniform model option for the recovery parameter α3 were 0.168 [CI

-0.36,0.57], 0.041 [CI -0.39,0.38] and 0.002 [CI -0.34,0.29] for learning, immediate retention

and delayed recall respectively. As the credible region for all the α3 parameters covers zero

there is virtually no recovery at six month post-chemotherapy, with posterior mean for

learning close to 0.17 or 17% but was no different to zero for immediate retention and delayed

recall being 4.1% and 0.2%.

At 18 months post-chemotherapy α4 was close to 1 for learning, that is, 1.01 [CI 0.59-1.64]

the Uniform prior option, so indicating a full verbal memory recovery for learning by that

time. However the degree of recovery for immediate retention and delayed recall were not as

strong with 0.59 [0.26,0.94] and 0.56 [0.28,0.85] for immediate retention and delayed recall

respectively, and note that the degree of recovery α4 was a little lower for delayed recall.

6.3.2 Latent class growth mixture models

Tables 6.3 and 6.4 present the Bayesian posterior parameter estimates for the two and three

class models respectively. All Bayesian estimated two and three class models converged with

all parameter estimates fulfilling the convergence criteria of R̂ < 1.1. Using the criterion of a

lower DIC indicating a better model fit, the three class models were preferred for learning and

immediate retention but the two class model for delayed recall. Nonetheless, in the following

where appropriate, all three class models will be considered together and similarly for two

class models in order for meaningful comparisons to be interpreted.

The classes identified in the two-class model are denoted as ‘Low’ and ‘High’, and in the

three-class model ‘Low’, ‘Mid’ and ‘High’ based on the posterior mean estimate of βk0. In the

two class model, for the three verbal memory outcomes the Low class demonstrated a

minimal or zero decline in recovery at 6 months post-chemotherapy -1.7%, -13.5%, -15.4%,

followed by a recovery to 49.4%, 39.6% and 35.9% of the initial score at 18 months

post-chemotherapy. The High class for the two class models demonstrated a higher degree of

recovery at 6 months post-chemotherapy 24.0%, 15.4%, 19.1% and 18 months

post-chemotherapy recovery 95.9%, 52.4%, 73.7%] for the three verbal memory outcomes.

For the three class models, for all verbal memory outcomes, the Low class showed minimal

change or decline in recovery at 6 months post-chemotherapy (posterior means for α3 of

6.3. Results 191

Table 6.3 Posterior parameter estimates for Bayesian latent basis two class mixture model;classes are denoted as Low and High

Low HighPosterior 95% Posterior 95%

Mean SD Cred Interval Mean SD Cred IntervalLearning α3 -0.017 0.274 -0.619 0.455 0.240 0.333 -0.502 0.834

α4 0.494 0.243 -0.013 0.960 0.959 0.361 0.100 1.565β0 44.860 0.839 43.140 46.420 55.740 0.830 54.150 57.390β1 3.809 1.088 1.792 5.979 2.662 1.042 0.413 4.579β0+β1 48.670 0.801 47.110 50.260 58.400 0.702 56.990 59.780σε 6.132 0.296 5.578 6.735 5.560 0.293 5.014 6.164n 62.260 3.653 55.000 69.000 57.740 3.653 51.000 65.000Prob 0.519 0.054 0.412 0.625 0.481 0.054 0.375 0.588deviance 3057.0 10.6 3040.0 3081.0DIC 3113.0pD 56.0

Immediate α3 -0.135 0.274 -0.754 0.335 0.154 0.349 -0.613 0.775Retention α4 0.396 0.248 -0.121 0.872 0.524 0.347 -0.271 1.140

β0 8.788 0.267 8.237 9.283 12.240 0.288 11.660 12.780β1 1.209 0.340 0.576 1.901 0.783 0.353 0.105 1.478β0+β1 9.997 0.281 9.445 10.540 13.020 0.263 12.520 13.550σε 2.052 0.099 1.866 2.254 1.652 0.113 1.433 1.875n 68.350 5.290 58.000 78.000 51.650 5.290 42.000 62.000Prob 0.568 0.062 0.445 0.687 0.432 0.062 0.313 0.555deviance 1963.0 9.8 1947.0 1985.0DIC 2010.5pD 47.6

Delayed α3 -0.154 0.247 -0.704 0.268 0.191 0.240 -0.344 0.610Recall α4 0.359 0.214 -0.081 0.772 0.737 0.216 0.310 1.164

β0 8.453 0.289 7.866 8.988 11.730 0.273 11.200 12.270β1 1.538 0.374 0.841 2.294 1.369 0.355 0.687 2.068β0+β1 9.990 0.284 9.431 10.540 13.100 0.287 12.550 13.670σε 2.199 0.104 2.001 2.411 1.692 0.116 1.466 1.922n 68.530 4.973 59.000 78.000 51.470 4.973 42.000 61.000Prob 0.570 0.060 0.449 0.686 0.430 0.060 0.315 0.551deviance 2011.0 9.854 1995.0 2033.0DIC 2059.3pD 48.6


Table 6.4 Posterior parameter estimates for Bayesian latent basis three class mixture model;classes are denoted as Low, Mid and High

Low Mid HighPosterior 95% Posterior 95% Posterior 95%

Mean SD Cred Interval Mean SD Cred Interval Mean SD Cred IntervalLearning α3 0.061 0.326 -0.649 0.650 -0.139 0.333 -0.853 0.460 0.294 0.340 -0.472 0.880

α4 0.290 0.328 -0.411 0.900 0.601 0.320 -0.071 1.200 0.961 0.402 -0.145 1.590β0 40.120 1.774 36.030 43.120 48.080 0.989 46.180 50.020 56.280 0.974 54.530 58.330β1 5.273 2.137 1.342 9.710 2.774 1.005 0.904 4.830 2.733 1.181 -0.103 4.820β0 + β1 45.390 1.835 41.440 48.610 50.850 0.941 49.150 52.800 59.010 0.783 57.480 60.510σε 6.458 0.643 5.248 7.820 4.819 0.417 4.024 5.660 5.408 0.313 4.823 6.050n 21.820 5.888 11 34 47.420 6.448 34 60 50.760 4.896 41 59Prob 0.186 0.059 0.083 0.310 0.394 0.068 0.262 0.530 0.421 0.060 0.301 0.530D 2971.0 15.273 2944.0 3004.0DIC 3087.5pD 116.6

Immediate α3 -0.200 0.386 -0.977 0.537 -0.028 0.256 -0.600 0.410 0.226 0.406 -0.657 0.946Retention α4 0.110 0.386 -0.692 0.838 0.584 0.229 0.130 1.042 0.385 0.419 -0.589 1.099

β0 7.610 0.411 6.742 8.378 9.852 0.412 9.113 10.730 12.750 0.344 12.100 13.460β1 0.967 0.491 0.010 1.965 1.354 0.350 0.693 2.061 0.603 0.409 -0.179 1.395β0 + β1 8.577 0.559 7.403 9.602 11.210 0.381 10.520 12.020 13.350 0.281 12.800 13.910σε 1.941 0.167 1.627 2.286 1.741 0.117 1.508 1.967 1.468 0.140 1.181 1.731n 26.560 7.410 14 43 56.330 6.678 42 68 37.110 6.424 23 48Prob 0.224 0.071 0.105 0.378 0.466 0.070 0.324 0.598 0.310 0.067 0.178 0.439D 1870.0 14.386 1845.0 1901.0DIC 1973.5pD 103.5

Delayed α3 -0.140 0.314 -0.804 0.433 -0.141 0.281 -0.758 0.350 0.262 0.240 -0.265 0.688Recall α4 0.096 0.323 -0.595 0.675 0.512 0.252 -0.011 0.998 0.738 0.222 0.303 1.172

β0 6.883 0.624 5.460 7.968 9.413 0.378 8.684 10.160 11.970 0.295 11.410 12.560β1 1.888 0.722 0.679 3.388 1.340 0.391 0.584 2.106 1.407 0.377 0.655 2.135β0 + β1 8.771 0.694 7.461 10.010 10.750 0.368 10.050 11.490 13.380 0.267 12.860 13.900σε 2.033 0.269 1.506 2.526 1.916 0.152 1.568 2.183 1.569 0.116 1.359 1.810n 22.100 8.576 6 41 56.030 8.579 36 70 41.870 4.701 33 51Prob 0.188 0.078 0.054 0.360 0.464 0.083 0.283 0.613 0.348 0.057 0.240 0.463D 1936.0 16.306 1908.0 1971.0DIC 2069.0pD 132.9

6.1%, -20.0%, -14.0% for learning, immediate retention and delayed recall respectively),

followed by a modest recovery at 18 months post chemotherapy (posterior means for α4 of

29.0%, 11.0%, 9.6% for the three verbal memory outcomes respectively).

In a similar manner the Mid class continued to decline at 6 month post chemotherapy

(-13.9%, -2.8%, -14.1%) followed by an improved recovery rate of 60.1%, 58.4% and 51.2%.

For the High class the 6 months recovery was markedly higher with posterior means for α3 of

29.4%, 22.6% and 26.2% and with the 18 months recovery of 96.1%, 38.5% and 73.8% for

the three outcomes (learning, immediate retention and delayed recall) respectively. Figure 6.3

presents posterior means with 95% credible intervals for the three class models for the set of

verbal memory outcomes.

The numbers of participants allocated to the Low class from the three class models ranged

between 22 (18%) and 26 (22%) participants, implying that of the order of 20% of

participants experienced a slower verbal memory recovery. A distinguishing feature of the

Low group was a lower level of response at time 2 (β0), where membership of the Low class

was determined by a score of learning between 36.0 and 43.1, or with the time 2 score of 6 to

6.3. Results 193

−1.

0−

0.5

0.0

0.5

1.0

1.5

Learning

Classes

Low Mid High

Deg

ree

of r

ecov

ery

at 1

8 m

onth

sw

ith 9

5% C

redi

ble

Inve

rval

−1.

0−

0.5

0.0

0.5

1.0

1.5

Immediate Retention

Classes

Low Mid High

−1.

0−

0.5

0.0

0.5

1.0

1.5

Delayed Recall

Classes

Low Mid High

Figure 6.3 Posterior means and 95% credible intervals for the estimated degree of recovery at18 months estimated under the three class latent growth mixture model for the threeverbal memory outcomes

8 for immediate retention or delayed recall.

The average agreement for membership of the three class models was 64.2%, and 74.2% for

the two class models; this indicated that these participants were allocated to the same class

over the three verbal memory outcomes. However the agreement in the 3 class situation for

learning and delayed recall was much stronger at 73.3%. Class membership agreement for

immediate retention and delayed recall with the two class situation was 90.8%.

The numbers of participants in the Low class of the three class models were similar for

learning (22) and delayed recall (23), but differed by 8 and 9 for the Mid and High classes

respectively for the three outcomes. Moreover, the three classes showed a different pattern for

immediate retention compared with the other two verbal memory measures, with the High

class having the α4 parameter estimate lower than that of the Mid class estimate; in contrast,

where the same parameter for the High class of Learning or Delayed Recall was always larger

than the corresponding parameter for the other classes.

A mean posterior probability for class membership for each of the k classes for each

participant is the proportion of times a participant is allocated to each of the k classes.

Table 6.5 presents the mean posterior probabilities for the two and three class models for each

class, for the allocated participant class, and for the proportion of participants who had mean

posterior probabilities (denoted as AVPP) less than 0.7 and less than 0.6 (but greater than 0.5).

194 CHAPTER 6. BAYESIAN ESTIMATION OF EXTENT OF RECOVERY30

4050

6070

Learning Low Class


Mea

n W

ord

Cou

nt

3040

5060

70Learning Mid Class


3040

5060

70

Learning High Class


46

810

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Immediate Retention Low


Mea

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Immediate Retention Mid


46

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Immediate Retention High


46

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Delayed Recall Low


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Delayed Recall Mid


46

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Figure 6.4 Sample mean scores by class membership for Learning, Immediate Retention andDelayed Recall for the three class latent basis model

6.4. Discussion 195

Table 6.5 Posterior Probability estimates for Bayesian latent basis two and three class mixturemodels, for each class, assigned class and proportion of participants with averageposterior probabilities less than 0.7 and 0.6

2 Class Models n Low High for allocated class propn propnMean Mean Mean SD SE AvPP<.7 AvPP<.6

Learning Low 62 0.955 0.045 0.955 0.096 0.012 0.048 0.016High 58 0.053 0.947 0.947 0.104 0.014 0.034 0.034

Immediate Low 68 0.946 0.054 0.946 0.182 0.105 0.044 0.015Retention High 52 0.077 0.923 0.923 0.138 0.019 0.115 0.077

Delayed Low 70 0.935 0.066 0.935 0.130 0.016 0.100 0.043Recall High 50 0.063 0.938 0.938 0.121 0.017 0.060 0.040

3 Class Models n Low Mid High for allocated class propn propnMean Mean Mean Mean SD SE AvPP<.7 AvPP<.6

Learning Low 20 0.856 0.144 0.000 0.856 0.156 0.035 0.250 0.050Mid 50 0.094 0.832 0.075 0.832 0.150 0.021 0.220 0.140High 50 0.000 0.059 0.941 0.941 0.101 0.014 0.040 0.020

Immediate Low 27 0.822 0.178 0.000 0.822 0.182 0.035 0.333 0.185Retention Mid 56 0.078 0.851 0.071 0.851 0.129 0.017 0.125 0.054

High 37 0.000 0.104 0.896 0.896 0.138 0.023 0.162 0.054

Delayed Low 20 0.815 0.186 0.000 0.815 0.149 0.033 0.200 0.100Recall Mid 57 0.102 0.848 0.050 0.848 0.114 0.015 0.105 0.053

High 43 0.000 0.092 0.908 0.908 0.139 0.021 0.116 0.070

For the two and three class models the mean posterior probabilities, for the allocated class

ranged from 0.923 to 0.955 and 0.815 to 0.941 respectively, indicating a good separation

between classes.

6.4 Discussion

The two aims of the paper were firstly to identify the degree of recovery at six and eighteen

months post-chemotherapy using Bayesian (single class) random effects latent basis trajectory

models, and secondly to estimate the extent of recovery for identified subclasses of women

who demonstrated different patterns of response, using a Bayesian latent basis latent class

growth mixture model.

In addressing the first aim of this paper three alternative non-informative prior distributions

were considered for the random effects of the single class model and found to give equivalent

results. The degree of recovery at six months post-chemotherapy was at best minimal with the

greatest degree of recovery being for the verbal learning outcome of 16 to 18% and virtually

zero for immediate retention and delayed recall. The degree of recovery at eighteen months


post-chemotherapy showed considerable improvement with learning virtually returned to

initial levels but immediate retention and delayed recall still only reached 59% and 56%

recovery. These results give weight to the often heard complaint of memory being a major

concern with the“chemo-brain” phenomena, and for the longer lasting effects of this problem

with immediate retention and delayed recall. Based on the results of this study, assuming a

proportional recovery, a return to initial scores, for retrieval of verbal information, would not

occur for another twelve months or of the order of 30 months post-chemotherapy. This

information may be useful in the design of future longitudinal studies assessing the impact of

chemotherapy on cognitive change, especially on the duration of the study and allocation of

assessment periods. Indeed it has resulted in a recent extension of the current study to five

years post-chemotherapy.

In considering the set of two class models, there was reasonable consistency of response for

each of the three memory measures in the Low class which exhibited a further minimal

decline six month post-chemotherapy and only a 39-49% recovery by eighteen months

post-chemotherapy. However the High class showed a recovery of between 15-24% at six

months and a nearly full (95%) recovery at eighteen months for Learning but still only a 52%

and 74% recovery for immediate retention and delayed recall, which was consistent with the

recovery rates of the single class model.

For the three class models, there was much more variation among the three classes and

between the three outcome verbal memory variables. Both the Low and Mid classes for the

three outcomes exhibited either no recovery or continued decline at six months

post-chemotherapy. The Low class showed minimal recovery, 29%, 11% and 9.6% at eighteen

months for Learning, Immediate Retention and Delayed Recall respectively, with the Mid

class exhibiting between 51-60% recovery. However the High class exhibited a recovery of

between 22 and 30% at six months but were more variable at eighteen months with Learning

again showing near to full recovery (96%), Immediate Retention as 38% and Delayed Recall

of 74%.

The identification of the Low class can have clinical significance in identifying patients who

may be at risk of reduced verbal memory recovery eighteen months post-chemotherapy. The

numbers of women in the Low classes are between 18.3 and 21.7% which are well within the

6.4. Discussion 197

range of estimates with previously published breast cancer literature [1, 5, 35, 46]. Since the

95% credible intervals for the initial score β0 + β1 were found to cover non-overlapping

regions of the parameter space for all three outcomes, a possible clinical indicator of class

membership is the initial verbal memory score. Under the three class model, the posterior

95% credible intervals for initial scores in the Low class were (41.4, 48.6) for learning, (7.4 to

9.6) for immediate retention and (7.4 to 10.0) for delayed recall. This may assist in the

identification for breast cancer patients who are at risk of longer term verbal memory

dysfunction, and therefore able to be targeted for cognitive rehabilitation programs.

The sample size of 120 subjects in this study is comparable to many other studies of cognitive

change after chemotherapy for breast cancer. Despite this, the relatively small sample size

restricts the ability to identify larger numbers of subgroups. More subgroups may be

identifiable with the use of informative prior distributions in the Bayesian analysis, as was the

case in the paper by Elliott [7]. The inclusion of covariates or mediating factors like age,

educational status, depression and anxiety measures may be able to refine the identification of

women at risk of long term memory impairment, but often although significant in predicting

class membership may have minimal impact on the class composition [33].

