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Weighted Kaplan-Meier Estimator for Adaptive Treatment Strategies in Two-Stage Randomization Designs

Sachiko Miyahara, Abdus S. Wahed, Ph.D Department of Biostatistics, Graduate School of Public Health, University of Pittsburgh

Summer 2007 Program on Challenges in Dynamic Treatment Regimes and Multistage Decision-Making , SAMSI, Research Triangle Park, NC Abstract

Notation Simulation Study

Simulation Results

Two-Stage Randomized Trials

References

Proposed Weighted Kaplan-Meier

Statistical inference for adaptive treatment strategies from two-stage randomization designs is mainly done through marginal mean models using inverse-probability-weighting (Murphy et al., 2001, JASA). In survival settings, Lunceford et al. (2002, Biometrics) proposed an estimating equation based semi-parametric approach for estimating the survival distribution for treatment strategies. A weighted risk set estimator (weighted Nelson-Aalen estimator) has recently been proposed in Guo and Tsiatis (2005, International Journal of Biostatistics). Although these methods have provided methodologies for consistently estimating parameters from two-stage randomization designs, their implementation is not as straightforward. In this study we propose a weighted Kaplan-Meier estimator and compare its properties to that of the estimating equations based estimator and the weighted risk set estimator through simulation. We analyzed a Leukemia data set to demonstrate the application of the weighted Kaplan-Meier estimator.

2. Weighted Risk Set Based Approach (Guo and Tsiatis, 2005)

where

.

1. Estimating Equation Based Approach (Lunceford et al., 2002)

Figure 1. Three Estimators with Modified Leukemia Data.

Table 1. Simulation Results: True Survival Rate and Three Survival Estimators with n=200 and 500, t=0.5 and 1.0, P(Response)=0.4 and 0.7

N T P(Resp) S(t) WKM(t)(MCSE(t))

EEB(t) WRSE(t) (MCSE(t)) (MCSE(t))

200 0.5 0.4 0.792 0.804 0.806 0.800

(0.051) (0.051) (0.045)

0.7 0.846 0.835 0.838 0.832

(0.051) (0.051) (0.044)

1.0 0.4 0.461 0.440 0.484 0.489

(0.114) (0.067) (0.056)

0.7 0.574 0.552 0.563 0.567

(0.097) (0.069) (0.057)

500 0.5 0.4 0.792 0.807 0.800 0.808

(0.032) (0.028) (0.032)

0.7 0.846 0.838 0.839 0.831

(0.032) (0.033) (0.028)

1.0 0.4 0.461 0.489 0.486 0.488

(0.042) (0.036) (0.043)

0.7 0.574 0.568 0.565 0.563

(0.042) (0.044) (0.036)

These two estimators are consistent; however, the implementation is not as user-friendly.

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Conclusions

If everyone in the sample were treated with the policy A1B1B2, their survival rate at time t could be estimated using the Kaplan-Meier

However, in the two-stage randomized trial settings, subjects receive a subsequent treatment based on the result of the first treatment. Therefore some patients will potentially receive treatment inconsistent with the policy (e.g., responders who receive B1’). Therefore, we need to adjust for the loss of these patients. We propose to use the weighted

Leukemia Data Analysis

At the first stage, patients are randomized into one of the initial treatments (A1 or A2). At the second stage, depending on how they respond to the initial treatment, they are randomized again for the second treatments (B1 or B1’, and B2 or B2’). With this design, there are eight treatment policies (A1B1B2, A1B1B2’, etc.), and the goal is to consistently estimate the survival function for each treatment policy.

The two-stage adaptive treatment regimes design is as follows:

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Two Existing Methods

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The following notations were used throughout this poster: R = Response status TR

= Time to second treatment C = Time to Censoring T = Time to death V = Minimum of T or C X1 = A1 treatment indicator Z1 = Indicator variable for the second treatment B1

(If Z1=1, then B1) Z2 = Indicator variable for the second treatment B2

(If Z2=1, then B2) Δ = Indicator variable for death pA1B1B2 = Probability that a subject will follow the policy A1B1B2

πA1 = Proportion of subject s with A1

πRA1 = Response rate for A1group πZ1 = Proportion of subjects receiving B1 within responders πZ2 = Proportion of subjects with B2 within non-responders

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All three estimators were approximately unbiased.

Absolute relative biases were very small and ranged from 0.038 to 0.061.

The Monte Carlo standard deviations for the estimators were comparable.

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The two stage double-blind placebo-controlled randomized clinical trial, Protocol 8923, was conducted by the Cancer and Leukemia Group (CALGB) . The data set contains 388 elderly patients with acute myelogenous leukemia. At the first stage, patients were randomized into one of the initial treatments (GM-CSF or Placebo following standard chemotherapy). At the second stage, the responders were randomized into one of two intensification treatments.

Data Modification:Since there was no second stage treatment for the non-responders, we have simulated the second treatment B2 using Bernoulli(0.5) distribution.

We proposed a weighted Kaplan Meier estimator to estimate the survival distribution of treatment strategies in two stage randomization designs.

The estimator performs well in terms of bias and variances compared to the existing estimators.

It is easy to implement using standard software packages.

Murphy S.A., van der Laan M.J., Robins J.M. & CPPRG (2001) Marginal Mean Models for Dynamic Regimes. JASA 96 1410-1423

Lunceford, J.K., Davidian, M., and Tsiatis, A.A. (2002) Estimation of Survival Distribution of Treatment Policies in Two-Stage RandomizationDesigns in Clinical Trials. Biometrics, 58, 48-57

Xiang Guo & Anastasios Tsiatis, 2005. "A Weighted Risk Set Estimator for Survival Distributions in Two-Stage Randomization Designs with Censored Survival Data," International Journal of Biostatistics, Berkeley Electronic Press, vol. 1(1), pages 1000-1000.

True population:R ~ Bernoulli(0.4) and Bernoulli(0.7) Z1 ~ Bernoulli(0.5) for R=1 only

Z2 ~ Bernoulli(0.5) for R=0 only

TR ~ Exp(1) C ~ Uniform(0, 2.5) TA1B1 = TR + Exp(1) --Time to death for patients receiving A1B1

TA1B2 = TR + Exp(0.5) --Time to death for patients receiving A1B2

T = (1-R)TA1B2 + RTA1B1

Acknowledgement

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estimator as follows:

Kaplan Meier estimator.

QA1B1B2 is defined at the estimator 1

We thank SAMSI, and Professor Steve Wisniewski of University of Pittsburgh for funding, and the CALGB group and Professor Butch Tsiatis of North Carolina State University for kind permission to use the leukemia dataset.

Monte Carlo mean and standard deviations are shown in Table 1 below.

0.2

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0.5

0.6

0.7

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0 100 200 300 400 500 600

Time to Death(days)

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WKM WRSE EEB

For this data set, the weighted risk set estimator (WRSE) estimates were always larger than the estimating equation based (EEB) and the weighted Kaplan-Meier (WKM) estimates.

The EEB and the WKM estimators followed each other very closely.

Leukemia Data Analysis Results

Applying the three methods to the modified CALGB 8923 data, thesurvival distributions of the treatment policy A1B1B2 were estimated. The results are shown in Figure 1 blow.

Sample: 5000 datasets 200 and 500 observations R, Z1, Z2, TR, C, TA1B1, TA1B2 are the same as above

TA1B1’, TA1B2’ are generated from exponential distributions

T = (1-R)[Z2 TA1B2+(1- Z2) TA1B2’ ]+R[Z1 TA1B1+(1- Z1) TA1B1’]

V = min(T, C)