Simulation of “forwards-backwards” multiple imputation technique in a

longitudinal, clinical dataset

Catherine Welch1, Irene Petersen1, James Carpenter2

1Department of Primary Care and Population Health, UCL2Department of Medical Statistics, LSHTM

Acknowledgements

• Steering Group:– Irwin Nazareth (UCL)– Kate Walters (UCL)– Ian White (MRC Biostatistics, Cambridge)– Richard Morris (UCL)– Louise Marston (UCL)

• This study was funded by the MRC

Overview

• Summary of motivation• “Forwards-backwards” algorithm• Issues that we have encountered

Introduction

• Most missing data techniques have been mainly designed for cross-sectional data

• “Forwards-backwards” multiple imputation (MI) algorithm has been developed to impute missing values in longitudinal databases

• We are in the process of applying this technique to The Health Improvement Network (THIN) primary care database

• Impute variables associated with incidence of cardiovascular disease (CVD)

Clinical databases

• Offer many opportunities that would be difficult and expensive to address using standard study design

• Designed for patient management

The Health Improvement Network (THIN)

• Primary care database• Longitudinal records of patients consultation with

General Practitioner (GP) or nurse• Data collected since early 90’s• 7 million patients to over 400 practices• Over 40 million person years of follow up• Systematically structured coding (Read codes)

Cardiovascular disease

• Clinical databases powerful data source for research e.g. cardiovascular disease

• New risk prediction models have caused much debate

• NICE recommends further research is required to validate models

• Important to have good measures of risk factors and consider missing data

Aims of this project…

• Explore the extent of missing data on health indicators (height, weight, blood pressure, cholesterol, smoking status, deprivation, alcohol consumption and ethnicity)

• Develop models for imputation of missing data

Survival models

1. Baseline – at practice registration

2. Age specific – extract data recorded at a specific age

3. Non-age specific – risk is constant across all ages

4. Time varying effect – risk varies across ages

50

Registration1 year following registration

60

Substantive model

• Include same variables as Framingham score plus deprivation (Townsend deprivation quintile) and BMI

• Poisson model to predict risk of Coronary heart disease• Explanatory variables without missing data: age, sex, left

ventricular hypertrophy (LVH), Type II diabetes• With missing data: deprivation, weight, height, total serum

cholesterol, high density lipoprotein (HDL) cholesterol, systolic blood pressure and smoking status

Imputation one year following registration

• Keep patients registered between 2005-2008 and with practice for at least one year

• Exclude patients that have coronary heart disease within the first year

• Average of all recorded measurements during the first year included in the analysis

• Select 50 practices with least missing data for systolic blood pressure and weight per person

• First step: understand structure and extent of missing data

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Townsend score Systolic blood pressure

Weight HeightTotal cholesterol HDL cholesterolSmoking status

Missing health indicator variables by age

72,759 patients registered to 50 practices between 2005 and 2008

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16-24 25-34 35-44 45-54 55-64 65-74 75-84 85-94 95+Age group (years)

Townsend score Systolic blood pressure

Weight HeightTotal cholesterol HDL cholesterolSmoking status

Missing health indicator variables by age

72,759 patients registered to 50 practices between 2005 and 2008

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16-24 25-34 35-44 45-54 55-64 65-74 75-84 85-94 95+Age group (years)

Townsend score Systolic blood pressure

Weight HeightTotal cholesterol HDL cholesterolSmoking status

Missing health indicator variables by age

72,759 patients registered to 50 practices between 2005 and 2008

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16-24 25-34 35-44 45-54 55-64 65-74 75-84 85-94 95+Age group (years)

Townsend score Systolic blood pressure

Weight HeightTotal cholesterol HDL cholesterolSmoking status

Missing health indicator variables by age

72,759 patients registered to 50 practices between 2005 and 2008

Missing health indicator variables by age

72,759 patients registered to 50 practices between 2005 and 2008

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16-24 25-34 35-44 45-54 55-64 65-74 75-84 85-94 95+Age group (years)