This study utilises a new approach to estimation of the degree of recovery for verbal memory

related aspects of cognitive change post-chemotherapy over time and the identification of

classes of differing trajectory responses which can have clinical implications to the targeting

of intervention responses.

Acknowledgements



Council of Queensland, the National Breast Cancer Foundation, and an Australian Research

Council Linkage Project. The authors would also like to thank all the oncologists, surgeons,

and research nurses who helped in the recruitment process, and the research assistants

involved in recruitment and data collection. Finally, the authors would also like to sincerely

thank all the women who participated in the study at such a distressing period in their life.


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Chapter 7

Conclusion

The overall contribution of this thesis has been to the development and application of

Bayesian statistical approaches to the identification of subgroups in longitudinal data with few

time points, using primarily Bayesian methodology.

This aim was addressed in two parts, namely in developing Bayesian statistical models and

techniques for the identification of subgroups in analysis of longitudinal data where the time

intervals are limited in number and secondly in applying these methodologies to a single case

study which investigates the neuropsychological cognition for early stage breast cancer

patients undergoing adjuvant chemotherapy treatment.

In this manner the research aims of identifying, tailoring and extending state-of-the-art

Bayesian latent class identification in the context for longitudinal studies with a limited

number of temporal observations and the application of these developed statistical

methodologies to a single case study were addressed. These developed statistical methods are

presented as a series of journal articles detailed in Chapter 3 to 6.

The papers in the thesis follow the development of a longitudinal process firstly in a classical

or non-Bayesian framework by identifying latent classes of trajectory patterns over a small

number (three) of time points and then extends to Bayesian latent class trajectory models over

four time points. The role of class predictor variables is also developed logically in the

sequence of papers, firstly in Chapter 3 paper as discriminating between the latent growth

classes, secondly in Chapter 4 as covariates influencing the class trajectory patterns directly,

and thirdly in Chapter 5 as covariates or predictors which influence the probability of class

203

204 CHAPTER 7. CONCLUSION

membership. Chapter 6 considers an alternative parameterisation of the trajectory model to

assess directly the degree of recovery of the response after a notable decline returning to the

initial levels.

Various models proposed covered a wide range of constructions of models over time, with the

identification of classes, the contribution of covariates and model parameterisation. All of the

models were able to be formulated by fitting in Winbugs [5] through the R interface,

R2WinBugs [6]. This provides a powerful linkage between the statistical and graphical

capabilities of R and the MCMC capability of WinBugs. As with any numerical approach,

issues of convergence with multiple simulated chains, adequate burnin iterations, and length

of runs of chains needed to be taken into consideration in the Bayesian analyses. The

advantage of WinBugs is that many of these features were able to be evaluated in a readily

acceptable manner. Similar issues of model choice needed to be considered for each of the

models. These were assessed using the WinBugs and R2WinBugs aspects with the deviance

information criterion (DIC). Finally this modelling approach allowed for quite complex

models to be fitted and the stability of posterior estimates could be explicitly assessed. The

combination of a Bayesian model formulation and the multi-chain Monte Carlo (MCMC)

approach allowed complex models to be estimated in this framework, in contrast to other

approaches using popular software. For example, the full random effects models in Chapter 6

were not successfully fitted in MPlus [3].

As discussed in Chapter 1 and throughout the thesis, there are many aspects which need to be

considered in complex longitudinal models, while some of these have been acknowledged, a

full analysis has been outside the primary scope of this thesis. For example the treatment of

missing data is an important issue in these types of analyses. In Chapter 1 the patterns of

missing data was discussed with the conclusion that subjects with data missing at earlier times

had lower initial verbal memory scores than subjects with non-missing data. However,

subjects undergoing attrition at later time points were found to be independent of initial level

of the primary verbal memory responses. In this case study, 34 subjects or 22% had missing

responses (Table 1.1), with 11% of subjects exhibiting attrition from times 2 and 3. Further

research could focus on this missingness.

The major findings of each of the papers presented in Chapters 3 to 6 are detailed in the

7.1. Research Findings 205

following section. All papers used complete data, Chapter 1 considered 130 subjects with

measurements at times 1 to 3 and Chapters 4, 5, 6 focused on 120 participants with all four

observational times. Scores at time 1 have been referred to as both ‘baseline’ or ‘initial’

scores.

7.1 Research Findings

In Chapter 3, piecewise linear random effects models with one known change-point were

developed for longitudinal models with three timepoints in a non-Bayesian framework and the

use of piecewise linear latent class growth models set the scene for the further research

developed in Chapters 4 and 5. The restricted numbers of time points impacted on the

complexity of the random effects employed and the relatively low sample size restricted the

numbers of latent classes able to be identified. Latent piecewise linear growth models were

fitted to responses at three time points: prior to chemotherapy, and one month and three

months after chemotherapy to identify classes of women who demonstrated different patterns

of response with respect to learning, immediate retention and delayed recall aspects of verbal

memory. Two trajectory classes were identified for learning and delayed recall, and three

classes for immediate retention. For all three verbal memory measures there was a difference

between the classes with respect to the level (intercept) of the score at the change point, one

month after completion of treatment. Significant decline was demonstrated for the first linear

component (before treatment to one month after treatment) for all classes with exception for

the third (lowest) class for immediate retention. However there was no significant change or

recovery for the second linear component (one month after treatment to six months after

treatment) for any of the verbal memory outcomes.

Classes, ordered by decreasing intercept or level, showed significant differences in numbers of

words of the order of 10, 3 and 3 at one month after completion of treatment between the first

and second classes for each of the outcomes respectively, and 2 words between the second and

third classes of immediate retention. The differences between classes typically were as

follows: compared to subjects in class 2, the subjects in class 1 had a higher word count at one

month after chemotherapy , were younger by two to three years (mean age of 47 to 48 versus

49 to 51 years), had higher NART scores (mean 112 to 113 versus 108 to 110) and higher


FACT Breast scores (mean 25 versus 23) and were in earlier stages of cancer 83% to 85%

versus 64% to 71% in Stages II and III). For immediate retention, the third class (lowest

intercept) was characterised by older subjects (mean 52 years), with lower NART scores

(mean 106) and contained proportionally fewer in later stages of cancer (59%). The paper in

this chapter provides a starting point for extending the analysis of verbal memory response

trajectories to incorporate a fourth time observation at 18 months post-chemotherapy using

Bayesian methodologies in the following chapter.

In chapter 4, Bayesian single class random effects latent growth models uses linear two-piece

process with a change-point at time two (one month after chemotherapy) once again to

explain the trajectory profile of all three outcomes of verbal memory (learning, immediate

retention and delayed recall). The two-piece linear model demonstrates a decline in

measurements from before chemotherapy to one month post chemotherapy (posterior slope

estimates -3.43, -1.23, -1.66 respectively) and a recovery phase from one month to eighteen

months post-chemotherapy (posterior recovery slope estimates 1.08, 0.25, 0.33). Investigation

of covariates incorporated into the trajectory part of the model showed that increasing years of

education and increased levels of cancer severity are associated with higher levels (intercepts)

of all three verbal memory responses. The more complex models, indicated that less fatigue

increased delayed recall memory ability and an increased HADS depression score measured

at 6 month post-chemotherapy reduced in the recovery response with the learning trajectory.

Three classes were preferred for learning and immediate retention and two for delayed recall

when Bayesian latent class growth mixture models with adjustments for education were fitted.

The resultant classes differed predominantly by the level of the response, and over all

outcomes, the decline in the first linear component was less steep for classes with higher

initial scores. Learning and delayed recall exhibited a faster rate of recovery for classes with

higher initial scores, however the recovery rate remained constant over all classes for

immediate retention. Age and baseline NART scores, and differing proportions of stages of

cancer reflected differences between the two classes of delayed recall. Older ages, lower

NART scores and a relatively larger proportion of subjects with less severe cancer (Stage I)

were indicative of the class with overall lower verbal memory performance levels. However

the differing proportions in stages of cancer was the only mediating variable which varied

7.1. Research Findings 207

between the three classes for learning and immediate retention.

Chapter 5 aimed to use Bayesian latent class growth mixture models to identify sub-classes of

women who demonstrated different patterns of response, to identify covariates or predictors of

these classes by assessing their impact on the probability of class membership and to

investigate the impact of quality of life time-varying trajectory covariates on the subgroup

structure. Two classes were consistently identified for each of the three outcomes of learning,

immediate retention and delayed recall. These classes were denoted as Low and High

depending on the level of the estimated intercept. For all the three verbal memory outcomes,

the decline was steeper from baseline to one month post-chemotherapy for the Low class

compared to the High class, with the Low class exhibiting a less rapid recovery from one

month to eighteen months post-chemotherapy. Increased age, lower NART estimates of

intellectual function, fewer years of education, being in Stage I, being more fatigued (lower

score), and having higher baseline indicators of anxiety or depression individually increased

the probability of being in the Low class for all three verbal memory outcomes. However

when included in combination, only the core covariates of age, education and stage were

substantive predictors for class membership, with being older, less educated and having Stage

I of the disease increasing the chance of being in the Low class for learning. In addition to the

core covariates set, baseline depression score was implicated for both immediate retention and

delayed recall, and anxiety for immediate retention.

The effect of changes in the time-varying covariates of estrogen producing ability, depression,

anxiety and fatigue resulted in minimal substantive impact, with only estrogen producing

ability at time 4 (18 months post-chemotherapy) being a substantive positive predictor of both

learning and delayed recall ability for the Low class at time 4 and depression scores at time 4

being a substantive negative predictor of learning for the High class.

Chapter 6 using Bayesian (single class) random effects latent basis trajectory models and

Bayesian latent basis latent class growth mixture model to identify the degree of recovery at

six and eighteen months post-chemotherapy for the single class and latent class models.

Although three alternative prior distributions were investigated, due to the consistency of the

results the Uniform prior results are discussed in detail. The degree of recovery for the single

class model at six months post-chemotherapy was at best minimal with the degree of recovery


for verbal learning of 16% to 18% and virtually zero for immediate retention and delayed

recall. The degree of recovery at eighteen months post-chemotherapy showed considerable

improvement with learning virtually returned to initial levels but immediate retention and

delayed recall still only reached 56% to 59% recovery. These results give weight to the often

heard complaint of memory being a major concern with the “chemo-brain” phenomena, and

for the long lasting effects of this problem with immediate retention and delayed recall. Based

on the results of this study, assuming a proportional recovery, return to baseline scores would

not occur for another twelve months or of the order of 30 months post-chemotherapy. This

information may be useful in the design of future longitudinal studies assessing the impact of

chemotherapy on cognitive change, especially on the duration of the study and allocation of

assessment periods. Indeed it has resulted in a recent extension of the current study to five

years post-chemotherapy. The sub-group analysis for three classes identified Low, Mid and

High classes for all three verbal memory outcomes. The Low classes of between 18 and 21%

of participants, showed further decline or no recovery at six months post-chemotherapy and

minimal recovery at eighteen months post-chemotherapy with posterior mean recoveries of

29%, 11% and 10% for the three outcomes respectively. The High classes, with 32% to 40%

of participants, exhibited between 22% to 30% recovery at six months post-chemotherapy, but

were more variable at eighteen months post-chemotherapy with 96% recovery for Learning,

38% for Immediate Retention and 74% recovery for delayed learning (from posterior means).

The posterior 95% credible intervals for the initial verbal memory score as an indictor of class

membership can have clinical implications in the identification of breast cancer patients who

may be at risk of longer term verbal memory disfunction, enabling the targeting of extra

assistance or intervention programs.

The results of the four chapters 3 to 6 as a whole, for the non-linear trajectory processes of the

verbal memory with the identification of latent classes or subgroups of participants having

distinctly different profiles and associated covariates undoubtedly assist in the identification of

breast cancer patients who may be at risk of higher levels of verbal memory impairment with

chemotherapy treatment. The ability to use Bayesian latent class growth mixture models with

a latent basis formulation to estimate the degree of recovery at specific timepoints, and the

identification of a verbal memory class exhibiting limited recovery can have clinical

7.2. Limitations of the Research 209

implications in the targeting of intervention responses.

7.2 Limitations of the Research

Although the sample size of 120 subjects in the Cognition in Breast Cancer Study is

comparable to many other studies of cognitive change after chemotherapy for breast cancer,

the relatively small number of patients restricts the ability to identify larger numbers of

subgroups. A larger sample size could enable greater sensitivity in the identification of

predictor or mediating variables.

The Bayesian models developed in this thesis were restricted to using only non-informative

prior distribution with model parameters, and more subgroups may be identifiable with the

use of informative prior distributions.

The three verbal memory variables are the main outcomes of the models addressed in chapters

3 to 6 and were considered as separate but related measures of an underlying process. The

multivariate nature of these responses together was not investigated as inferences relating to

the individual aspects of verbal memory were important for both neuropsychological and

clinical reasons.

Autocorrelated time responses were not considered in the Bayesian models investigated in this

thesis with the justification that six months between measurement occasions reduced the

likelihood of such processes being of substantive importance. However for longitudinal data

of short duration the restrictions imposed on temporal variance/covariance can be an

important consideration for close measurement occasions.

In general terms, the limitations of the research were mainly due to the restrictions of the

single case study in targeting the statistical methodology development.

7.3 Possible Future Research

Hierarchical modelling for measurements repeated over time can include autocorrelated time

responses, where measurements at adjacent times are more alike than measurements taken

with a longer temporal separation. The models considered in this thesis made no allowance

for these possible effects and so the inclusion of more complex temporal variance/covariance


structures would be a possible future research direction. In doing so obtain an understanding

of the temporal distances at which autocorrelated responses where having a meaningful effect

for cognitive measurement.

The Bayesian analyses in this thesis were restricted to incorporating non-informative prior

distributions. The work of Elliott et. al. used informative prior distributions in a Bayesian

growth mixture model for relatively small numbers of subjects (46) and was able to identify

up to 6 latent classes for a dual trajectory process of daily affective mood scores and binary

adverse event indicators [1]. The utilisaton of informative prior distributions may be able to

identify further subgroups, which would be especially useful in models where predictors of

class membership were included. Since as found in the results of Chapter 5 and in the work of

Muthen [2] and Nagin [4] the inclusion of predictors of class membership often reduced the

number of classes identified when compared to the unconditional model.

Throughout this thesis the three aspects of verbal memory were considered as separate

although related outcome variables, primarily as the characteristics of learning and the other

verbal memory outcomes of immediate retention and delayed recall behaved differently.

There are established neuropsychological reasons for this which involves different parts of the

brain being stimulated by the different verbal memory aspects. However these three aspect of

verbal memory could be considered as a multivariate response, so future research directions

could be considering different ways to model the multivariate responses and the temporal

nature of the data with Bayesian structural equation modeling techniques and latent class

growth trajectories.

Verbal memory is only one aspect of cognitive function, and as many studies of cognition

explore several important aspects of cognitive function, including, executive function,

working memory, and visual memory, the challenge for future research would be to model a

range of cognitive constructs measured each by multiple outcomes over multiple times

generated by multiple latent classes or subgroups of subjects. Thus modelling latent variables

at many levels: subgroups, cognitive construct, growth function, variance/covariance structure

and other possible higher order sampling structures.

BIBLIOGRAPHY 211

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1.4.



Appendices

213

Appendix A

Cognition Joint Paper 1

This is co-authored paper with Katharine Vearncombe, a registered psychologist, as first

author and Margaret Rolfe as second author, which has been accepted for publication by the

Journal of the International Neuropsychological Society (2009), 15, 1 - 12.

This paper illustrates the aspect of the PhD training in collaborative interdisciplinary research

in the Cognition in Breast Cancer Study. My contribution to the paper includes advice

regarding the implementation of the statistical methods, development of initial syntax for

running the statistical analyses in SPSS, overseeing the writing of the statistical methods and

results and undertaking a general editorial role.

