Abstract

1. 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 Leukemia Data Analysis Simulation Results Abstract Two Existing Methods 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. 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. Monte Carlo mean and standard deviations are shown in Table 1 below. 1. Estimating Equation Based Approach (Lunceford et al., 2002) . where Leukemia Data Analysis Results Applying the three methods to the modified CALGB 8923 data, the survival distributions of the treatment policy A1B1B2 were estimated. The results are shown in Figure 1 blow. Two-Stage Randomized Trials is a consistent estimate of p from the sample data. The two-stage adaptive treatment regimes design is as follows: 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 2. Weighted Risk Set Based Approach (Guo and Tsiatis, 2005) • 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. where Conclusions These two estimators are consistent; however, the implementation is not as user-friendly. • 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. Proposed Weighted Kaplan-Meier 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. Figure 1. Three Estimators with Modified Leukemia Data. If everyone in the sample were treated with the policy A1B1B2, their survival rate at time t could be estimated using the Kaplan-Meier • 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. estimator as follows: 1 If t < t1 Acknowledgement If t ≥ t1 Notation Simulation Study 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. 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 True population: where • 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 tj , j = 1, 2, .., are distinct ordered failure times. 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 References 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 Randomization Designs 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. Kaplan Meier estimator. 1 If t < t1 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) If t ≥ t1 where QA1B1B2 is defined at the estimator 1