Sample size calculation for the proportional hazards model with a time-dependent covariate

The Cox proportional hazards (PH) model with time-dependent covariates (referred to as the extended PH model) has been widely used in medical and health related studies to investigate the effects of time-varying risk factors on survival. Theories and practices regarding model estimation and fitting have been well developed for the extended PH model. However, little has been done regarding sample size calculations in survival studies involving a time-varying risk factor. A novel sample size formula based on the extended PH model is proposed by investigating the asymptotic distributions of the weighted log-rank statistics under the null and local alternative hypotheses. The derived sample size formula is an extension of the sample size formula for the standard Cox PH model. The performance of the proposed formula is evaluated by extensive simulations, and examples based on real data are given to illustrate the applications of the proposed methods.