Title: Predictors of Cognitive Decline After Chemotherapy in Breast Cancer Patients

Journal of the International Neuropsychological Society (2009), 15, 1 – 12 . Copyright 2009INS. Published by Cambridge University Press. Printed in the USA.doi:10.1017/S1355617709990567

Authors: Katharine J. Vearncombe1, 2, Margaret Rolfe3, 4, Margaret Wright5, Nancy A.Pachana1, Brooke Andrew1, 2 And Geoffrey Beadle5

1The University of Queensland , School of Psychology , Brisbane , Australia2The Wesley Research Institute , Wesley Hospital , Brisbane , Australia3Graduate Research College , Southern Cross University , Lismore , Australia4School of Mathematical Sciences , Queensland University of Technology5 Queensland Institute of Medical Research , Brisbane , Australia

Abstract

The objective of this study is to identify whether decline in cognitive functioning afterchemotherapy in women with breast cancer is associated with health/disease, treatment, and

215

216 APPENDIX A. COGNITION JOINT PAPER 1

psychological variables. Neuropsychological performance, health/disease, andtreatment-related information of 136 women with breast cancer (age M = 49.38; SD = 7.92;range = 25.25–67.92) was assessed pre-chemotherapy and 1-month post-chemotherapy. TheReliable Change Index corrected for practice (RCIp) identified women whose performancesignificantly declined, while Pearson correlations assessed the relationship between cognitivechange and predictor variables. A total of 16.9% of women showed significant declinepost-chemotherapy, with affected domains including verbal learning and memory, abstractreasoning, and motor coordination. Decline in hemoglobin levels and increased anxiety overthe course of chemotherapy was found to significantly predict impairment in multiplecognitive measures. Change in specific cognitive measures was significantly associated withbaseline fatigue, depression, and functional well-being (r = 0.23 to 0.33; p = .01to < .001).Although the effects are small, there is evidence that psychological and health factors mayincrease vulnerability to cognitive dysfunction after chemotherapy for breast cancer.Significant associations reported in this study may be useful in the identification and treatmentof at-risk individuals.Keywords : Adjuvant chemotherapy , Breast cancer , Cognitive domains , Cognitiveimpairment , Neurotoxicity , Memory

217

Predictors of Cognitive Decline After Chemotherapy in BreastCancer Patients

INTRODUCTION

Cytotoxic drugs, or chemotherapy, have been linked to varying degrees of cognitive deficits inbreast cancer patients. Commonly referred to as chemo-brain by patients, typical complaintsinvolve difficulties with memory and concentration (Castellon et al., 2004). However, ascancer treatment usually comprises many systemic drugs administered concurrently, it is stilluncertain which chemotherapy drugs are neurotoxic. In addition, it is also possible thatgenetic variability, tumor biology, or the immune systems reaction to a tumor may increase anindividuals vulnerability to chemotherapy- induced cognitive changes (Ahles & Saykin,2007). In fact, some researchers have suggested that it is premature to attribute the observeddeclines directly to chemotherapy at all, instead preferring cancer-treatment-related decline(Hurria, Somlo, & Ahles, 2007).Evidence from previous research suggests that cancer treatment-related cognitive dysfunctiononly occurs in a subgroup of women, with reports generally ranging between 15 and 50%(Vardy & Tannock, 2007). These declines in cognitive performance are subtle, with the mostcommonly affected domains being verbal memory, language, visual memory/ spatial abilityand executive functioning (for meta-analyses, see Faletti, Sanfilippo, Maruff, Weih, &Phillips, 2005; Jansen, Miaskowski, Dodd, Dowling, & Kramer, 2005; Stewart, Bielajew,Collins, Parkinson, & Tomiak, 2006). However, reports of affected domains are variable, withsome studies finding global difficulties (e.g., Schagen et al., 1999 ; Scherwath et al., 2006 ;Wieneke & Dienst, 1995) and some finding more specific deficits after chemotherapy (e.g.,Bender et al., 2006 ; Quesnel, Savard, & Ivers, 2009), while others have reported no deficits(e.g., Donovan et al., 2005 ; Hermelink et al., 2007 ; Hermelink, Henschel, Untch, Bauerfeind,Lux, & Munzel, 2008). Methodological differences between studies include inconsistencies inthe definition of cognitive impairment, lack of a baseline/pre-chemotherapy assessment andlarge variations in the time since treatment (Donovan et al., 2005 ; Hurria et al., 2007 ).However, while the majority of studies have reported cognitive dysfunctionpost-chemotherapy in at least a proportion of patients, the reason for this cognitive decline islargely unknown. There is some evidence for chemotherapy having a direct effect onneurological function, as imaging studies have identified cerebral atrophy, corticalcalcification (Verstappen, Heimans, Hoekman, & Postma, 2003), and decreased metabolicactivity (Silverman et al., 2007) in numerous brain regions after chemotherapy. Additionally,a dose-dependent relationship has been found, with higher doses associated with poorerneuropsychological performances (van Dam et al., 1998). However, there is also evidence thatpatients exhibit cognitive dysfunction before receiving chemotherapy (Ahles et al., 2008 ;Wefel, Lenzi, Theriault, Buzdar, Cruickshank, & Meyers, 2004a ), which suggests that other(non-chemotherapy) factors may also play a role.To date, the exploration of relationships between cognitive functioning and health/disease andtreatment-related factors in breast cancer patients has been limited. Most treatment andhealth/disease-related factors (e.g., time since treatment and use of hormone replacementtherapy) have not been significantly associated with cognitive dysfunction after chemotherapy.On the other hand, the majority of these factors have been compared to neuropsychologicalperformance in only one or two studies, many of which used a cross-sectional design. Onlytwo factors have been reported to be significantly associated with cognitive dysfunctionfollowing chemotherapy for breast cancer, namely, longer treatment duration (Wieneke &


Dienst, 1995) and use of adjuvant endocrine therapy (Bender et al., 2006 ; Castellon et al.,2004 ; Collins, Mackenzie, Stewart, Bielajew, & Verma, 2009), although the evidence isconflicting. Anemia, as measured by hemoglobin levels, has also been implicated in theoccurrence of cognitive dysfunction after chemotherapy, with cancer patients who becameanemic (defined as hemoglobin levels falling below 12g/dL) showing significant declines inperformance on tests of attention and visual memory (Jacobsen et al., 2004 ). However, onlyone study has examined the impact of anemia on cognition after breast cancer treatment, andno significant relationship to cognitive functioning was reported (Tchen et al., 2003).Nevertheless, the examination of all these factors is far from extensive and requires systematicinvestigation.

Many studies investigating chemotherapy-related cognitive decline have also evaluated theimpact of fatigue, mood (particularly anxiety and depression), and quality of life (QOL) oncognitive dysfunction, with mixed results. Fatigue is the most frequently investigated factor,with only a few studies reporting significant associations between fatigue and objectiveneuropsychological performance, particularly in the domains of attention, working memory,and verbal memory (Cimprich, 1992 , 1993 ; Mehlsen, Pedersen, Jensen, & Zachariae, 2009 ;Mehnert et al., 2007). Higher levels of depression have been found to be associated withcognitive dysfunction after chemotherapy in several studies (Bender et al., 2006 ; Schagen etal., 2002 ; Stewart et al., 2008 ; Wefel, Lenzi, Theriault, Davis, & Meyers, 2004b), althoughthis not consistent (e.g., Castellon et al., 2004 ; Schagen, Muller, Boogerd, Mellenbergh, &van Dam, 2006 ; van Dam et al., 1998 ; Wieneke & Dienst, 1995). On the other hand, anxietygenerally has not been found to predict declines in cognitive functioning, with only onecross-sectional study reporting that higher levels of anxiety were associated with worse verbalmemory performance 2–5 years after a breast cancer diagnosis (Castellon et al., 2004 ).

Similarly, there is little evidence to suggest that QOL impacts on cognitive functioning, withthe majority of breast cancer studies finding no significant associations between QOL andcognitive functioning (e.g., Schagen et al., 2002 ; Tchen et al., 2003 ; Wefel et al., 2004b).However, two recent small studies have reported significant relationships. Mehnert andcolleagues (2007) found that declines in specific cognitive domains were associated withpoorer social, emotional, and physical functioning, while another study reported that cancerand cardiac patients with higher life satisfaction and social support performed better onprocessing speed and verbal memory tasks, respectively (Mehlsen et al., 2009). However,measurement of all these factors has been somewhat restricted, with only two studiesinvestigating whether change in possible covariates is associated with cognitive change(Collins et al., 2009; Stewart, Collins, Mackenzie, Tomiak, Verma, & Bielajew, 2008).Therefore, the investigation of the relationship between health/disease, treatment, andpsychological variables and objective cognitive performance has been both limited and hasyielded inconsistent results, warranting further research.

The current study aims to explore whether health/disease (hemoglobin, stage of cancer,estrogen receptor status, baseline menopausal status), treatment (type of surgery, number ofchemotherapy courses), and psychological variables (depression, anxiety, fatigue, and QOL)contribute to acute cognitive decline after chemotherapy for breast cancer. While the findingsfrom recent research have been inconsistent, we expect to find significant cognitive decline onseveral specific cognitive measures (particularly in the verbal memory and executive functiondomains), as well as significant associations between cognitive decline and depression. Basedon previous research, no significant results were expected for fatigue, baseline menopausalstatus, anxiety, QOL, stage of cancer, type of surgery, or number of chemotherapy courses.Given that there has been little investigation into chemotherapy-induced anemia in the

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existing literature, it is unclear how this variable may impact on cognitive functioning.However, as anemia is a common side effect of chemotherapy it was deemed an importanthealth factor to examine by means of hemoglobin levels.

METHODS

Participants

Data are from the Cognition in Breast Cancer (CBC) study, a longitudinal study examiningthe causes of variation in cognitive functioning, health and well-being in women up to 2 yearspost-chemotherapy. Eligible participants were required to be between 18 and 70 years old;proficient in English; and have no previous history of cytotoxic drug treatment, neurologicalor psychiatric symptoms, or current use of medications that might affect neuropsychologicaltest performance. All participants provided written, informed consent, and this study wasapproved by the following ethics committees; the Queensland Institute of Medical Research,the University of Queensland, and all participating hospitals (Wesley Hospital, RoyalBrisbane and Womens Hospital, Redcliffe Hospital, Princess Alexandra Hospital, MaterHospital, St Vincents Hospital, and St Andrews Hospital).Two groups of early breast cancer patients were recruited from hospitals across south-eastQueensland, Australia; patients scheduled to have chemotherapy treatment and patientsscheduled for other forms of breast cancer treatment (i.e., endocrine treatment and/orpostoperative radiotherapy). Patients were approached by their oncologist/ surgeon or aresearch nurse after definitive surgery, and those that initially agreed to participate received aphone call from a psychologist, who described the purpose and procedures of the study. Thepsychologist also discussed the eligibility criteria, and those patients who were eligible andwilling to participate were scheduled to sign informed consent forms and complete theassessment battery (approximately 2.5 hours in duration). Neuropsychological testing wasadministered both before commencement and after completion of chemotherapy, while thenon-chemotherapy group was assessed at similar time points.Of the 192 women initially recruited to the study, 11 withdrew before the first assessment, twodid not finish chemotherapy, and 20 withdrew due to illness/personal reasons or were unableto complete the post-chemotherapy assessment. The women who withdrew from the study didnot differ from the rest of the sample in age, education, estimated intellectual functioning,menopausal status, type of surgery, or number of planned chemotherapy courses. They alsodid not differ from women who remained in the study on any of the psychological measuresand the majority of cognitive measures before the commencement of chemotherapy. However,it was found that women who withdrew were significantly more likely to have lower stagecancers ( p < .001) and perform more poorly on an executive functioning measure (matrixreasoning; p < .01). The final sample consisted of 159 women (age M = 49.95; SD = 8.09;range = 25.25–67.92). One group comprised 138 participants scheduled to receive standarddose adjuvant chemotherapy (with or without endocrine treatment and radiotherapy). Asecond group included 21 women with breast cancer scheduled to receive no chemotherapy(i.e., endocrine treatment, radiotherapy, and/or surgery only).

Procedure

Participants were assessed either in a quiet room at a participating hospital or in their ownhome. Participants completed a demographic interview and neuropsychological assessmentbattery at two time points: at baseline (after surgery but before commencement of


chemotherapy – T1) and approximately 4 weeks after administration of the last course ofchemotherapy (T2). The second group of women were assessed at similar time points. Eachof the neuropsychological assessments was individually administered and all participantscompleted the test battery in the same order. Clinical information was collected beforechemotherapy and at chemotherapy completion by clinical research nurses.

Measures

Neuropsychological tests and self-report measuresThe neuropsychological, mood and QOL measures used in the current study are presented inTable A.1. The cognitive battery was designed to assess a variety of cognitive domains,namely verbal learning/ memory, visual memory, processing speed, as well as differentaspects of attention and executive functioning. As the tests used in the current research yieldmultiple outcome measures, Table 1 also lists the specific variables used in the analyses.Quality of life was measured using the Functional Assessment of Cancer Therapy–General(FACT-G), along with the fatigue subscale. The FACT-G comprises 27 items covering fourQOL domains: physical, emotional, social/family, and functional well-being. The fatiguesubscale comprises 13 items measuring the disruptiveness and intensity of fatigue, forexample, I feel listless (washed out). Participants rate each item on a five point scale, rangingfrom not at all to very much. A higher score indicates more satisfaction/ well-being and lessfatigue on the QOL scale and fatigue scale respectively.Self-reported depression and anxiety was measured using the Hospital Anxiety andDepression Scale (HADS), a 14-item rating scale assessing the presence and prominence ofdepressive and anxious symptoms over the week before test administration. Separate scoresfor depressive and anxious symptomatology are calculated, with higher scores indicatinghigher levels of depression or anxiety.Age, education level (maximum 20 years), and general cognitive ability (Full Scale IQ, FSIQ)were collected as covariate information because these variables have been found to affectperformance on objective neuropsychological tests (Schagen et al., 2002). FSIQ wasestimated using the National Adult Reading Test, version 2 (NART-2; Nelson & Willison,1991), which is a validated reading test. Participants are required to read 50 irregularly spelledwords, and accuracy of pronunciation is used to predict IQ (Strauss et al., 2006).Clinical variablesTime-invariant and time-variant health, disease, and treatment information were alsocollected. Time-invariant data included stage of cancer, estrogen receptor status (positive ornegative), type of surgery (breast conserving or mastectomy), number of chemotherapycourses, and baseline menopausal status. Stage of cancer is a predictor of survival anddescribes how much the cancer has spread. It takes into account size of the tumor andinvolvement of axillary lymph nodes. Due to the small number of participants diagnosed withstage III cancer (n = 9), stages II and III were combined in the current study. Baselinemenopausal status was divided into estrogen producing and not estrogen producing. Womenwere classified as estrogen producing if they had experienced menstruation within the past 12months at the time of diagnosis, while women who had not menstruated within the past 12months were considered nonestrogen producing. Time-variant clinical data was hemoglobinlevel, which is an indicator of anemia. Statistical Analysis Statistical Package for SocialSciences (SPSS) for Windows, versions 15 and 16 were used for all analyses. Raw scoreswere used in the current analyses and all noncontinuous

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Table A.1 Neuropsychological and self-report measures and outcome variablesDomain Measure Variables (abbreviation)

NEUROPSYCHOLOGICALVerbal Learning Auditory Verbal Learning Test Total number of words remembered in trials

and Memory (Geffen & Geffen, 2000) * 1–5 (AVLT)Total number of words remembered after a30 minute delay (AVLT8)

Visual memory a) WMS-IIIa Visual Reproduction a) Total correct immediately after seeing eachimmediate design (VR1)b) WMS-III Visual Reproduction b) Total correct 30 minutes after being showndelayed designs (VR2)c) WMS-III Visual Reproduction c) Total number of designs correctlyrecognition identified (VRrecog)

Working memory WAIS-IIIb Backward Digit Span∗ Total number of trials correctly completed (BDS)Processing Speed Symbol Digit Modalities Test, Total number completed in 90 seconds (SDMT)

oral version (Smith, 1982 )Attention a) TEAc Visual Elevator∗ a) Total time taken per switch (TEA-VE)

b) TEA Telephone Search∗ b) Total time taken without distraction. (TEA-TS)Executive function a) WAIS-III Matrix Reasoning a) Total correct (MR)

b) Stroop (Golden & b) Total number correct in color word conditionFreshwater, 2002 ) (Stroop)c) DKEFSd Card Sorting Task∗ c) Total correct in free-sorting condition (Card Sort)d) Controlled Oral Word d) Total number of words across phonemicAssociation Test (Lezak, 1995)∗ verbal fluency condition (COWAT)

Motor coordination Purdue Pegboard (Tiffin, 1968) Total number of pegs constructed in assemblycondition. (PPassembly)

SELF-REPORT QOLFunctional Assessment of Chronic Total Physical well-being subscale scoreIllness Therapy – Breast scale Total Emotional well-being subscale score(Brady et al., 1997 ) Total Social/Family well-being subscale score

Total Functional well-being subscale scoreFatigue Functional Assessment of Chronic Total Fatigue subscale score

Illness Therapy – fatigue scale(Yellen, Cella, Webster,Blendowski, & Kaplan, 1997)

Mood Hospital Anxiety and Depression Total depression scoreScale (Zigmond & Snaith, 1983) Total anxiety score

a WMS-III = Wechsler Memory Scale-Third Edition (Wechsler, 1997a ).b WAIS-III = Wechsler Adult Intelligence Scale-Third Edition (Wechsler, 1997b).c TEA = Test of Everyday Attention (Robertson, Ward, Ridgeway, & Nimmo-Smith, 1994).d DKEFS = Delis-Kaplan Executive Function Scale (Delis, Kaplan, & Kramer, 2001).∗ Alternate forms used.


Statistical Analysis

Statistical Package for Social Sciences (SPSS) for Windows, versions 15 and 16 were used forall analyses. Raw scores were used in the current analyses and all noncontinuous variableswere dichotomized. Statistical inspection of the data revealed two cases that were multivariateoutliers. These were excluded from all analyses, leaving 136 participants in the chemotherapygroup. No differences were observed between women who had and had not commencedendocrine treatment or those who did and did not contribute complete hematologicalinformation. Thus, all cases were included in all analyses.Two separate analyses were performed to evaluate whether health/disease, treatment, andpsychological factors contributed to change in the neuropsychological data. First, to increasecomparability between the current study and previous research, dichotomous impaired/notimpaired classifications for each patient were calculated for specific cognitive tests. Thecontribution of the predictor variables on the impaired/ not impaired classifications were thenevaluated by multiple binary logistic regressions. Second, the association between change incognitive performance (irrespective of impaired/ not impaired classifications) and predictorvariables were assessed using Pearson correlations. Given the high number of comparisons,the statistical significance cutoff was arbitrarily set a priori at p <.01 for all analyses.Impaired versus not impaired classificationsImpairment on specific cognitive tests were defined as significant decline identified using theReliable Change Index (corrected for practice, RCIp), while ”Multiple Test Decline” wasdefined as significant decline on two or more cognitive tests. The RCIp was proposed byChelune and colleagues (1993) and uses test-retest reliability and the standard error of thedifference (Sdi f f ) to establish whether the change between baseline and follow-up scores issignificant. Given the small control sample, test-retest or delayed alternate forms reliability(AVLT variables only) coefficients were based on published data to increase stability of thecorrelations. As alternate forms of the AVLT were used in the current study, the delayedalternate forms reliability coefficients were deemed to provide a better indication of retesteffects over time when alternate forms were used. Mean change between assessments in thenonchemotherapy group was used to control for practice effects, and the cutoff used todetermine impairment in each cognitive outcome measure was a decline of more than 1.96standard deviations. The formulae used in the current study can be seen below:

RCI + practice = (S Edi f f )(±1.64) + practice effect

Definitions and Formulae for Reliable Change Indices

S Edi f f =√

2(S E)2

S E = S D√

1 − rxx

S D = Standard deviation from published norms

rxx = Reliability coefficient from published norms.