Townsend score Systolic blood pressure

Weight HeightTotal cholesterol HDL cholesterolSmoking status

72,759 patients registered to 50 practices between 2005 and 2008

Missing health indicator variables by gender

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Female MaleGender

Townsend score Systolic blood pressure

Weight HeightTotal cholesterol HDL cholesterolSmoking status

Problems with ‘ad-hoc’ imputation

• ‘Ad hoc’ imputation methods (e.g. complete case analysis, LOCF) result in bias results and potentially incorrect conclusions

• Multiple imputation is now established as an alternative method to deal with missing data

Multiple imputation

• Assume Missing At Random• Use the relationship between the variables to

impute a valid estimate for a missing value• Multiple estimates are combined using Rubins

Rules to produce unbiased estimates of coefficients and standard errors

• This takes account of uncertainty and variation in the data

Multiple imputation model

• All variables in substantive model included in imputation model

• Exponential survival model so indicator for CHD and variable for time to event or censoring

• MI applied 5 times and results combined

Results for health indicators at baseline

Complete case Imputed data

Townsend score quintile, %

1 (least deprived) 13.72 13.65

2 14.05 13.95

3 24.77 24.84

4 30.46 30.59

5 (most deprived) 17.00 16.98

Height (m), mean (SE) 1.70 (0.00041) 1.70 (0.00041)

Weight (kg), mean (SE) 72.6 (0.06644) 72.8 (0.06583)

Systolic blood pressure (mmHg), mean (SE) 123.8 (0.06707) 123.8 (0.05866)

Total serum cholesterol (mmol l-1), mean (SE) 5.16 (0.01024) 5.05 (0.00882)

HDL cholesterol (mmol l-1), mean (SE) 1.40 (0.00401) 1.43 (0.00545)

Smoking status, % Smoker 30.29 30.32

Non-smoker 69.71 69.68

Survival models

1. Baseline – at practice registration

2. Age specific – extract data recorded at a specific age

3. Non-age specific – risk is constant across all ages

4. Time varying effect – risk varies across ages

50

Registration1 year following registration

60

Considerations when applying MI to longitudinal clinical data

• Longitudinal and dynamic structure of the data• Imputing cross-sectionally is not appropriate• Imputations need to produce a logical sequence

of values over time• Introduction of new quality measures which have

improved data recording

Example of THIN data

Practice ID Sex Age (years)

Cholesterol (mmol/l)

Weight (kg)

1 1 M 65 5.2 80

1 1 M 66 ?.? 86

1 1 M 67 6.0 89

1 1 M 68 6.0 95

1 2 F 65 3.4 60

1 2 F 66 3.6 60

1 2 F 67 3.6 ??

1 2 F 68 4.0 70

“Forwards-backwards” technique

• Based on the fully conditional specification method of MI

• Takes into account the dynamic, longitudinal structure of the data

• Does not require measurements at equally spaced time points

Nevalainen et al. Missing values in longitudinal dietary data: A multiple imputation approach based on a fully conditional specification. Statist. Med. 2009; 28:3657–3669

Fully conditional specification (FCS)

• Based on a flexible selection of univariate imputation distributions

• Impute one variable at a time using a distribution conditional on all the other variables

• Procedure iterates over the variables in cycles until assumed convergence

• Appropriate for non-normal distributions

A graphical illustration of the “forwards-backwards” FSC procedure

Within-time iteration

Among-time iteration

),,,|( 1,1 ijijiimisij YXXXXf

Example

Practice ID Sex Age (years)

Cholesterol (mmol/l)

Weight (kg)

1 1 M 65 5.2 80

1 1 M 66 ?.? 86

1 1 M 67 6.0 89

1 1 M 68 6.0 95

1 2 F 65 3.4 60

1 2 F 66 3.6 60

1 2 F 67 3.6 ??

1 2 F 68 4.0 70

Example

Practice ID Sex Age (years)

Cholesterol (mmol/l)

Weight (kg)

1 1 M 65 5.2 80

1 1 M 66 ?.? 86

1 1 M 67 6.0 89

1 1 M 68 6.0 95

1 2 F 65 3.4 60

1 2 F 66 3.6 60

1 2 F 67 3.6 ??

