Summarizing Quality of Life in the Presence of Limited Survival

Gerhardt Pohl, Li LiEli Lilly and Company

ObjectivesIn this talk we focus on a special case of informative missing data, patients who cease to provide longitudinal patient reported outcomes (PRO’s) due to death. We compare various commonly used methods for analyzing such data and propose an approach based on the proportion of patients at various levels.

Overview• Problem Statement• Limitations of Current Methods• (Brief Aside on Plotting Individual Patient

Profiles)• Proposal• Summary and Discussion of Future

Directions

Problem Statement• Consider QoL or other PRO’s collected over

time.• We desire to summarize the mean profile of

the scores at the various time points.• However, patients often fail to complete all

assessments due to early death.• Further complicating the situation is

informative censoring. Patient’s scores decline as they approach death, but they also often fail to report scores as they decline.

Non-Informative vs. Informative Censoring

Solid lines indicate observed data; and dotted, missing data.

Non-Informative Informative

Commonly Used Methods

• Mixed Model Repeated Measures (MMRM)

• Area Under the Curve (AUC)• Survival Methods• Latent Effects Models

Limitations of MMRM• Consider the model with unique mean and between-

patient variability at each time point with possible within-patient correlation in the scores over the time.

• Underlying assumption is that patients share same trajectory of score over time with some patients only contributing a portion of the profile.

• Variability is modeled only in outcome and not in time of observation which is assumed fixed with common mean outcome at each observation time.

• However, in reality, patients are experiencing accelerated time to failure with informative censoring.

MMRM Unbiased in the Non-Informative Setting

Solid lines indicate observed data; and dotted, missing data.

Non-Informative Informative

Complex Profile of Real-Life PRO’s in Oncology

Burden

Time

Untreated

Treated

Chemotherapy

Complex Profile Befuddles Time to Worsening Analyses

• Time to event analysis appears ideal for handling right-censored data.

• However, of worsening in treated occurs immediately at outset of cytotoxic chemotherapy.

Causal Diagram

Treatment

PROAE

Disease

Cycle 1

Treatment

PROAE

Disease

Cycle 2

Etc.

DiseaseMeasure

AEMeasure

DiseaseMeasure

AEMeasure

Plotting Individual Patient Profiles: Spaghetti Plots

Week

0 1 2 3 4 5 6 7

Sym

ptom

Sco

re

0

1

2

3

4

5

Symptom scores (discrete 0-4) for 300 patients versus time.

Plotting Individual Patient Profiles: Lasagna Plots

• Bruce J. Swihart, Brian Caffo, Bryan D. James, Matthew Strand, Brian S. Schwartz, and Naresh M. Punjab. “Lasagna Plots: A Saucy Alternative to Spaghetti Plots”. Epidemiology, Vol. 21, Number 5, Sept. 2010.

• Remap intensity of score from vertical axis to a color and use the location on vertical axis to denote individual patient.

Each Row is a PatientWeek 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7

Each Row is a PatientWeek 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7

Hint of early tolerability burden

Sorted by Treatment GroupWeek 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7

Control

Treated

Sorted by Treatment Group and Duration of Follow-Up

Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Duration

Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Duration

Sorted by Treatment Group and Duration of Follow-Up

Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Duration

Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Duration

Poorer scores near termination

Additional Features• Sorting by characteristics of plotted data and/or by

external characteristics• Annotation of discrete events and events with duration.• Filtering rows to subsets of patients• Automated aggregation of patients with similar profiles

to allow more than one patient per horizontal band.• Side panels showing related data, e.g., Kaplan-Meier

plots, proportion of data plotted.• Special thanks to Wei Wang, Eli Lilly and Co.,

Advanced Analytics Visualization.

Limitations of AUC Methods

Score

Time0 1 2

0

0.5

1.0Patient 1

Patient 2

Two AUC-Equivalent Patients

• An approach to compensate for varying lengths of survival is to calculate area under the curve or score time values (cf. QALY).