Practice effect = Mean difference between the follow-up and baseline scorein the breast cancer control group.

The two groups (chemotherapy and nonchemotherapy) were compared by means ofindependent group t-tests and χ2 analyses to ensure sufficient similarity on demographic and

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cognitive baseline measures. The RCIp was then computed and used to identify participantswho were cognitively impaired and those that had not changed or improved. Multiple testimpairment was calculated by adding the number of tests that reliably declined more than 1.96standard deviations for each participant, then dichotomized into ”less than 2” or ”2 or more”tests. Binary multiple logistic regression (with backward stepwise selection) was performedon each of the impaired/not impaired cognitive variables to determine whether thehealth/treatment or psychological variables predicted significant cognitive decline afterchemotherapy.Cognitive change irrespective of impaired/ not impaired classificationCognitive change was calculated by taking the difference between Time 2 and Time 1 (T2-T1)for each cognitive test. Pearson correlations between clinical variables, psychologicalvariables (mood and QOL), and cognitive change scores were used to determine whetherthese factors were associated with cognitive change. Significant associations betweencognitive performance and age, IQ, and education level were partialled out of analyses.

RESULTS

Dichotomous Classifications of Impaired/ Not ImpairedComparisons between chemotherapy and control groupThe characteristics of the two groups at baseline are shown in Table 2. Independent groupt-tests yielded no significant differences in age, education and baseline FSIQ. However, thetest.retest interval was found to be significantly different, with the control group having alonger interval between assessments. The χ2 analyses also found significant differencesbetween the two groups in baseline menopausal status and stage of cancer, with women in thecontrol group more likely to be postmenopausal and have stage 1 cancers. However, as stageof cancer is an indication of severity/ aggressiveness, differences on this variable are expectedas it is a determinant for recommendations about adjuvant chemotherapy. The two groups didnot significantly differ in surgery type, estrogen receptor status, or marital status. In addition,no significant differences were found in baseline cognitive, mood or QOL performancebetween the chemotherapy and nonchemotherapy groups (data not shown), suggesting that thetwo groups were matched adequately for estimated practice effect information to beextrapolated.Reliable Change Index corrected for practice (RCIp)Published reliability coefficients for each cognitive task, as well as the means and standarddeviations for both groups are presented in Table A.3.Paired t-tests showed significant differences in the chemotherapy group, with significantdeclines found in the verbal memory measures, and significant improvements seen in thevisual memory, processing speed, and attention domains. No significant changes were seen inthe nonchemotherapy group (at the p < .01 level).Table A.4 shows the results of the RCIp. Only four measures showed a substantial number ofparticipants who were classified as impaired (decline of > 1.96 SD), namely AVLT, AVLT8,MR, and PPassembly. Multiple Test Decline defined as a reliable decline on two or morecognitive measures, was found in 16.9% of all participants who had received chemotherapy.Based on the results of the RCIp analyses, subjects were then dichotomized into impaired andnot impaired classifications and analyzed with binary multiple logistic regression models(with backward stepwise selection). Factors included in the model were baseline menopausalstatus, stage of cancer, type of surgery, number of courses, estrogen receptor status, as well aschange on time-variant psychological and health factors (anxiety, depression, fatigue, QOL


Table A.2 Demographic and treatment related characteristics of the study sampleChemo mean (SD) % Non-chemo mean (SD) % t/χ2

Age in years 49.38 (7.92) 53.98 (8.24) -2.46FSIQ 110.75 (8.32) 112.62 (10.76) -0.92Years of education 13.07 (3.35) 13.52 (3.94) -0.57Marital status 5.65

Singlea 22 16.2 8 38.1Marriedb 114 83.8 13 61.9

Menopausal statusc 10.32∗∗Pre/peri-menopausal 99 68.3 7 33.3Postmenopausal 44 30.4 14 66.7Unknown 2 1.4

Stage of cancer 23.29∗∗I 37 27.2 17 81.0II/III 99 72.8 4 19.0

Surgery 5.87Breast conserving 77 56.6 17 81.0Mastectomy 59 43.4 3 14.3Unknown 1 4.8

Estrogen receptor status 3.43Negative 30 22.1 1 4.8Positive 106 77.9 20 95.2

Chemotherapy regimenFEC 70 44.6FEC + Taxotere 5 3.2FEA 1 0.6CAF 14 8.9CA 8 5.1CA + Taxol 30 19.1CA + Taxotere 1 0.6CEA 5 3.2CMF 1 0.6C + Taxotere 1 0.6

Number of courses3 1 0.74 15 11.05 4 2.96 89 65.47 1 0.78 26 19.1

Mean test-retest interval 5.23 (1.08) – 6.37 (0.69) – -4.89∗∗months (SD) range 3 – 10.13 range 5.16 – 8.07Days since last treatment 42.37 (17.93) – – –cycle

Note. FSIQ = Full Scale IQ, F = 5-fl uorouracil, E = epirubicinC = cyclophosphamide, A = Adriamycin M=methotrexate.∗ Significant at p < .01.∗∗ Significant at p < .001.a Includes divorced and widowed participants.b Includes defacto couples.c Baseline measurement

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Table A.3 Means, standard deviations, and reliability estimates for Time 1 and Time 2 cogni-tive variables in the chemotherapy and non-chemotherapy groups

Chemotherapy group Non-chemotherapy groupDomain Variable Time 1 Time 2 Time 1 ) Time 2 ) Reliability

Mean (SD) Mean (SD) t (135) Mean (SD) Mean (SD) t (20) r

Verbal memory AVLT 52.21 (7.37) 49.62 (8.06) 4.40∗∗ 51.19 (9.23) 46.90 (8.58) 2.21 .77a

AVLT8 11.15 (2.39) 9.63 (2.55) 7.54∗∗ 10.62 (2.13) 9.57 (1.83) 2.06 .70a

Visual Memory VR1 85.41 (11.86) 88.14 (10.78) -3.22∗ 81.81 (11.35) 82.24 (15.61) -0.15 .79b

VR2 66.38 (22.69) 73.91 (20.80) -4.85∗∗ 64.86 (20.17) 68.43 (16.87) -0.91 .77b

VRrecog 44.83 (2.36) 45.58 (2.39) -3.81∗∗ 44.76 (2.49) 45.29 (2.00) -1.14 .75b

Working memory BDS 7.83 (2.17) 7.76 (2.10) 0.49 7.24 (2.63) 7.19 (2.56) 0.10 .65c

Processing speed SDMT 58.36 (9.09) 60.15 (9.38) -3.58∗∗ 56.43 (7.49) 58.38 (6.31) -1.52 .76d

Attention TEA-VE 4.25 (0.95) 3.81 (0.82) 6.42∗∗ 3.86 (0.57) 3.75 (0.91) 0.60 .79e

TEA-TS 2.98 (0.58) 2.90 (0.55) 2.24 3.10 (0.54) 3.05 (0.43) 0.53 .86e

Executive function MR 17.46 (4.64) 17.49 (4.51) -0.10 16.38 (4.30) 16.57 (4.03) -0.25 .69b

Stroop 46.40 (9.26) 46.76 (8.61) -0.76 44.00 (8.60) 46.86 (9.71) -1.71 .73 f

Card sort 9.38 (1.90) 9.31 (2.60) 0.41 9.90 (2.02) 9.86 (1.59) 0.14 .60g

COWAT 43.45 (12.64) 45.01 (12.24) -2.22 45.67 (13.46) 47.05 (12.88) -1.06 .72d

Motor coordination PPassembly 33.30 (7.07) 33.61 (7.30) -0.63 30.81 (6.43) 31.86 (6.83) -0.88 .81d

Note. Cutoff p < .01. AVLT = Auditory Verbal Learning Test, VR = Visual Reproduction, BDS = Backward Digit Span,SDMT = Symbol Digit Modalities Test, TEA = Test of Everyday Attention, MR = Matrix Reasoning,COWAT = Controlled Oral Word Association Test.∗p < .01.∗∗ p < .001.aGeffen, Butterworth, & Geffen, (1994).bTulsky et al., (1997).cWaters & Caplan (2003).dStrauss et al., (2006).eRobertson et al., (1996 ).f Golden & Freshwater, (2002).gDelis et al., (2001) .

Table A.4 Classifications of impaired, no change, and improved after chemotherapyDomain Measures N (%) showing N (%) showing N (%) showing

negative change no change positive change

Verbal memory AVLT 28 (20.6) 99 (52.9) 9 (6.6)AVLT8 26 (19.1) 108 (65.4) 2 (1.5)

Visual Memory VR1 0 (0.0) 128 (94.1) 8 (5.9)VR2 0 (0.0) 125 (91.9) 11 (8.1)VRrecog 2 (1.5) 127 (93.4) 7 (5.1)

Working memory BDS 3 (2.2) 129 (94.9) 4 (2.9)Processing speed SDMT 0 (0.0) 133 (97.8) 3 (2.2)Attention TEA-VE 3 (2.2) 124 (91.2) 9 (6.6)

TEA-TS 0 (0.0) 134 (98.5) 2 (1.5)Executive function MR 10 (7.4) 114 (83.8) 12 (8.8)

Stroop 0 (0.0) 135 (99.3) 1 (0.7)Card sort 0 (0.0) 136 (100.0) 0 (0.0)COWAT 2 (1.5) 132 (97.1) 2 (1.5)

Motor coordination PPassembly 11 (8.1) 112 (82.4) 13 (9.6)No. of tests declined 0 76 (55.9)

1 37 (27.2)2 19 (14.0)3 3 (2.2)4 1 (0.7)

Multiple test decline 2+ tests impaired 23 (16.9)

Note. AVLT = Auditory Verbal Learning Test, VR = Visual Reproduction, BDS = Backward Digit Span,SDMT = Symbol Digit Modalities Test, TEA = Test of Everyday Attention, MR = Matrix Reasoning,COWAT = Controlled Oral Word Association Test.


Table A.5 Means and standard deviations for cognitive change (T2-T1) in the chemotherapygroup

Domain Variable Change (T2-T1)N Mean (SD)

Verbal memory AVLT 136 -2.59 (6.86)AVLT8 136 -1.52 (2.35)

Visual Memory VR1 136 2.73 (9.90)VR2 136 7.54 (18.14)VRrecog 136 0.76 (2.30)

Working memory BDS 136 -0.07 (1.76)Processing speed SDMT 136 1.79 (5.81)Attention TEA-VE 136 -0.43 (0.79)

TEA-TS 136 -0.08 (0.42)Executive function MR 136 0.03 (3.44)

Stroop 136 0.36 (5.50)Card sort 136 -0.07 (2.09)COWAT 136 1.56 (8.20)

Motor coordination PPassembly 136 0.31 (5.72)

Note. AVLT = Auditory Verbal Learning Test, VR = Visual Reproduction,BDS = Backward Digit Span, SDMT = Symbol Digit Modalities Test,TEA = Test of Everyday Attention, MR = Matrix Reasoning,COWAT = Controlled Oral Word Association Test.

domains, and hemoglobin). In addition, given the high number of analyses, a significancecutoff of p < .01 was used.No health/disease, treatment, psychological or QOL factors were identified to significantlycontribute to impairment on specific cognitive measures. However, the binary multiplelogistic regression analysis retained two factors for the multiple test impairment. Impairmenton two or more tests was jointly predicted by declines in hemoglobin level betweenassessments (Wald = 4.14; p < .05, odds ratio [OR] = 1.04, 95% confidence interval [CI] =

1.00–1.09) and increases in anxiety from time 1 to time 2 (Wald = 4.31; p < .05, OR = 1.15;95% CI = 1.01–1.31) These factors together explain 11.2% of the variance in theclassification of multiple test impairment ( χ2 = 9.04; p = .01).Factors associated with cognitive change irrespective of classificationDifference scores (T2-T1) were computed for each of the cognitive variables to investigatecognitive change over the course of chemotherapy. These change scores were correlated withbaseline measurements of predictor variables as well as change scores on time-variantpsychological and clinical factors (anxiety, depression, fatigue, QOL domains, andhemoglobin). Means and standard deviations for these difference scores are shown inTable A.5, while the means and standard deviations for baseline and change (T2-T1)psychological and health variables are presented in Table A.6.Three cognitive tasks (SDMT, COWAT, and PPassembly) were significantly associated withage, education, and/or FSIQ, and these effects were partialled out before running the analyses.No time-variant health or psychological factors were associated with changes in cognition.However, higher levels of fatigue and depression as well as lower functional wellbeing atbaseline were significantly associated with change in cognitive measures, with correlationsshown in Table A.7.Trends were also found between several other variables above the p < .01 cutoff: BDS withbaseline emotional functioning (r = 0.21; p < .02), where decline in working memoryperformance was associated with poorer initial emotional functioning; estrogen receptor statuswith TEA-TS (r = .0.21; p < .02), where estrogen receptor negative breast cancers were

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Table A.6 Baseline and change (T2-T1) means and standard deviations for the psychological,health, and treatment factors in the chemotherapy group

Time 1 Change (T2-T1)Domain Variable N Mean (SD) N Mean (SD)

Mood Depression 136 3.12 (2.42) 136 0.45 (2.92)Anxiety 136 6.45 (3.74) 136 -0.53 (3.81)

Quality of life Physical well-being 136 22.49 (3.75) 136 -0.36 (4.56)Social well-being 136 24.29 (3.44) 136 -1.37 (4.15)Emotional well-being 136 18.76 (3.78) 136 0.68 (3.21)Functional well-being 136 20.51 (5.01) 136 0.12 (5.08)

Fatigue Fatigue 136 38.74 (8.86) 136 -4.29 (10.28)Anemia Hemoglobin g/L 132 130.00 (11.21) 132 -12.92(14.35)

Table A.7 Pearson correlations between change in cognitive measures (T2-T1) and health andpsychological measures

Domain Measure Fatigue(N) Depression (N) Functional well-being (N)

Attention TEA-TS -0.25 ∗ (136) 0.14 (136) -0.23 ∗ (136)Executive function Card Sort 0.27 ∗∗ (136) -0.17 (136) 0.19 (136)

COWAT 0.33∗∗ (127) -0.26 ∗(127) 0.26 ∗(127)Note. TEA-TS is a timed score, therefore, a decrease in score indicates an improvement in performance.

TEA = Test of Everyday Attention, COWAT = Controlled Oral Word Association Test.∗p <.01.∗∗ p <.001.


Table A.8 Correlations between psychological and clinical change variables (T1-T2)Physical Emotional Functional Social Hemoglobin

Anxiety Depression Fatigue QOL QOl QOl Qol

Anxiety 1Depression 0.43∗∗ 1Fatigue -0.26∗ -0.48∗∗ 1Physical QOL -0.22∗ -0.52 ∗∗ 0.65 ∗∗ 1Emotional QOL -0.45∗∗ -0.39∗∗ 0.35∗∗ 0.33∗∗ 1Functional QOL -0.36∗∗ -0.62∗∗ 0.57∗∗ 0.60∗∗ 0.42∗∗ 1Social QOL -0.11 -0.22 0.21 0.16 0.18 0.32∗∗ 1Hemoglobin -0.05 -0.03 -0.05 -0.04 -0.16 0.03 -0.03 1

Note. Higher scores on anxiety and depression measures indicate higher depression and anxiety.Higher scores on fatigue and quality of life domains indicate less fatigue and better well-being. QOL = quality of life.∗ p < 0.01.∗∗ p < 0.001.

associated with worse performance; and VR1 with change in hemoglobin levels (r = 0.20; p <.02), where decline in immediate visual memory was associated with decline in hemoglobinlevels.Interrelationships between predictor variablesThe relationships between predictor variables were evaluated using Pearson correlations(shown in Table A.8). High correlations were found between depression, anxiety, fatigue, andaspects of QOL (physical, emotional, and functional wellbeing). Surprisingly, changes insocial well-being were relatively independent from the other self-report measures, with only asignificant positive association with change in functional well-being found. Change inhemoglobin was not significantly related to any self-report measure.

DISCUSSION

The main goal of this study was to investigate whether health/disease, treatment factors,mood, and quality of life (QOL) significantly contributed to the cognitive dysfunction that hasbeen frequently reported after chemotherapy for breast cancer. Similar to previous research, asmall proportion (16.9%) of breast cancer patients treated with chemotherapy were found todecline on multiple cognitive measures (Collins et al., 2009; Quesnel et al., 2009 ). Consistentwith our hypothesis, the cognitive domains that showed the greatest decline were verballearning and memory, although only abstract reasoning showed any of the expected declinesin the executive function domain. The observed improvement in some measures, notably inthe visual memory and executive function domains, were consistent with practice effects.Surprisingly, no significant practice effects were observed in the control group, althoughnon-significant declines were evident on the verbal memory task. This questions the utility ofrecruiting healthy women as controls for research of this nature, as controlling for practiceeffects based on this group may lead to an overestimation of patients experiencing cognitivechanges.In line with previous research, the current study found little evidence to suggest that increasesin depression, and fatigue, as well as declines in well-being significantly affect cognitivefunctioning shortly after completion of chemotherapy (Collins et al., 2009 ; Stewart et al.,2008 ). However, it was found that decline in hemoglobin (conjointly with increases in thelevel of anxiety) significantly predicted impairment on multiple (two or more) cognitivemeasures. While these results are not overly strong, they are consistent with previous research

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that suggests that anemia may detrimentally affect cognitive performance (Jacobsen et al.,2004 ), which has been largely overlooked in the extant literature. Moreover, hemoglobin wasfound to be independent of self-report measures and may provide a useful clinical indicatorfor risk of cognitive impairment. However, caution is required when interpreting these resultsas the occurrence of blood transfusions was not recorded in the current study, andconsequently it is not possible to determine whether the performance of patients who requiredblood transfusions declined more than those who did not. Nevertheless, these findings suggestthat sub-clinical anemia may detrimentally affect cognitive functioning and warrants furtherinvestigation.