1 2 F 68 4.0 70

Example

Practice ID Sex Age (years)

Cholesterol (mmol/l)

Weight (kg)

1 1 M 65 5.2 80

1 1 M 66 ?.? 86

1 1 M 67 6.0 89

1 1 M 68 6.0 95

1 2 F 65 3.4 60

1 2 F 66 3.6 60

1 2 F 67 3.6 ??

1 2 F 68 4.0 70

Example

Practice ID Sex Age (years)

Cholesterol (mmol/l)

Weight (kg)

1 1 M 65 5.2 80

1 1 M 66 ?.? 86

1 1 M 67 6.0 89

1 1 M 68 6.0 95

1 2 F 65 3.4 60

1 2 F 66 3.6 60

1 2 F 67 3.6 ??

1 2 F 68 4.0 70

Example

Prac ID Sex Age (years)

Cholesterol 66 (mmol/l)

Cholesterol 65 (mmol/l)

Cholesterol 67 (mmol/l)

Weight 66 (kg)

Weight 65 (kg)

Weight 67 (kg)

1 1 M 66 ?.? 5.2 6.0 86 80 89

1 2 F 66 3.6 3.4 3.6 60 60 ??

Example

Practice ID Sex Age (years)

Cholesterol (mmol/l)

Weight (kg)

1 1 M 65 5.2 80

1 1 M 66 5.8 86

1 1 M 67 6.0 89

1 1 M 68 6.0 95

1 2 F 65 3.4 60

1 2 F 66 3.6 60

1 2 F 67 3.6 ??

1 2 F 68 4.0 70

Apply “forwards-backwards” algorithm to THIN

• Select patients registered to 50 THIN practice from 2005 to 2008

• Apply algorithm at all ages• Extract imputations for 11,614 patients aged 60

years old

Preliminary results

Complete case Imputed data

Townsend score quintile, %

1 30.05 28.67

2 24.56 24.76

3 18.69 18.71

4 14.86 15.75

5 11.83 12.11

Height (m), mean (SE) 1.68 (0.00130) 1.67 (0.00091)

Weight (kg), mean (SE) 80.25 (0.23961) 79.39 (0.15976)

Systolic blood pressure (mmHg), mean (SE) 136.18 (0.18086) 135.86 (0.21134)

Total serum cholesterol (mmol l-1), mean (SE) 5.26 (0.01616) 5.40 (0.01482)

HDL cholesterol (mmol l-1), mean (SE) 1.44 (0.00667) 1.47 (0.00738)

Smoking status, % Smoker 29.13 27.92

Non-smoker 70.87 72.08

11,614 patients aged 60 years old registered to 50 practices between 2005 and 2008

Discussion

• Potential to develop this method further• Validation:

– using simulations– investigate distributions of longitudinal values– external information

• What would be the best way to include outcome in the “forwards-backwards” imputation model?

• Interactions

FCS using longitudinal data

• Y – fully observed outcome variable

• X =(X1, . . . , Xq ) where Xi =(Xi1, . . . , Xip), q repeated measures of p explanatory variables intended to be collected

• Xobs and Xmis denote the observed and the missing elements in X

• Need the specify a suitable imputation model f (Xmis|Xobs,Y,θ)

• The FCS of the imputation model in which imputations are made one variable at a time using a series (j =1, . . . , p) of conditional densities

• denoted as

• have been imputed k+1 times

• have been imputed k times.

),,,...,,,...,|( )1()1(1 ijipjijiimisij YXXXXXf

),|( , YXXf ji

misij

)1(1,..., jii XX

ipji XX ,...,)1(

FCS using longitudinal data

• At time i impute conditional onand the outcome Y.

• Rather than condition only on the observed data, we generate appropriate values for from the fully conditional imputation model

• One iteration (within-time iteration) runs over the variables j =1, . . . , p.

• The inter-correlation among repeatedly measured variables is also of importance, we have a second imputation iteration among the index i (among times).

misiX

misiX

obsi

obsji

obsi XXX 1,1 ,,

misijX

),,,|( 1,1 ijijiimisij YXXXXf

FCS using longitudinal data