A Weighted Geometric Average Hazard Ratio Based Sample Size Formula for Non‐Proportional Hazards and Its Application to a Smoothed Piecewise Model in Cancer Immunotherapy Trial Design

Cancer immunotherapy trials often present unique challenges for time-to-event data modeling and analysis due to heterogeneous treatment effects across time or subgroups. A few known scenarios are delayed treatment effects and cure rates, where survival and hazard functions often take complex shapes. Sophisticated survival models like piecewise model are commonly used to capture the shapes of the functions. However, the time of hazard transition for individual patients varies, making it necessary to model it as a latent random variable. In this study, we propose a smoothed piecewise model to account for the random hazard transition time, based on the linear relationship between hazard function and the distribution function of the individual transition time within the smoothing window. We then develop the weighted geometric average hazard ratio (wgAHR) to estimate sample size and power based on the non-centrality parameter of the weighted log-rank statistics under non-proportional hazard (non-PH). We demonstrate that the wgAHR is not only directly linked to the sample size formula, but can also be interpreted as a measure of treatment effect, even when the proportional hazard assumption is violated. Additionally, we provide maxCombo, a combination of weighted log-rank statistics, for robust testing across different non-proportional hazard scenarios. Simulation studies and real trial examples illustrate the performance and robustness of the wgAHR based method in power and sample size estimation in cancer immunotherapy trials.