Multiple associations between baseline psychological and QOL factors and performance oncognitive measures were also found in the current study. Although many of the larger,prospective studies have generally not found any significant relationship betweenpsychological variables and objective cognitive performance, our results are consistent withresearch that have investigated different aspects of QOL and fatigue. Two recent studies havereported significant associations between fatigue, domains of QOL, and specific cognitivedomains, one of which was conducted over the same time frame as the current study (Mehlsenet al., 2009; Mehnert et al., 2007). Importantly, these studies differ from the majority ofresearch as they have compared specific domains of QOL and fatigue to objectiveneuropsychological performance. As the current study found that social well-being was notsignificantly associated with other areas of QOL, and that areas of QOL may differentiallyaffect performance on cognitive tests, it is possible that previous studies using global measuresof QOL may have overlooked these subtle effects. However, these studies also containnumerous limitations such as not containing pre-chemotherapy assessments, small samplesizes, and multiple comparisons (increasing type 1 error). Notably, while the causality ofresults cannot be determined due to their correlational nature, these results may be useful inidentifying patients at greater risk of cognitive impairment after chemotherapy.

While the overall level of impairment found in the current study is in agreement with previousresearch (Vardy & Tannock, 2007 ), the significant relationships found between health andpsychological factors diverge from the majority of longitudinal studies in this area. Thesedifferences may be due to sample size, with previous research mainly comprising smallersamples (range, 18–101) and possibly lacking the power to detect these associations(Hermelink et al., 2007 ; Wefel et al., 2004b ). Alternatively, due to the large number ofcomparisons performed, it is possible that some of these significant associations could havearisen by chance. However, we adopted a more stringent statistical significance level, makingthis unlikely. A more likely explanation may be that many previous studies calculatedcognitive impairment by combining the performance on cognitive tasks into one globalimpairment score (e.g., Schagen et al., 2006 ; Tchen et al., 2003 ; van Dam et al., 1998 ;Wieneke & Dienst, 1995 ). This may have masked significant associations as the current studysuggests that these health/treatment, psychological, and well-being factors may havedifferential effects depending on cognitive domain.

While these results are revealing, the RCIp results in particular must be interpreted withcaution due to differences in the test–retest interval between groups, with thenonchemotherapy group found to have a significantly longer reassessment interval (by 1.14months) than the chemotherapy group. This is problematic as the magnitude of the practiceeffects on neuropsychological tests tends to decrease with time (Lezak, 1995 ), and levels ofimpairment identified through the Reliable Change Index may be an overestimate of the truelevels of impairment after the administration of chemotherapy. On the other hand, previousresearch has also reported that practice effects on neuropsychological tests do not significantly


differ over a 2–16 month test–retest interval (e.g., McSweeny, Naugle, Chelune, & Luders,1993 ; Temkin, Heaton, Grant, Dikmen, 1999 ), suggesting that practice effects may notdecrease too much over the time periods investigated in the current study. In addition, aspublished practice effects generally involve very short test–retest intervals (1 week to 1month) and the two groups were relatively well matched on demographic, cognitive, andpsychological factors, the non-chemotherapy group was deemed to be the best estimate ofpractice effects available. Furthermore, as the RCIp is vulnerable to artifacts associated withregression toward the mean, it is currently unclear whether the current findings are due toclinically significant changes.Strengths of the current study include its longitudinal research design, comprehensiveneuropsychological assessment, large sample size, and use of specific test measurementsrather than global scores. In addition, as very few differences were found between participantswho did and did not withdraw, these results can be viewed as relatively representative ofbreast cancer patients, although there will always be selection bias due to voluntaryparticipation in cognitive research. However, as this study focused on the acute effects ofchemotherapy, some potentially important factors were not assessed such as use of adjuvantendocrine treatment and chemotherapy induced menopause. In addition, whereas the ReliableChange Index is useful for investigating individual change, the high level of correlation andcomplexity within this kind of research may require more complex analyses to appropriatelycontrol for interrelationships, such as complex systems analysis.To further elucidate the relationships identified in this study, future studies comprising clinicalcontrol groups (such as patients with chronic diseases) are required. This is particularlyimportant as expected practice effects in the control group were not found in the current study,suggesting that other factors (disease or other treatments) can have subtle adverse effects oncognition in this population, even in the early stages. In addition, as causality between thesefactors and cognitive changes cannot be inferred in the current study as participants were notrandomized to conditions, investigators were not blinded, and correlations were used, theseresults should be hypothesis-building with future experimental studies required to furtherinvestigate these relationships. Furthermore, as prognostic variables such as estrogen receptorstatus came close to significance, it is recommended that a sample receiving morehomogeneous chemotherapy regimens should be studied to attempt to obtain a clearer view ofthe role of these factors.In conclusion, the current study demonstrates associations between objectiveneuropsychological performance and psychological and health factors over the time period ofchemotherapy administration that previously have not been reported by large studies with apre-chemotherapy assessment. In particular, as sub-clinical declines in hemoglobin werefound to significantly predict impairment on multiple neuropsychological tests, it is importantto monitor declines that are above the threshold for a blood transfusion. These findings mayhave important implications for identification of at-risk individuals as well as rehabilitation ofcognitive difficulties post-chemotherapy, with chemotherapy-induced anemia, fatigue, mood,and quality of life warranting further attention.

ACKNOWLEDGMENTS

We thank Drs. Toni Jones, Donna Spooner, and Elena Moody for their input in the design andimplementation of the study. Also, thank you to all oncologists, surgeons, and research nurseswho helped in the recruitment process, and the research assistants involved in recruitment anddata collection. The authors thank Professor Kerrie Mengersen and Dr. Jahar Choudhury for

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their statistical advice. Finally, the authors thank all the women who participated in the studyat such a distressing period in their life. No conflicts of interest were identified by the authors.This research was generously supported from various sources: The Wesley Research Institute(200320), the Cancer Council of Queensland and the National Breast Cancer Foundation(406900), and the Australian Research Council (LPO669670).

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Appendix B


This is co-authored paper with Katharine Vearncombe, a registered neruopsychologist, as firstauthor and Margaret Rolfe as second author, which has been submitted as an original article to”Menopause: the Journal of the North American Menopause Society” on 19 June 2009.This paper illustrates the aspect of the PhD training in collaborative interdisciplinary research.My contribution to the paper includes advice regarding the implementation of the statisticalmethods, development of initial syntax for running the statistical analyses in SPSS, overseeingthe writing of the statistical methods and results and undertaking a general editorial role.

Cognitive effects of chemical menopause and adjuvant endocrine treatment in earlybreast cancer

Authors: Katharine J. Vearncombe1, 2, Margaret Rolfe3, 4, Brooke Andrew1, 2, Nancy A.Pachana1, Margie Wright5 and Geoffrey Beadle5


ABSTRACT

Objective: To examine the effects of chemotherapy-induced (chemical) menopause andendocrine treatment on cognitive functioning in women with early breast cancer.Methods: The neuropsychological performance of 122 breast cancer patients scheduled toreceive chemotherapy was assessed pre-chemotherapy, one month and six monthspost-chemotherapy. Demographic, treatment and psychological information was alsocollected at each time point. Neuropsychological performance of 13 women who receivedendocrine treatment (tamoxifen or anastrozole) was assessed at similar time points.Results: This study was conducted in two stages, both of which were analysed using linearmixed modelling. The first stage investigated the cognitive effects of type of menopause (pre-,chemical, and post-menopause) and it was found that chemically menopausal women treatedwith endocrine treatment performed significantly worse than chemically menopausal womenwho did not receive endocrine treatment on an abstract reasoning task. The second stage

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236 APPENDIX B. COGNITION JOINT PAPER 2

evaluated the cognitive effects of type of treatment (chemotherapy only, chemotherapy plusendocrine, and endocrine treatment only). Patients receiving endocrine treatment only werefound to perform significantly worse on a measure of verbal learning than patients whoreceived chemotherapy. No other significant interactions remained in either experiment aftercorrection for multiple comparisons and covariates.Conclusions: There was little evidence to suggest that chemical menopause or endocrinetreatment significantly affected cognitive functioning acutely after treatment administration.However, as the majority of participants in the chemotherapy group had only commencedendocrine treatment by the final assessment, longer follow-up assessments are warranted toassess the long-term effects of combined chemotherapy and endocrine treatment.Keywords (6): Induced menopause, neurotoxicity, breast cancer, memory, tamoxifen,aromatase inhibitors.

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Cognitive effects of chemical menopause and adjuvant endocrinetreatment in early breast cancer

INTRODUCTION

Currently the effect of oestrogen on neurological functioning is poorly understood.Neurobiological research has found that oestrogen has many neuromodular andneuroprotective properties, and oestrogen receptors are found in many areas of the brainimportant to cognition including the hippocampus, amygdala and neocortex1. In addition,oestrogen replacement therapy has been associated with improvements in verbal memory,working memory and attention2,3, although results are inconsistent across studies1. Breastcancer treatment is also associated with changes in endocrine functioning, through eitherchemotherapy-induced menopause or use of adjuvant endocrine treatments that serve todeplete oestrogen levels. However, while cognitive dysfunction following administration ofbreast cancer treatments has been widely reported in the literature, there has been limitedinvestigation as to whether endocrine-related changes contribute to these cognitive difficulties.Chemotherapy-induced menopause, or chemical menopause, occurs in 70.5-77% of allpremenopausal women given cytotoxic treatment for breast cancer4. It involves an abruptdisruption of ovarian steroid production due to toxicity of cytotoxic drugs on ovarianfunction5 and has been associated with more favourable outcomes (e.g. reduced relapses andimproved survival) in younger breast cancer patients6. However, chemical menopause hasalso been linked with early development of osteoporosis and cardiovascular disease7. Inaddition, it has been postulated that chemical menopause may play a role in cognitivedysfunction after chemotherapy, but this issue has not been thoroughly investigated5.There are significant limitations in studies that have investigated the contribution ofmenopausal status to cognitive dysfunction after chemotherapy. First, most are cross-sectionaland retrospective, making it difficult to accurately determine timing and occurrence ofchemical menopause8−13. Second, three of these studies combined naturally postmenopausalwomen with women who had experienced a chemical menopause8−10, making the relationshipbetween cognitive dysfunction and induced menopause difficult to disentangle. Third, threestudies explored the impact of menopausal status indirectly by measuring menopausalsymptoms12−14. While measurement of menopausal symptoms is useful when consideringissues such as fatigue and quality of life, breast cancer patients treated with adjuvantendocrine treatment usually experience an increase in menopausal symptoms (particularlyvasomotor) regardless of whether they have already gone through the menopausal transition15.Therefore, these studies provide little information regarding the effect of chemical menopauseon cognitive functioning.To date, the effects of chemical menopause on cognitive functioning in early breast cancer hasbeen directly investigated in three studies. Jenkins and colleagues16 administered acomprehensive neuropsychological assessment to 85 women scheduled to receivechemotherapy for the treatment of breast cancer, of which 32 underwent a chemicalmenopause. Chemically menopausal patients were reported to be more likely to experiencedecline on multiple measures both 1 month and 6 months post chemotherapy completion (2.6times and 1.51 times respectively), although these declines were not statistically differentfrom women who were already postmenopausal at diagnosis. In addition, Hermelink andcolleagues17,18 assessed 101 patients with breast cancer before and during chemotherapy aswell as 92 patients approximately 7 months after completion of chemotherapy. In contrast tothe findings of Jenkins and colleagues16, they reported that chemical menopause was


associated with an improvement in verbal fluency performance compared to women whosemenopausal status remained stable. Finally, Schagen and colleagues19 assessed women priorto commencement and six months post completion of chemotherapy, with results indicatingno differences in cognitive performance between women who experienced chemicalmenopause and women whose menopausal status did not change.

While these three studies have not found any significant contribution of chemical menopauseon cognitive dysfunction, there are a number of methodological limitations that should beacknowledged. First, two of these studies compared women who underwent chemicalmenopause with a combined group of premenopausal and naturally postmenopausal breastcancer patients18,19. Combining the pre- and post-menopausal groups may mask groupdifferences due to the possible neuroprotective effects of oestrogen1. Second, it is unclearwhether these studies made the distinction between chemotherapy-induced ammenorhea (i.e.temporary cessation of menses due to chemotherapy) and chemotherapy-induced menopause(permanent cessation of menses). No definition of chemical menopause was provided inJenkins et al.16 while only a short-term follow-up assessment was conducted in the other twostudies18,19. As circulating oestrogen levels may be quite different between these two states, itis important to be able to differentiate between temporary and permanent cessation ofmenstruation through blood tests or longer follow-up assessments.

In addition to experiencing hormonal changes due to chemotherapy, endocrine treatments thatsuppress oestrogen production or conversion are indicated for women with oestrogen-receptorpositive breast cancers20. Two classes of drugs are usually prescribed in the treatment ofoestrogen-related breast cancer, namely selective oestrogen receptor modulators (SERMS e.g.tamoxifen) and aromatase inhibitors (AIs e.g. anastrazole or Arimidex). Tamoxifen is anoestrogen receptor antagonist in cells in the breast, while having mixed agonistic andantagonistic properties in other organs in the body20. It readily crosses the blood brain barrier,but little is known about its effect on oestrogen receptors in the brain21. On the other hand,AIs suppress tumour growth by inhibiting the conversion of androgens into oestrogen,effectively limiting the level of circulating oestrogen available in postmenopausal women22.Similar to tamoxifen, little is known about the potential neurological effects of AI’s21.

A number of studies have assessed the impact of endocrine treatment after chemotherapy,with mixed results. The majority of studies have found no significant association betweencognitive dysfunction and use of endocrine treatment14,16−18,23. However, a number of studieshave reported that women treated with both chemotherapy and endocrine drugs perform worseon cognitive tasks than other breast cancer patients24−26. Castellon et al.25 found that womenreceiving chemotherapy and endocrine treatment were significantly more impaired in thevisual memory, visuospatial and verbal learning domains compared to breast cancer patientstreated with surgery only. In addition, Bender et al.24 reported that while patients treated withchemotherapy only declined on verbal memory measures, those patients with combinedchemotherapy and tamoxifen experienced significant decline in performance on both verbaland visual memory tasks. Similarly, in the study of Collins and colleagues26 the performanceof chemotherapy patients receiving endocrine therapy was worse for measures of processingspeed and verbal memory than patients treated with chemotherapy only.

While the effect of endocrine treatment without concurrent adjuvant chemotherapy has rarelybeen studied, there is a small body of research that suggests that these drugs may have adetrimental effect on cognitive functioning. For example, Palmer and colleagues21 recentlyreported that 23 premenopausal women using tamoxifen performed significantly worse onvisual and verbal memory, verbal fluency, visuospatial and processing speed tasks thanage-matched healthy controls. In addition, Shilling, Jenkins, Fallowfield and Howell27 found

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that 94 women prescribed endocrine treatment (tamoxifen or anastrozole) were significantlymore impaired on tasks assessing verbal memory and processing speed than 35 healthycontrols. Finally, Bender and colleagues28 reported that women receiving anastrozole hadpoorer verbal and visual memory than women who received tamoxifen.The impact of endocrine-related changes due to breast cancer treatment on cognition is stilluncertain due to methodological limitations in the extant literature. Therefore, the aims of thisstudy were twofold: to investigate the cognitive consequences of 1) chemical menopause and2) adjuvant endocrine treatment. The cognitive status of women diagnosed with breast cancerwas assessed before commencing adjuvant chemotherapy, as well as one month and sixmonths after completion (and at similar times for participants receiving endocrine treatmentonly). It was hypothesised that women who experienced a chemical menopause afterchemotherapy treatment would decline more between time points in verbally based tasks thanthose who remained pre- or post-menopausal throughout the assessment period, and havesignificantly less improvement than those participants who remained premenopausal. Inaddition, it was expected that the combined effect of chemotherapy and endocrine treatmentwould be more detrimental to cognitive functioning in specific domains (verbal memory,visual memory and processing speed domains) than endocrine treatment or chemotherapyonly26,27.