• Note that death is mapped to zero score.• AUC yields a complete ordering of score and survival.• Exchangeability of quality and time is questionable.• Induces linearity in PRO scale that may not be realistic.

Probability-Based Methods• Rather than average scores, summarize as

proportion of patients at various levels at each time point.

0 1 2 3 40%

20%

40%

60%

80%

100%

Score=4

Score=3

Score=2

Score=1

Time

Percent of Patients

n = 10 10 8 6 6

0 1 2 3 40.000.501.001.502.002.503.003.504.00

Average

Time

Scoren = 10 10 8 6 6

vs.

Incorporating Survival• Death can be appended to low end of score.

0 1 2 3 40.000.501.001.502.002.503.003.504.00

Average

Time

Scoren = 10 10 8 6 6

vs.

0.00

0.20

0.40

0.60

0.80

1.00

0 1 2 3 4

Survival Prob.

Time

0 1 2 3 40%

20%

40%

60%

80%

100%

Score=4

Score=3

Score=2

Score=1

Dead

Time

Percent of Patients

n = 10 10 10 10 10

Summaries of Categorical Probabilities

• Cumulative Proportion of Time in Category – One can “integrate” over time to obtain the

cumulative proportion of time the group spends in each PRO level.

0 1 2 3 40%

20%

40%

60%

80%

100%

Score=4

Score=3

Score=2

Score=1

Dead

Time

Percent of Patients

Marginal0%

20%

40%

60%

80%

100% Proportion of Group

Time

Underlying Nature of DataPatient Time in Category

1 (1.0, 2.0, 1.0, 1.0, 0.0)2 (0.0, 0.0, 1.0, 4.0, 0.0)3 (0.0, 1.0, 1.0, 3.0, 0.0)4 (0.0, 0.0, 1.0, 1.0, 3.0)

Group-Level (1.0, 3.0, 4.0, 9.0, 3.0)

Proportion of time-person spent in each CategoryGroup-Level (1/20, 3/20, 4/20, 9/20, 3/20)

Treatment (p ) (p1, p2, p3, p4, p5 )

Control (q ) (q1 , q2 , q3 , q4, q5)

∑𝑖=1

5

𝑝𝑖=1𝑎𝑛𝑑∑𝑖=1

5

𝑞𝑖=1.𝑝5=𝑝𝑟𝑜𝑏 .(𝑠𝑝𝑒𝑛𝑡 𝑡𝑖𝑚𝑒𝑖𝑛 h𝐷𝑒𝑎𝑡 )

Need an Ordering Metric for Ranking which Summary Vectors are “Better”

3 possible methods, each has pros and cons.

1. Majorization Order

• Introduce majorization order over all cumulative levels of the state categories

• Treatment , p , as being better than that of control, q , iff

• Pro: no need to decide weight; includes requirement that patients in treated group survive longer than control group.

• Con: too strong condition.

2. Utility or Cost function• Introduce a utility or cost function for each category.

• Treatment group, p , as being better than that of control group, q , iff

where• Pro: reduces comparison of vectors to a single dimension

• Con: • One needs to assign weight between PRO states;• No explicit requirement that survival be better for

treated than control.

3. Pseudo Increasing Convex Order (P-ICX order)

– Treatment , p , as being better than that of control, q , iff :

,

Pro: (1) less strong condition than majorization; (2) no need to consider weight between death and PRO states; (3) includes requirement that patients in treated group survive longer than control group.

Con: One needs to assign weight between PRO states.

A 3-state example3 health states Good Bad Death

Proportion of time spent in each state

Treatment (p) P1 P2 P3

Control (q) q1 q2 q3

Difference (d=p-q) d1 d2 d3

Treatment group is better in good state.

Treatment group is better in bad state.

