Introduction

1. Sample Size Calculation for Comparing Strategies in Two-Stage Randomizations with Censored Data Zhiguo Li and Susan Murphy Institute for Social Research and Departments of Statistics, University of Michigan, Ann Arbor Introduction Sample size calculation Test statistics: comparing two strategies • Assuming proportional hazards • Using asymptotic distribution of test statistics under local alternative hypothesis • Clinical trials with two-stage randomizations become increasingly popular, especially in areas such as cancer research, substance abuse, mental illness, etc. • Patients are first randomized to a primary therapy. Then non-responders (defined by some criterion) are further randomized to a second stage treatment, and responders are treated with a maintenance therapy (depending on the area, sometimes responders are randomized instead of non-responders). • Interest is in comparing treatment strategies (combinations of first stage treatment and second stage treatment if eligible) and select the best strategy. • A question of interest is the determination of the necessary sample size to achieve a certain power for testing the equivalence of two strategies. • In particular, our interest is in cases where time to some event may be censored. Notation: Strategy “11”: get A1 first, and if no response then get B1. Strategy “22”: get A2 first, and if no response then get B2. T: time to event, S: time to response, C: censoring time X=I(A1), Z=I(B1), p=P(X=1), q=P(Z=1) R: response indicator=I(S<T, S<C) T11: time to event under strategy “11”, T22: time to event under strategy “22” : the time of the end of study : survival probability under policy jj • Using test statistics based on estimation of • Using weighted log-rank test • Test statistics based on estimation of Each subject is associated with a weight when estimating survival probabilities: inverse of the probability that a subject is consistent with a strategy : Significance level, : power, : log hazard ratio : asymptotic variance of (numerator of) test statistic • Most important issue: guess of variance based on prior knowledge before data collection: usually get an upper bound—conservative sample size • Difficulty: variance depends on quantities like the following, which involves time to response and time to response is correlated with time to event • Weighted Kaplan-Meier estimator: • Weighted Aalen-Nelson estimator: Guo and Tsiatis (2006): • Weighted sample mean: Lunceford et al. (2002) • Test statistic: Illustration of a two-stage randomization R = randomization Using martingale property, this can be bounded by Guess at the upper bound is relatively easy Simulation results Sample sizes calculated from the test based on the weighted Kaplan-Meier estimator and power of different tests under this sample size • Weighted log-rank test Failure time and time to response are generated from a Clayton copula model with a positive association parameter Failure time and time to response are generated from a Frank copula model with a negative association parameter