Methods

Participants

Data are from the Cognition in Breast Cancer (CBC) study, which is longitudinally examiningthe causes of variation in cognitive functioning, health and well-being in women up to 2 yearspost-chemotherapy. However, as the current study is interested in the degree of decline in theacute phase of chemotherapy, only the first three assessments are analysed in the currentpaper. Eligible participants were required to be between 18 and 70 years old, proficient inEnglish, and have no previous history of cytotoxic drug treatment, neurological or psychiatricsymptoms or currently use medications that might affect neuropsychological test results. Allparticipants provided written, informed consent, and the study was approved by the followingethics committees; the Queensland Institute of Medical Research, the University ofQueensland, and all participating hospitals (the Wesley Hosptial, Royal Brisbane andWomen’s Hosptial, Redcliffe Hosptial, Princess Alexandra Hospital, the Mater Hospital, StVincent’s Hospital, and St Andrew’s Hospital).Two groups of early breast cancer patients were recruited from hospitals across south-eastQueensland, Australia; patients who were scheduled for chemotherapy treatment (with orwithout endocrine treatment and post-operative radiotherapy) and patients scheduled foradjuvant endocrine treatment only (with or without post-operative radiotherapy but nochemotherapy). Patients were approached by their oncologist or a research nurse afterdefinitive surgery, and those who agreed to participate received a phone call from apsychologist, who discussed the purpose and procedures of the study. The psychologist alsodiscussed the eligibility criteria, and those patients who were eligible and willing to participatewere scheduled to sign informed consent forms and complete the neuropsychologicalassessment battery (approximately 2.5 hours in duration). Neuropsychological testing wasadministered before commencement of chemotherapy (after definitive surgery), and at onemonth and six months after completion of chemotherapy, or at similar timepoints.One hundred and eighty-three participants scheduled to receive systemic breast cancer


treatment were recruited. Of these, 8 withdrew prior to the first assessment, 9 withdrew due toillness/ personal reasons, 8 had incomplete assessments, and 9 were not contactable forreassessment or did not finish treatment. The women who withdrew from the study did notdiffer from the rest of the sample in terms of age, education, estimated intellectualfunctioning, menopausal status, marital status, type of surgery, stage of cancer, or number ofchemotherapy courses. Neither did they differ on baseline measurements of anxiety,depression, fatigue or quality of life (QOL). However, it was found that women who withdrewperformed significantly worse on most of the cognitive measures at baseline compared towomen who did not withdraw from the study (data not shown). The final sample consisted of135 women (age M = 50.22, SD = 8.25, range = 25.25-67.92).

Measures

Neuropsychological tests and self-report measuresThe neuropsychological, mood and QOL measures used in the current study are presented inTable 7.1. The cognitive battery assessed a variety of different cognitive domains, i.e. verballearning/ memory, visual memory, cognitive and motor processing speed, as well as differentaspects of attention and executive function. As the tests utilised in the current research yieldmultiple outcome measures, Table B.1 also lists the 14 specific variables used in the analyses.Quality of life (QOL) was measured using the Functional Assessment of Cancer Therapy -General (FACT-G), along with the fatigue subscale. The FACT-G comprises 27 itemscovering four QOL domains, specifically physical, emotional, social/family, and functionalwell-being. The fatigue subscale comprises 13 items measuring the disruptiveness andintensity of fatigue, e.g. ”I feel listless (washed out)”. A higher score indicates moresatisfaction/ well-being and less fatigue on the QOL and fatigue scale respectively.Self-reported depression and anxiety was measured using the Hospital Anxiety andDepression Scale (HADS), a 14-item rating scale that screens for the extent and severity ofdepressive and anxious symptoms over the week prior to test administration. Separate scoresfor depressive and anxious symptomatology were calculated, with higher scores indicatinghigher levels of depression or anxiety.Age, education level (maximum 20 years) and an estimate of general cognitive ability (FullScale IQ- FSIQ) were collected as covariate information because these variables have beenpreviously found to affect performance on objective neuropsychological tests23. FSIQ wasestimated using the National Adult Reading Test, version 2 (NART-2), which is a validatedreading test41. Participants are required to read 50 irregularly spelt words, and accuracy ofpronunciation is used to predict FSIQ42.Clinical variables:Time-invariant treatment and health information was also collected. Time-invariant dataincluded stage of cancer, oestrogen receptor status (positive or negative), type of surgery(breast conserving or mastectomy), chemotherapy regimen, and number of chemotherapycourses. Due to small numbers in some groups, endocrine treatment (e.g. tamoxifen andanastrozole) was dichotomised into 2 levels (use/ no use). In addition, women were classifiedas pre-, chemical, or post-menopausal based on the four assessments of the larger study.Women were classified as premenopausal if they had regular, active menstruation throughoutchemotherapy or recovered cycles prior to the 18 months post completion assessment. Womenwere regarded as postmenopausal if they had not menstruated within the past 3 months priorto diagnosis. A classification of chemically induced menopause was determined by the patternof menstruation over a 2 year period, and comprised women who were premenopausal atdiagnosis (active menstruation) and became postmenopausal over the 18 months of follow up

241

Table B.1 Neuropsychological and self-report measures and outcome variables

Domain Measure Variables (abbreviation)

NEUROPSYCHOLOGICALVerbal Learning Auditory Verbal Learning Total number of words remembered in trials

Test29∗ 15 (AVLT)Total number of words remembered after a30 minute delay (AVLT8)

Visual memory a) WMS-IIIa Visual a) Total correct immediately after seeing eachReproduction immediate design (VR1)b) WMS-III Visual b) Total correct 30 minutes after being shownReproduction delayed designs (VR2)

Working memory WAIS-IIIb Backward Digit Total number of trials correctlySpan∗ completed (BDS)

Processing Speed Symbol Digit Modalities Test, Total number completed in 90 secondsoral version32 (SDMT)

Attention a) TEAc Visual Elevator∗ a) Total time taken per switch (TEA-VE)b) TEA Telephone Search∗ b) Total time taken without distraction. (TEA-TS)

Executive function a) WAIS-III Matrix Reasoning a) Total correct (MR)b) Stroop34 Total number correct in color word condition

(Stroop)c) DKEFSd Letter-number c) Total time taken to complete switching taskswitching task (L-N switching)

d) Controlled Oral Word d) Total number of words across phonemicAssociation Test36∗ verbal fluency condition (COWAT)e)DKEFSd Card Sorting e) Total correct in free-sorting conditionTask∗ (Card Sort)

Motor coordination Purdue Pegboard36 Total number of pegs constructed in assemblycondition. (PPassembly)

SELF-REPORT QOLFunctional Assessment of Total Physical well-being subscale scoreChronic Illness Therapy Total Emotional well-being subscale scoreBreast scale38 Total Social/Family well-being subscale score

Total Functional well-being subscale scoreFatigue Functional Assessment of Total Fatigue subscale score

Chronic Illness Therapyfatigue39

Mood Hospital Anxiety and Total depression scoreDepression Scale40 Total anxiety score

a WMS-III: Wechsler Memory Scale-Third Edition30, b WAIS-III: Wechsler Adult Intelligence Scale-Third Edition31,cTEA = Test of Everyday Attention 33,d DKEFS: Delis-Kaplan Executive Function Scale35

∗ Alternate forms used.


assessments (i.e. amenorrhea occurred during treatment and without recovery of menstruationpost chemotherapy).

Procedure

Participants were interviewed in a quiet room at a participating hospital or in their homes.Participants completed a demographic interview and neuropsychological assessment battery atthree time points: at baseline (after surgery but prior to commencement of chemotherapy -T1), approximately 1 month (T2), and 6 months post chemotherapy completion (T3). Each ofthe neuropsychological assessments was individually administered by psychologists (trainedat the postgraduate level) and all participants completed the test battery in the same order.Clinical information was collected before chemotherapy and at chemotherapy completion byclinical research nurses at the participant’s hospital.

Statistical Analysis

Statistical Package for Social Sciences (SPSS) for Windows, version 15 was used for allanalyses. Raw scores were used in the current analyses. Statistical inspection of the datarevealed three cases that were multivariate outliers, and were excluded from all analyses. Fivecognitive measures, comprising the subtests from the Test of Everyday Attention (TelephoneSearch and Visual Elevator), DKEF’s Number-Letter Switching, and the Visual Reproductionsubtests (1 and 2) did not conform to a normal distribution of residuals. These measures weretransformed, using log transformation for the TEA subtests and DKEFS Number-LetterSwitching, and square transformation for the Visual Reproduction subtests to achievenormality.Chemotherapy-induced MenopauseMenopausal status was divided into three levels (pre-, chemical, and post-menopausal).Women who had experienced surgical menopause (through bilateral oophorectomy) or whosemenopausal status could not be determined (e.g. due to hysterectomy) were excluded fromanalysis (n = 11). Independent groups t-tests and chi-square tests found no difference betweenpremenopausal women and women who experienced transient cessation of menstruation onany demographic or cognitive measure across all three time-points. Thus, these women werecombined in the premenopausal group to increase statistical power. The final sample consistedof 122 participants.Demographic and clinical information was compared using ANOVA or chi-squared asappropriate over the three groups, with statistical significance set at p=0.05. Linear mixedeffects modelling was performed in order to evaluate the effects of menopausal status, withtime random effects modelled by first-order autoregressive covariance. Repeated measuresANOVA’s evaluated the potential covariates (age, FSIQ, education, depression, anxiety,fatigue, QOL and use of endocrine treatment) with factors showing significant time and groupinteractions included in the model. Post hoc tests using the bonferroni adjustment were usedto investigate significant main effects and interactions.Endocrine treatmentType of treatment was divided into three levels (chemotherapy only, chemotherapy plusendocrine, and endocrine only). Demographic and clinical information were compared acrossgroups using ANOVA or chi-squared as appropriate over the three groups, with statisticalsignificance set at p=0.05. Linear mixed effects modelling was conducted to evaluate theeffects of treatment on cognitive performance over time, with time random effects modelledby first-order autoregressive covariance. Repeated measures ANOVA’s evaluated the potential

243

Table B.2 Demographic and treatment related characteristics of the menopausal groupsPremenopausal Chemical Postmenopausal

(n=26) menopause (n=41) (n=55) F/χ2

Mean (sd) Mean (sd) Mean (sd)

Age in years 39.72 (5.11) 47.13 (3.32) 56.22 (5.68) 109.00***FSIQ 108.50 (8.89) 112.54 (6.61) 110.29 (8.96) 2.02Years of Education 14.15 (3.83) 13.15 (2.98) 12.42 (3.56) 2.28Marital status 3.49

Singlea 2 10 8Marriedb 24 31 47

Stage of cancer 0.65I 8 9 14II/ III 18 32 41

Surgery 0.80Breast conserving 16 21 32Mastectomy 10 20 23

Use of Endocrine 5.33treatment- Time 3

None 10 6 17Anastrozole 3 13 19Tamoxifen 12 22 17Other 1 0 2

Chemotherapy 7.62∗regimen

FEC based 12 30 26CA based 14 11 29

Number of courses 2.733-5 5 3 106-8 21 38 45

∗ sig at p < 0.05 ∗∗∗ sig at p < 0.001aincludes divorced and widowed participants bincludes defacto couplesF = 5-Fluorouracil E = Epirubicin C = Cyclophosphamide A = Adriamycin

covariates (age, FSIQ, education, depression, anxiety, fatigue, and QOL) with factors showingsignificant time and group interactions included in the model. Post hoc tests using thebonferroni adjustment were used to investigate significant main effects and interactions.

Results

Effects of chemical menopauseOf the 122 patients scheduled to receive chemotherapy, 55 were postmenopausal at time ofdiagnosis. Of the remainder, 26 (21%) continued to report regular menses during thefollow-up period, although 14 (11.9%) reported transient cessation of menstrual activity.Forty-one (34%) developed permanent amenorrhea (12 months without menstruation) aftercompletion of chemotherapy. Demographic information for these three groups is provided inTable B.2. One-way ANOVA’s or chi-squared tests showed no significant differences betweengroups on estimated FSIQ, education, marital status, stage of cancer, type of surgery, ornumber of chemotherapy courses received. Endocrine treatment at Time 3 was examined asuse at Time 2 was negligible, and no difference was found between groups. However, themean age across groups was found to be different, with Games-Howell post-hoc testsrevealing significant differences in age between all groups (p<0.001). The chemicalmenopause group was also found to be significantly more likely to receive FEC-basedcytotoxic treatment than the other chemotherapy groups. This remained significant aftercorrection for multiple comparisons and participant age.


Table B.3 Means (M) and standard deviations (SD) for cognitive functioning measures at base-line (Time 1: T1), 1 month post chemotherapy (Time 2: T2) and 6 months postchemotherapy completion (Time 3: T3).

Cognitive variables Premenopausal Chemical menopause Postmenopausal

T1 T2 T3 T1 T2 T3 T1 T2 T3M (SD) M (SD) M (SD) M (SD) M (SD) M (SD) M (SD) M (SD) M (SD)

Verbal memory AVLT 54.88 52.81 53.46 52.76 49.59 50.85 51.53 48.69 48.96(6.91) (7.33) (8.39) (7.48) (7.79) (8.77) (7.04) (7.94) (8.08)

AVLT8 12.00 10.12 10.46 11.00 9.76 9.78 11.07 9.31 9.69(2.79) (2.23) (2.47) (2.30) (2.34) (3.00) (2.12) (2.64) (2.75)

Visual memory VR1 89.23 93.65 94.12 87.56 88.76 89.90 83.25 87.04 87.91(12.35) (7.73) (6.82) (10.93) (11.06) (9.13) (12.71) (9.85) (8.86)

VR2 77.58 85.50 87.96 70.23 75.07 81.63 59.93 68.44 75.36(20.22) (18.27) (15.24) (20.80) (22.44) (15.90) (23.42) (20.34) (18.78)

Working memory BDS 7.92 7.42 8.42 8.20 8.07 7.98 7.87 7.85 7.84(2.12) (1.94) (2.18) (2.22) (2.02) (2.14) (2.29) (2.30) (2.24)

Processing speed SDMT 63.96 67.15 67.38 60.54 62.44 64.78 55.00 56.25 57.58(7.54) (7.58) (6.75) (7.99) (8.09) (8.16) (8.70) (8.67) (8.88)

Attention TEA-VE 4.07 3.50 3.16 4.35 3.92 3.53 4.24 3.90 3.53(0.99) (0.65) (0.61) (0.89) (1.01) (0.73) (0.88) (0.70) (0.67)

TEA-TS 2.77 2.46 2.44 2.82 2.82 2.74 3.20 3.15 3.05(0.56) (0.38) (0.42) (0.41) (0.45) (0.36) (0.64) (0.53) (0.55)

Executive function MR 18.38 19.12 20.35 18.90 18.07 18.78 16.16 16.80 18.13(3.91) (3.12) (3.29) (3.32) (3.98) (3.90) (5.15) (4.94) (4.33)

Stroop 47.08 48.50 48.46 48.68 48.37 49.18 45.25 45.96 45.78(9.60) (8.43) (8.20) (9.03) (9.34) (8.97) (9.24) (8.78) (10.13)

N-L switching 52.67 49.49 47.90 61.04 59.88 57.22 71.38 68.03 67.66(9.33) (10.82) (7.76) (16.36) (16.21) (14.98) (20.11) (20.36) (18.91)

COWAT 44.12 46.77 46.15 47.27 49.02 48.22 41.05 42.15 43.60(12.51) (11.39) (11.03) (12.19) (11.59) (10.24) (11.73) (11.03) (11.37)

Card Sort 10.04 10.00 11.50 9.61 9.59 11.37 8.93 8.78 10.20(1.48) (2.38) (1.53) (1.91) (2.07) (2.20) (2.11) (3.03) (2.16)

Motor PPassembly 37.31 37.64 39.04 35.34 34.00 34.00 30.98 31.55 31.64(6.55) (7.13) (6.48) (5.40) (6.40) (5.92) (6.87) (6.82) (6.52)

Table B.3 presents the means and standard deviations for each cognitive measure for all threemenopausal groups. Significant time by group interactions were found for depression and useof endocrine drugs, thus these variables were included as covariates. Linear mixed modelling,controlling for endocrine use, depression and FSIQ showed significant interactions on onlytwo of 14 cognitive variables.A significant three-way interaction was found between time, endocrine use and menopausalstatus on Matrix Reasoning, a measure of abstract reasoning/ executive functioning, F(4,127)= 3.044, p=0.02. After correcting for multiple comparisons, post hoc tests found thatchemically menopausal women treated with endocrine drugs performed significantly worse atT3 than chemically menopausal women that were not treated with endocrine drugs (p<0.05,see Figure B.1. A significant difference in performance was also seen in the postmenopausalgroups at Time 1, with women who later had endocrine treatment performing significantlyworse at baseline than those not undergoing endocrine therapy (p=0.04). However, only thechemical menopause interaction remained significant after correction for age (p<0.04).Second, a significant group by time interaction was found for the Telephone Search subtest ofthe Test of Everyday Attention, a measure of selective attention and distractibility, F(4,121) =

3.22, p<0.02 (see Figure B.2). After correction for multiple comparisons, the post hoc testsshowed no significant changes in the chemical menopause or postmenopausal groups over thethree assessment points. However, premenopausal women showed significant improvementbetween T1 and T2 (p< 0.001), performing significantly better than the other groups at bothT2 and T3 (p< 0.02 to 0.001). Postmenopausal women also performed significantly worsethan the other two groups at T2 and T3 (p<0.01 to 0.001), while chemically menopausalwomen performed intermediately between the pre and postmenopausal women. However, this

245

05

1015

2025

Assessment time

Mat

rix R

easo

ning

(to

tal c

ount

)

T1 T2 T3

chemical menopause without endocrinepostmenopausal without endocrinechemical menopause with endocrinepostmenopausal with endocrine

Figure B.1 Interaction between menopausal status, endocrine treatment and time (pre-menopausal data not shown).