Treatment group is better in survival (less proportion of time spent in the state of death).Because:

Connecting each approach with d. Treatment group is better than control group:

• Cost function (f): c1>c2> c3=0 ( ) ,,

Cost function: ,

• Majorization:

• P-ICX:,

3 health states

Good Bad Death

Proportion of time spent

in each state

Trt(p)

P1 (0.29)

P2 (0.71)

P3 (0.0)

Con (q)

q1 (0.30)

q2 (0.5)

q3 (0.5)

Diff. (d=p-q)

d1 (-0.01)

d2 (0.21)

d3 (-0.5)

Majorization: reject that treatment is better than control

P-ICX: accept that treatment is better than control at c=2

3 health states

Good Bad Death

Proportion of time spent

in each state

Trt(p)

P1 (0.49)

P2 (0)

P3 (0.51)

Con (q)

q1 (0.0)

q2 (0.97)

q3 (0.03)

Diff. (d=p-q)

d1 (0.49)

d2 (-0.97)

d3 (0.48)

P-ICX: Reject that treatment is better than control at c=2trol

Cost Function: Accept that treatment is better at c=2an

P-ICX order is in the middle of majorization and Cost function method regarding acceptance of good PRO performance.

Both majorization order and P-ICX order consider survival benefit.

Example• Simulated Data

– Two arms: treatment vs. control (1:1)– Sample size: 300.– Survival:

• treatment arm has longer survival rate than control (To show contrast, treatment arm survival rate ~ 1).

– Planned visits: 6 bi-monthly visit. Follow PRO until death or completion of visits, follow patients until death or completion of study (720 days).

Survival curve

• Simulated Data (continued)– Longitudinal categorical QoL scores

• True trend:– Treatment arm has worse QoL score than control

at the first 2-3 cycles, decreased to more tolerable score than control with time going on.

– Control arm has an increasing trend over time– Health status declines faster (PRO score

increases) as they approach death.

• Observed trend (Average of Available Data):– Missing due to death or inability to conduct survey

due to approaching death.

True Curve

Method 1: Naïve Estimator • Average of score at each visit among available patients.• Observed curve gives impression that control arm is better than

treatment arm.

Method 2: MMRM– Treat score as continuous dependent.– Model separate means at each visit (treatment by visit

interaction) with exchangeable covariance within-patient and independent between-patient.

– Profile is similar to naïve estimator.

Method 3: AUC method• Area under curve up to 14 months.• [Conclusion]

x1 N Mean Std Dev Std Err Pr > |t|

Control 142 278.3 123.0 10.3236

Treatment 158 542.4 126.1 10.0342 <.0001

P-value: two-sample t-test

Proposed Method• Select a time period of interest– e.g., 14 months.

• Collapse 5 categories (raw categories: 0-4) to 2 categories (0 or category for scores of 1-4).

• Incorporate death as the worst PRO level.

• Integrate over time to obtain the proportion of time the group spends in each PRO level.

• Adopt P-ICX order to compare PRO and select weight of treatment effect in each level: weight (2,1)->state (0, 1-4).

Results• Proportion of time spent in each level during 12 months

Weight for raw states: (5,4,3,2,1)

d=(-0.06, 0.20)

Quality of Life StatusArms 0 1-4 DeathTreatment 0. 26 0.73 0.0001Control 0.32 0.53 0.15Difference -0.06 0.20 -0.15

– One can conduct formal hypothesis test :

– Calculate the vector statistics– gives impression that treatment arm is better than control arm within 12 months since baseline.– P-value: chi square test.

( 0.26

Summary and Discussion:

• Proposed probability based method to compare PRO between treatments may avoid need for weighting scores in some cases (majorization).

• 3 possible ranking methods for comparing vectors.– Majorization: strongest condition– Cost function: simple concept– P-ICX order: cost function+ improved survival requirement

• Future research: How to choose weight? P-ICX share the same question with cost function.

A formal definition of ICX• the distribution of a random variable Y is larger than the

distribution of a random variable X in the increasing convex order, i.e.

X Y , if and only if E{f(X)} E{f(Y )}

holds for all non-decreasing convex functions f for which expectations are defined.

• Insurance and actual science application