Sample size calculation for mixture model based on geometric average hazard ratio and its applications to nonproportional hazard.

With the advent of cancer immunotherapy, some special features including delayed treatment effect, cure rate, diminishing treatment effect and crossing survival are often observed in survival analysis. They violate the proportional hazard model assumption and pose a unique challenge for the conventional trial design and analysis strategies. Many methods like cure rate model have been developed based on mixture model to incorporate some of these features. In this work, we extend the mixture model to deal with multiple non-proportional patterns and develop its geometric average hazard ratio (gAHR) to quantify the treatment effect. We further derive a sample size and power formula based on the non-centrality parameter of the log-rank test and conduct a thorough analysis of the impact of each parameter on performance. Simulation studies showed a clear advantage of our new method over the proportional hazard based calculation across different non-proportional hazard scenarios. Moreover, the mixture modeling of two real trials demonstrates how to use the prior information on the survival distribution among patients with different biomarker and early efficacy results in practice. By comparison with a simulation-based design, the new method provided a more efficient way to compute the power and sample size with high accuracy of estimation. Overall, both theoretical derivation and empirical studies demonstrate the promise of the proposed method in powering future innovative trial designs.