interaction became non-significant after correction for age.Inspection of effects of time showed significant cognitive decline in only two measure, namelyverbal learning on the Auditory Verbal Learning task (AVLT), F(2, 140) = 10.82, p<0.001 andthe delayed recall trial on the Auditory Verbal Learning Task (AVLT8), F(2, 141) = 31.32,p<0.001. All groups showed significant decline between T1 and T2, with no significantchange between T2 and T3. All other cognitive variables were either stable or demonstratedan improvement in performance over time. Finally, differences in performance betweenwomen of different menopausal status were seen in a number of tasks, with premenopausalwomen performing significantly better in SDMT, TEA-TS, VR1, VR2, PPassembly, N-Lswitching and Card Sort than postmenopausal women. The performance of chemicallymenopausal women was at a level between the other two groups.Endocrine treatment The effect of type of treatment on cognitive functioning was evaluated,with women classified as receiving chemotherapy only, chemotherapy plus endocrinetreatment, and endocrine treatment only (demographics are shown in Table B.4). Thirteenwomen received endocrine treatment only, the majority (77%) of whom were postmenopausalat the time of diagnosis.A one-way ANOVA revealed a significant difference in age between groups, withGames-Howell post hoc comparisons indicating that patients receiving endocrine treatmentonly were significantly older than chemotherapy only patients (p<0.05). In addition,endocrine only patients were significantly more likely to have stage one cancers. However, asstage of cancer is an indication of severity/ aggressiveness, differences on this variable areexpected as it is a determinant for recommendations about the administration of adjuvantchemotherapy.Repeated measures ANOVAs did not show any significant interactions between type of


Table B.4 Demographic and treatment related characteristics for each of the systemic treatmentgroups

Chemotherapy Chemotherapy Endocrine F/χ2

only plus Endocrine only(n =36) (n = 97) (n=13)

Mean (sd) Mean (sd) Mean (sd)

Age in years 48.39 (8.60) 49.93 (7.70) 55.56 (7.48) 3.96∗FSIQ 109.92 (8.52) 111.18 (8.07) 111.00 (12.46) 0.28Years of Education 13.53 (3.87) 12.88 (3.21) 12.38 (3.33) 0.71Marital status 2.23Singlea 6 14 4Marriedb 30 83 9Baseline menopausal status 13.06∗

Premenopausalc 23 68 3Postmenopausal 13 17 10Unknown 0 2 0

Stage of cancer 10.78∗∗I 11 24 9II/ III 25 73 4

Surgery 4.44Breast conserving 23 52 10Mastectomy 13 45 2Unknown 0 0 1

Type of Endocrine treatmentAnastrozole - 38 7Tamoxifen - 56 6Other - 3 0

Chemotherapy regimen 0.05FEC based 20 56 -CA based 16 41

Number of courses 0.053-5 5 15 -6-8 31 82

∗ sig at p < 0.05 ∗∗ sig at p < 0.01aincludes divorced and widowed participantsb includes defacto couples c includes perimenopausalF = 5-Fluorouracil E = Epirubicin C = Cyclophosphamide A = Adriamycin

247

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Assessment time

Log

tran

sfor

mat

iono

f TE

A te

leph

one

sear

ch

T1 T2 T3

premenopausalchemical menopausepostmenopausal

Figure B.2 Significant time by menopausal group interaction (across time) for the telephonesearch subtest.

treatment and psychological variables (anxiety, depression, fatigue, and QOL). Therefore,only age and FSIQ were included as covariates in the linear mixed model analyses. Table B.5presents the results for the Time 1 to Time 3 objective cognitive measures for each treatmentgroup. A significant time by treatment interaction was found in AVLT, a measure of verballearning, F(4, 161) = 2.93, p=0.02. The results of post hoc tests showed that patients treatedwith endocrine treatment only performed significantly worse than women treated withchemotherapy only at Time 2 (p=0.21) and worse than both chemotherapy groups at Time 3(p<0.01 and p=0.01 for chemotherapy only and chemotherapy plus endocrine treatmentrespectively). No differences were seen between chemotherapy groups, although a significantdecline was observed in the chemotherapy plus endocrine group over time (see Figure B.3).Time by type of treatment interactions were also found for both attentional measures,although these did not remain significant after correction for multiple comparisons.

Discussion

Chemical (or chemotherapy-induced) menopause and adjuvant endocrine treatment arepurported to be important mechanisms underlying cognitive dysfunction occurring after breastcancer treatment. However, the results of the current study suggest these factors seem to havelittle effect on cognitive functioning. The investigation into chemical menopause revealedonly one significant interaction, suggesting that women who experienced a chemicalmenopause were not more likely to experience cognitive decline than pre- or post-menopausalwomen in most areas of cognitive functioning. This is consistent with the research of Jenkinset al.16 and Schagen et al.19, but not with that of Hermelink et al.18, who reportedimprovements in verbal fluency after chemical menopause. In the present study, a measure of


Table B.5 Means (M) and standard deviations (SD) for cognitive functioning measures at base-line (Time 1: T1), 1 month post chemotherapy (Time 2: T2) and 6 months postchemotherapy completion (Time 3: T3).

Cognitive variables Chemotherapy alone Chemotherapy and Endocrine Endocrine aloneT1 T2 T3 T1 T2 T3 T1 T2 T3

M (SD) M (SD) M (SD) M (SD) M (SD) M (SD) M (SD) M (SD) M (SD)

Verbal memory AVLT 53.64 51.28 51.64 52.40 49.47 49.96 50.77 43.31 41.69(6.01) (6.51) (7.95) (7.36) (8.17) (8.52) (10.05) (7.44) (9.64)

AVLT8 11.92 9.78 10.25 11.06 9.73 9.75 10.08 9.15 8.08(2.12) (2.23) (2.45) (2.38) (2.58) (2.84) (1.94) (1.35) (3.12)

Visual memory VR1 87.58 92.03 91.44 85.49 87.97 89.59 80.15 79.62 81.08(11.48) (8.63) (8.11) (12.22) (10.25) (8.95) (9.35) (15.46) (9.91)

VR2 69.09 75.67 83.64 66.43 74.44 79.15 66.00 67.62 73.42(23.71) (20.25) (16.23) (21.92) (21.16) (17.68) (19.76) (19.07) (13.50)

Working memory BDS 7.86 7.67 7.72 7.92 7.84 8.04 6.46 6.77 6.92(2.40) (1.66) (2.25) (2.13) (2.23) (2.16) (2.11) (2.62) (1.98)

Processing speed SDMT 60.83 63.42 64.31 58.13 59.81 61.52 57.15 57.38 61.15(9.84) (8.76) (8.38) (8.37) (9.17) (9.26) (8.98) (6.40) (6.24)

Attention TEA-VE 3.96 3.78 3.23 4.32 3.82 3.53 3.80 3.85 3.26(0.68) (0.71) (0.57) (0.99) (0.84) (0.71) (0.54) (1.01) (0.63)

TEA-TS 3.08 2.82 2.67 2.96 2.93 2.88 2.92 2.91 2.91(.075) (0.57) (0.51) (0.51) (0.53) (0.50) (0.37) (0.47) (0.23)

Executive function MR 18.22 18.39 18.97 17.51 17.60 18.78 14.54 15.23 14.62(3.74) (3.80) (3.85) (4.70) (4.46) (4.01) (3.73) (3.19) (2.36)

Stroop 46.36 48.17 47.50 46.91 46.96 47.74 44.46 46.00 46.25(8.55) (8.83) (7.98) (9.27) (8.75) (10.19) (10.08) (9.65) (9.77)

N-L switching 62.46 57.84 58.68 64.01 61.94 59.95 76.69 72.29 65.80(21.07) (12.99) (16.85) (18.52) (19.98) (17.77) (21.11) (24.75) (20.22)

COWAT 39.75 42.31 43.11 45.19 46.58 46.59 44.46 45.77 46.92(11.31) (10.28) (10.64) (12.75) (12.42) (11.29) (15.35) (14.67) (15.11)

Card Sort 9.50 9.19 10.92 9.42 9.44 10.75 9.62 9.92 11.17(1.83) (2.84) (2.06) (2.00) (2.51) (2.16) (2.26) (1.80) (2.25)

Motor PPassembly 35.28 35.59 36.03 33.25 33.27 33.36 30.92 30.38 31.67(7.21) (7.03) (6.75) (6.81) (7.07) (6.91) (6.37) (6.74) (5.85)

010

2030

4050

60

Assessment time

Per

form

ance

on

AV

LT o

ver

five

tria

ls

T1 T2 T3

chemotherapychemotherapy and endocrineendocrine

Figure B.3 Performance on the Auditory Verbal Learning test (total recalled over 5 trials) indifferent treatment groups (over time).

249

abstract reasoning was the only one to show a significant interaction with menopause aftercontrolling for effects of age, with women who experienced a chemical menopause and werealso treated with endocrine drugs performing significantly worse than those women withchemical menopause without endocrine treatment. However, the number of women in thelatter group was very small (n = 7), therefore the robustness of these results are uncertain.

In addition, there is no evidence to suggest that the combination of chemotherapy plusendocrine treatment had a cumulatively detrimental effect on cognitive functioning. The lackof significant differences between the two chemotherapy groups is supported by the majorityof studies investigating the effects of endocrine treatment within patients who have alsoundergone chemotherapy14,16,18,23. However, two longitudinal, prospective studies have alsofound support for increased cognitive dysfunction in chemotherapy patients also receivingendocrine treatment24,26, although one of these comprised a very small sample24. Differencesbetween these studies may be resolved by assessing time since commencing endocrinetreatment, as these drugs are associated with vasomotor symptoms (such as hot flushes) thatcan disrupt sleep, particularly during the first year of administration15.

The current findings suggest that women who receive endocrine treatment only are morelikely to experience verbal memory difficulties over time than patients treated withchemotherapy. However, as the endocrine group in the current study comprised only 13patients and this result is not consistent with the findings of previous research26, therobustness of this finding may be questionable. In addition, the effects of different types ofendocrine drugs could not be determined due to statistical power considerations. Hence, theseresults should be considered as preliminary and hypothesis-building.

Overall, most cognitive variables demonstrated significant improvements over time,presumably due to practice effects. Previous research has indicated decline in a number ofdomains, including processing speed, verbal memory and working memory43,44. However, thepresent results indicate that cognitive dysfunction after chemotherapy may be quite specific,with decline only seen within the verbal memory domain, which is supported by recentresearch24. This domain-specific cognitive dysfunction is becoming increasingly recognisedas more longitudinal studies containing a pre-chemotherapy assessment are conducted.However, to suggest that cancer diagnosis and treatment have no effect on cognitivefunctioning is premature, as significant rates of cognitive impairment have been reported instudies comparing breast cancer patients to healthy women or normative data18−10,13−14,19.Only one study to date has included patients with another chronic disease, namely cardiacdisease, and this study was limited by a small sample size45. However, Mehlsen et al.45

revealed interesting and previously unreported relationships between fatigue, quality of lifeand cognitive functioning. Therefore, to further elucidate relationships between cancer-relatedcognitive decline and decline related to other factors (e.g. depression due to chronic illness),comparisons with more varied clinical control groups are required.

The current study has a number of limitations. First, although the overall sample size is one ofthe largest to date, some of the groups contained small numbers. Second, the confounding ofage with both chemical menopause and use of endocrine treatment complicates statisticalinterpretation; older premenopausal women were more likely to undergo a chemicalmenopause than younger premenopausal women, and the cognitive domains postulated to beaffected by induced menopause may also be vulnerable to age effects. Age effects are alsoinherent in types of treatment recommended, as breast cancer in younger women is generallymore aggressive indicating the need for chemotherapy. The current study attempted toovercome these difficulties using statistical methods that are designed to incorporatecorrelated data46, although this approach lacks the ability to look at individual change that


may be required to determine characteristics of at-risk individuals. Third, there weredifferences between participants who withdrew and those who remained in the study,suggesting that these results may not be generalisable to the wider population.Previous research has suggested that a subgroup of breast cancer patients experience cognitivedysfunction after chemotherapy. The results of the current study suggest that these declinesare not specifically associated with mechanisms related to change in endocrine functioning, atleast in the acute phase of treatment. However, the long-term cognitive consequences of thesefactors, particularly adjuvant endocrine treatment, still need to be evaluated. In addition,future research should address other possible mechanisms that may be associated with thisdecline, particularly psychological variables that are associated with chronic life-threateningillnesses, genetic vulnerability to cognitive decline after chemotherapy, and neurotoxicity ofdifferent chemotherapeutic regimens.

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Acknowledgements

This research was generously supported from various sources: The Wesley Research Institute(200320), the Cancer Council of Queensland and the National Breast Cancer Foundation(406900), and the Australian Research Council (LPO669670). We would like to thank DrsToni Jones, Donna Spooner, and Elena Moody for their input in the design andimplementation of the study. Also, thank you to all oncologists, surgeons, and research nurseswho helped in the recruitment process, and the research assistants involved in recruitment anddata collection. The authors would like to thank Professor Kerrie Mengersen and Dr Jahar

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Choudhury for their statistical advice. Finally, the authors would also like to sincerely thankall the women who participated in the study at such a distressing period in their life. Noconflicts of interest were identified by the authors.

Appendix C


This is co-authored paper with Katharine Vearncombe, a registered neruopsychologist, as firstauthor and Margaret Rolfe as third author, which has been submitted as an BriefCommunication to the Journal of the International Neuropsychological Society July 2009 forpublication.This paper illustrates the aspect of the PhD training in collaborative interdisciplinary researchin the Cognition in Breast Cancer Study. My contribution to the paper includes adviceregarding the implementation of the statistical methods, development of initial syntax forrunning the statistical analyses in SPSS, overseeing the writing of the statistical methods andresults, contributing to the structure of the paper and undertaking a general editorial role.

Evaluating methods of detecting cognitive change using a sample of breast cancerpatients.

Authors: Katharine J. Vearncombe1, 2, Nancy A. Pachana1, Margaret Rolfe3, 4, KerrieMengersen4, And Geoffrey Beadle5


ABSTRACT

Background: The Reliable Change Index (RCI) is the most commonly used individual changemethod for investigating cognitive changes after chemotherapy for breast cancer. However,the RCI has many limitations and change may be more appropriately investigated using othermethods. This study compared three change analytic techniques, namely standardisedpercentage change scores (PCS), RCI and RCI corrected for practice (RCIp) and astandardised regression-based approach (SRB). Methods: Neuropsychological performance of139 women was assessed pre- and one-month post-chemotherapy. A non-chemotherapy breastcancer control group (N=21) was assessed at similar time-points. Findings: The morecomplex methods found significantly greater levels of impairment, but also generated more

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256 APPENDIX C. COGNITION JOINT PAPER 3

extreme scores (> ±3SD). Conclusions: When only a small, adequately matched controlgroup is available, simple individual change analyses provide better estimates of levels ofimpairment. This research highlights the importance of sufficiently sized control groups forgreater accuracy of common analytic methods and more informative findings.Keywords: Chemotherapy, cognitive impairment, regression, percentage change, reliablechange index.

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Evaluating methods of detecting cognitive change using a sample ofbreast cancer patients.

INTRODUCTION

Recent research has generally supported the assertion that breast cancer treatment isassociated with cognitive difficulties. Both cross-sectional and longitudinal studies have foundsuch declines, with the proportion of women experiencing impairment ranging from 15-50%(Vardy & Tannock, 2007). While the cause of these difficulties is still contentious, it has beenpostulated that results may be due to differences in chemotherapy regimen or other factorsincluding endocrine treatment, depression, fatigue, type of surgery, general anaesthesia andgenetic predisposition (Hurria et al., 2007). However, variation in results may also be partlyattributed to differences in the method of analysis (Shilling et al., 2006).Determining significant levels of change has always been problematic, as test results willinevitably differ over repeated administrations even in people who have not experiencedcognitive change. Normal variations in performance are reflected in imperfect test-retest/alternate forms reliability, practice effects, test-retest interval length, regression towards themean, familiarity with the testing situation, and state changes in psychological factors (such asaffect, motivation and arousal). Detecting significant change is further complicated by thelimited amount of normative change data and the lack of an accepted definition of cognitiveimpairment or decline.Traditional statistical methods investigating mean differences may mask change when only asmall subgroup is affected (Trster et al., 2007). However, a number of statistical methods havebeen proposed to evaluate individual change, three of which will be the focus for the currentstudy, namely Percentage Change Scores (PCS), the Reliable Change Index (with and withoutcorrection for practice - RCIp and RCI) and standardised regression-based scores (SRB). Thesample sizes of control groups within the breast cancer literature are generally small, with themajority of control groups comprising less than 36 people (range =12-100). As these threemethods differ in the degree to which they rely on a control group in their calculation, it isappropriate to investigate which method of analysing change provides the most robust resultsbased on small control groups.The simplest method for determining change is standardised percentage change scores (PCS).PCS’s are commonly used in clinical trials and significant change is usually determined byusing an arbitrary cut-off of more than 50% decline or improvement (Jensen et al., 2003). Thisapproach does not utilise information from a control group; instead PCS is typicallycalculated by subtracting the second assessment score (or treatment outcome score) from theinitial score, then dividing by the initial score (x 100). To make it comparable to the othermethods, the PCS has been standardised (converted into a z score) by dividing the percentagechange score by the standard deviation of the percentage change score.This method controls for initial performance, which has been found to be highly correlatedwith subsequent performances (Temkin et al., 1999). However, PCS does not control for othersources of error, such as practice effects or regression to the mean.The Reliable Change Index (RCI) also evaluates significant individual change and is thepredominant method used in the breast cancer literature. The RCI utilises test-retest reliabilityto establish whether the change between baseline and follow-up scores is significant. Achange score is typically considered significant if it lies outside the 90% confidence interval.Although there are number of different formulae for the calculation of the RCI, two variationswere used: the formula revised by Christensen and Mendoza (1986) and the RCI corrected for


practice effects (RCIp) proposed by Chelune and colleagues (1993). These formulae can bewritten as:

RCI + practice = (S Edi f f )(±1.64) + practice effect

Definitions and Formulae for Reliable Change Indices

S Edi f f =√

2(S E)2

S E = S D√

1 − rxx

S D = Standard deviation from published norms

rxx = Reliability coefficient from published norms.

