Tests for treatment group differences in the hazards for survival, before and after the occurrence of an intermediate event

In many settings, one would expect that the hazard for a terminal event would change with the occurrence of an intermediate event. For example, in an AIDS clinical trial, it is of interest to assess whether there is a difference between treatments in the hazards for death prior to drop in Karnofsky performance score and in the hazards subsequent to the drop in Karnofsky score. Tests for the effect of treatment on these hazard functions, separately or jointly, are useful in conjunction with tests of overall survival. We consider four Cox regression models for the hazard function, constructed by allowing for various combinations of time-dependent stratification and time-dependent covariates, both of which are based on the occurrence of the intermediate event. Assuming a Markov transition model from the intermediate to the terminal event, partial likelihoods can be used for inference, enabling the use of standard statistical software for computation. We develop analytic approximations for the power of the derived score tests for treatment differences in the hazard functions and evaluate them through simulations. We apply our results to AIDS Clinical Trials Group (ACTG) protocol 021.