Practice effect = Mean difference between the follow-up and baseline scorein the breast cancer control group.

The RCI method has been criticised for being too conservative (i.e. producing wideconfidence intervals) as well as requiring similar variances on both testing occasions andhaving poor prediction accuracy, although prediction accuracy is vastly improved with thecorrections for practice effects (Temkin et al., 1999).The third method of determining change is a standardised regression-based approach (SRB).While there are multiple ways of calculating SRB, this approach is concerned with examiningthe difference between the participant’s actual score (obtained score) and the score predictedby the regression equation (Crawford & Garthwaite, 2006). The method utilised by Temkinand colleagues (1999) in which linear regression of the control group’s retest scores on theirinitial scores is used to generate a regression equation (i.e. slope and intercept) was adopted.This prediction equation is then applied to the chemotherapy group and the predicted scoresare compared to the obtained scores (Temkin et al., 1999). An obtained retest score isconsidered significantly different if the difference between the observed and predicted valuesfalls outside the 90% confidence interval calculated from the control group.SRB corrects for both practice and regression towards the mean, and it has been found to bemore accurate than the RCI in detecting meaningful cognitive change in areas such assports-related concussion and epilepsy (e.g. McCrea et al., 2005; Temkin et al., 1999). It hasalso been utilised to identify cognitive impairment in a single study in the breast cancerliterature, with covariates also being included in the regression model (Stewart et al., 2008).However, it is recognised that the SRB approach may be problematic when the regressionequation is built using a small sample (Crawford & Garthwaite, 2006).The fact that these three methods differ in the extent to which they rely on a control group isimportant as research examining cognitive dysfunction after breast cancer treatment isnotorious for small sample sizes. Although PCS does not rely on a control sample, the RCIrequires control-group-based calculations of test-retest reliability (and practice effects), andthe SRB approach relies on a control group for calculation of its numerator, denominator andestimation of the difference.The detection of cognitive impairment is important for determining the neurocognitive effectsof cancer treatments. While the majority of longitudinal breast cancer studies have utilised theRCI, there is evidence that regression-based methods may be more appropriate. In addition,the PCS does not control for sources of measurement error, while the RCI controls for errorsdue to test-retest reliability and practice, and the SRB method accounts for practice and

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regression towards the mean. With these strengths and limitations in mind, the aim of thecurrent study was to compare the three different change methods (i.e. PCS, RCI and RCIp,and SRB) that rely differentially on a control group when only a small control sample isavailable, as is typical in such studies. The null hypothesis was that these methods would haveequivalent performance. However, given the small sample size in the current research, it waspostulated that the SRB would produce more extreme scores and hence less accurate changedata.

Methods

ParticipantsData are from the ”Cognition in Breast Cancer” study, a longitudinal study examining thecauses of variation in cognitive functioning in breast cancer patients. Participants were 18-70years old, proficient in English, had no history of chemotherapy, neurological or psychiatricsymptoms or use of psychotropic mediations. All participants provided written, informedconsent, and this study was approved by the following ethics committees; the QueenslandInstitute of Medical Research, the University of Queensland, and all participating hospitals.Two groups of early breast cancer patients were recruited; chemotherapy recipients andpatients scheduled for other forms of treatment (i.e. endocrine treatment and/ or radiotherapy).

MeasuresA sample of the neuropsychological measures was included in the current study forillustration purposes, namely; the Auditory Verbal Learning Test (learning, delayed recall),Visual Reproduction (WMS-III; immediate, delayed, recognition), Symbol Digit ModalitiesTest (SDMT), Stroop test, Card Sorting task (DKEFS) and the Controlled Oral WordAssociation Test (COWAT).

Self-reported depression and anxiety was measured using the Hospital Anxiety andDepression Scale (HADS). Age, education level (maximum 20 years) and general cognitiveability (FSIQ) were collected as these variables have been found to affect neuropsychologicaltest performance (Schagen et al., 2002). FSIQ was estimated using the National AdultReading Test (NART).

ProcedureParticipants completed a neuropsychological assessment battery at two time points:chemotherapy patients completed an assessment at baseline (after surgery but prior tocommencement of chemotherapy - T1) and approximately 4 weeks post-chemotherapycompletion (T2). The non-chemotherapy group completed the same assessment at similartimepoints, namely 2-3 weeks post surgery (T1) and 6 months post surgery (T2).

Data AnalysisStatistical analyses were performed using the Statistical Package for the Social Sciences(SPSS) version 15. Differences in demographic characteristics and baseline cognitiveperformance between groups were assessed using independent group t-tests and chi-squaredanalyses. Differences between baseline and follow-up cognitive outcome measures weredetermined using paired t-tests, for each group. The three methods of evaluating individualchange were performed (PCS, RCI and RCIp, SRB) and McNemar tests were used to evaluatelevel of impairment. Due to the number of comparisons, a significance cut-off of p<0.01 wasused.


Table C.1 Means and standard deviations for each of the cognitive measures for both thechemotherapy and non-chemotherapy groups.

Chemotherapy Non-Chemotherapy

Time 1 Time 2 Time 1 Time 2Domain Measure Mean (SD) range Mean (SD) range t Mean (SD) range Mean (SD) range t

Verbal memory AVLT learning 52.09 (7.48) 36 49.39 (8.61) 54 4.57** 51.19 (9.23) 30 46.90 (8.58) 28 2.21AVLT delayed recall 11.14 (2.40) 11 9.59 (2.64) 14 7.75** 10.62 (2.13) 8 9.57 (1.83) 7 2.06

Visual memory VR immediate 85.36 (11.99) 50 87.99 (10.95) 51 -3.15* 81.81 (11.35) 45 82.24 (15.61) 67 -0.15VR delayed 66.32 (22.73) 95 73.91 (20.78) 96 -4.97** 64.86 (20.17) 65 68.43 (16.87) 60 -0.91VR recognition 44.82 (2.35) 13 45.55 (2.40) 10 -3.74** 44.76 (2.49) 9 45.29 (2.00) 7 -1.14

Processing speed SDMT 58.24 (9.21) 45 59.99 (9.47) 46 -3.56** 56.43 (7.49) 30 58.38 (6.31) 20 -1.52Executive function Stroop 46.33 (9.22) 42 46.80 (8.55) 42 -1 44.00 (8.60) 32 46.86 (9.71) 40 -1.71

Card Sort 9.36 (1.93) 12 9.27 (2.65) 13 0.53 9.90 (2.02) 7 9.86 (1.59) 6 0.14

Range = minimum score - maximum score∗ sig difference p < .01; ∗∗ sig. difference p < .001

Results

Analysis of Group DataThe two groups were reasonably comparable in terms of demographic characteristics: therewere no significant differences in age (chemotherapy M=49.43, SD=7.99; non-chemotherapyM=53.98, SD=8.24), education (chemotherapy M=13.04, SD=3.34; non-chemotherapyM=13.52, SD=3.94) and baseline FSIQ (chemotherapy M=110.71, SD=8.27;non-chemotherapy M=112.62, SD=10.76). However, the non-chemotherapy group wassignificantly more likely to be postmenopausal (χ2=10.32, p<0.001), have stage 1 cancers(χ2=23.25, p<0.001) and have a longer retest interval (χ2=-6.46, p<0.001). However, as stageof cancer is an indication of severity/ aggressiveness, differences on this variable are expectedas it is a determinant for chemotherapy recommendations. Independent groups t-tests wereperformed to compare baseline cognitive performance, anxiety and depression. No significantgroup differences at the p<0.01 level were observed at T1, suggesting that the two groupswere adequately matched (data not shown).

Predicting retest scoresThe means and standard deviations of the scores at each time-point for the chemotherapy andcontrol groups are shown in Table C.1. Paired t-tests found the difference between T1 and T2in the chemotherapy group to be significant on six of the measures, with significant declinefound in verbal memory, and significant improvements found in the visual memory andprocessing speed domains. Paired t-tests found no significant differences between T1 and T2in the control group for any measure. However, all of the ranges of the cognitive outcomemeasures (maximum minus minimum score) were smaller at baseline in the control groupcompared to the chemotherapy group.

Comparison of change methodsThe results of all three approaches are shown in Table C.2. The PCS analysis detectedsignificantly less total impairment than both the RCIp (McNemar test P=0.001) and SRB(McNemar test P<0.001), although this method was not significantly different from RCI. Inaddition, RCI identified significantly less overall impairment than SRB (McNemar testP=0.001). There was no difference between levels of total impairment identified throughRCIp and SRB. Inspection of the affected cognitive domains also varied between methods,with values for several tests (particularly on the Card Sorting, and Stroop tasks) for the morecomplex methods (SRB and RCIp) being quite extreme (> ±3SD; see Figure C.1).

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Table C.2 Number of participants classified as not impaired (no decline) and impaired (decline)by the three methods using the 90% confidence interval cut-off.

Domains Measures PCS RCI RCI p SRBdecline no decline decline no decline decline no decline decline no decline

Verbal memory AVLT learning 12 127 7 132 17 122 5 134AVLT delayed recall 19 120 14 125 27 112 17 122

Visual Memory VR immediate 1 138 2 137 2 137 0 139VR delayed 0 139 2 137 0 139 7 132VR recognition 3 136 6 133 2 137 16 123

Processing speed SDMT 0 139 0 139 0 139 9 130Executive Function Stroop 2 137 2 137 0 139 13 126

Card Sort 8 131 22 117 22 117 28 111Total cases that declined 36 103 46 93 54 85 67 72

Percentage impaired 26.4 33.8 39.7 49.3

Method of change analysis

RCIpRCIPCS

Dis

trib

uti

on

of

z sc

ore

s

5.0

2.5

0.0

-2.5

-5.0

Figure C.1 Standardised distribution of change on DKEFs Card Sorting Task for the three dif-ferent change methods (PCS, RCI, RCIp, SRB).


Discussion

The methods that least relied on a control group identified the lowest number of impairedcases. However, these scores may be the most accurate estimate of impairment when only asmall control group is available. This study found that the more complex statistical methodsrequire a more robust sample size to ensure accuracy.In addition, the change identified by SRB (and to a lesser extent, RCIp) contained someextreme (> ±3SD) scores. While these methods may be more accurate than the RCI and PCSin accounting for practice effects, changes following chemotherapy have found to be quitesubtle in meta-analyses (Stewart et al., 2006). Again, the wisdom of using the SRB and RCIpmethods are called into question when only a small control sample is available.The results of SRB analyses in the current sample are contrary to previous research findings,which have generally supported the use of the SRB method for detecting significant change inneuropsychological measures (e.g. McCrea et al., 2005; Temkin et al., 1999). The SRBmethod has a number of theoretically important advantages such as controlling for practiceeffects, regression to the mean and ability to include covariates that should make this methodsuperior to simpler methods. In the current study, poor performance of SRB in predictingchange is most likely due to the small sample size of the control group. While the controlgroup appeared adequately matched to the chemotherapy group, ranges on many of thecognitive variables in the control group were restricted, thus not allowing an accurateprediction equation to be generated for scores outside this range in the chemotherapy group.Only one other author has reported using the SRB method with breast cancer patients (Stewartet al, 2008); however, they had one of the largest control groups to date (N=51) in thisliterature. The sample size in the current study, while relatively modest, is comparable tomany other studies of cognitive change after chemotherapy for breast cancer and hence isrelevant in the evaluation of different change methods.As well as the exploration of different change methods, the results of this study alsoemphasizes two important points regarding the choice of control groups for future researchinto cognitive change post-chemotherapy. First, the breast cancer control group demonstratedsmall non-significant declines between assessments on neuropsychological measures. Thelack of significant practice effects on measures in the control group is reasonable due tocancer-treatment-related changes (e.g. radiotherapy-related fatigue) and supports the moveaway from using healthy controls as the normative sample in cancer studies. Second, asignificantly longer test-retest interval in the control group has implications for the level ofchange detected in the RCIp and SRB methods, with level of significant change in thechemotherapy group possibly an overestimation of true change (McCaffrey et al., 2000).However, previous research has found no significant differences in practice between 2-16months on some neuropsychological measures (Temkin et al., 1999), suggesting that practiceeffects may not decrease greatly over the time periods investigated in the current study.Therefore, methods utilising the control group to account for practice are deemed to be a moreappropriate method for detecting individual change compared to those methods in which noattempt to correct for practice is made.Studies within the breast cancer literature utilise varying statistical methods, makingcomparisons between studies difficult. Determining the most accurate method of prediction ofchange post-chemotherapy is the first step in disentangling potential causes for cognitivedecline found in previous research. When only a small control sample is available, simplermethods are recommended. On the other hand, SRB, used with an adequate control group,may provide not only a more accurate prediction model, but also a highly effective method ofexamining possible factors associated with cognitive change.

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Acknowledgements

This research was generously supported by the Wesley Research Institute (200320), theCancer Council of Queensland and the National Breast Cancer Foundation (406900) and theAustralian Research Council Linkage Project (LPO669670). The authors would also like to


thank Brooke Andrew and Dr Jahar Choudhury for their comments.

Appendix D

Cognition Joint Poster

Presented is the abstract of a co-authored poster with Dr Geoffrey Beadle, as first authorwhich was presented at the San Antonio Breast Cancer Symposium December 13-16 2007. DrBeadle in his capacity as the consulting oncologist is one of the Principal Investigators withthe Cognition in Breast Cancer Study being conducted by the Wesley Research Institute,Brisbane.This abstract illustrates the aspect of the PhD training in collaborative interdisciplinaryresearch in the Cognition in Breast Cancer Study. My contribution to the poster included thedesign and implementation of statistical analyses and had an editorial role in the presentationof these methods and results.

30th Annual San Antonio Breast Cancer Symposium

Abstract Number: 550448Presenting Author: Geoffrey BeadleAuthor for Correspondence: Geoffrey F BeadleDepartment/Institution: Wesley Medical CentreAddress: Wesley Medical Centre, Suite 39, Level 3, 40 Chasely StreetCity/State/Zip/Country: Auchenflower, Queensland, 4066, AustraliaPhone: 617 3870 4255 Fax: 617 3870 4305E-mail: [email protected] Categories: 36. Psychosocial AspectsPresentation format: Poster presentationDISCLOSURE:Name of Presenter: Geoffrey Beadle

ABSTRACT

Title: Memory loss after adjuvant chemotherapy for breast cancer: a preliminary

analysis of mediating variables utilizing cross-sectional correlations and multilevel

longitudinal analysis.

Geoffrey Beadle, Margaret Rolfe, Katharine Vearncombe, Brooke Andrew, Kerrie Mengersen

and Margaret Wright.

1Translational Research Laboratory, Queensland Institute of Medical Research, Brisbane,Queensland, Australia, 4000; 2School of Mathematics, Queensland University of Technology,

265

266 APPENDIX D. COGNITION JOINT POSTER

Brisbane, Queensland, Australia, 4000; 3Department of Psychology, University ofQueensland, Brisbane, Queensland, Australia, 4000; 4Department of Psychology, Universityof Queensland, Brisbane, Queensland, Australia, 4000; 5School of Mathematics, QueenslandUniversity of Technology, Brisbane, Queensland, Australia, 4000 and 6Genetic EpidemiologyLaboratory, Queensland Institute of Medical Research, Brisbane, Queensland, Australia, 4000.

BACKGROUND

Cognitive impairment is a well recognized complication of adjuvant chemotherapy but further

research is required to identify factors that mediate cognitive change in breast cancer

survivors.

METHODS

This study investigated cognitive change in verbal memory in 119 women aged less than 70

years before, at completion of, and 6 months after adjuvant chemotherapy. Verbal memory

was assessed with the auditory verbal learning test trials 1-5 (AVLT1-5) and executive

processing of immediate and delayed recall with the AVLT7 and the AVLT8 respectively.

Cross-sectional correlations were performed with time invariant variables of age, years of

education and general cognitive ability utilizing the national adult reading test (NART).

Correlations with time varying variables included quality of life measures (HADS, FACT-B

and FACT-F) and changing hormonal phenotype (cessation of hormone replacement therapy

after diagnosis of breast cancer and changing menstrual function after chemotherapy).

Unconditional random intercept quadratic regression models were fitted to AVLT1-5, AVLT7

and AVLT8, with temporal and subject level variances estimated by restricted maximum

likelihood.

RESULTS

In this exploratory analysis, age, NART and years of education were significantly correlated

with AVLT1-5, AVLT7 and AVLT8 at all time points (all p values <0.05). Quality of life

correlates were inconsistent at most time points but statistically significant when all time

points were combined (HADS depression <0.05 and FACT fatigue <0.05 for AVLT8;

FACT-B <0.05 for AVLT1-5, and <0.01 for AVLT7 and AVLT8). Age, NART and years of

education were significant predictors of these changes (p <0.01). In particular, a high NART

predicted a less steep decline of memory over time. There was no evidence of a statistically

significant association between AVLT and self-report measures of quality of life or changing

hormonal phenotype after adjustment for age and NART.

267

CONCLUSION

A significant decline of the AVLT occurred during and after treatment with adjuvant

chemotherapy. Age, NART and years of education were strongly associated with AVLT at all

time points but not quality of life or changing hormonal phenotype. Further investigation of

memory and executive functioning is currently underway in a larger sample of patients

followed over a longer time. Multilevel longitudinal analysis is a promising tool for

investigating longitudinal data that contain multiple changing covariates.

Reference: Beadle G.F., Rolfe M., Vearncombe K., Andrew B., Mengersen K., Wright M.

Memory loss after adjuvant chemotherapy for breast cancer: a preliminary analysis of

mediating variables utilizing cross-sectional correlations and multi-level longitudinal analysis

(poster presentation). San Antonio Breast Cancer Symposium. December 13-16, 2007.

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