Complex survival trial design by the product integration method.

Nonproportional hazards (NPHs) are often observed in survival trials such as the immunotherapy cancer trials. Under NPH, the classical log-rank test can be inefficient, and the estimated hazards ratio from the Cox model is difficult to interpret. The weighted log-rank test, and the tests for comparing the restricted mean survival time or the milestone survival become increasingly popular in handling NPH. The sample size calculation for these tests may require high-dimensional numerical integration. We present a sample size determination method for survival trials via product integration on the basis of a continuous-time multistate Markov model. The main challenge of the method lies in the design of the multistate model under a complex NPH pattern, and this is illustrated for NPH induced by delayed effect with individual heterogeneity in the lag duration, cure fractions, and treatment switching due to disease progression or noncompliance. Numerical examples are presented to demonstrate the accuracy of the proposed method. We obtain the following findings. The powers of the tests for milestone survival and RMST depend on both the trial duration and milestone timepoint, and may not increase as the milestone timepoint increases. If the milestone timepoint is appropriately chosen, the RMST test can be more powerful than the conventional log-rank test in the presence of diminishing treatment effect or in the proportional hazards cure model. In general, the RMST test yields lower power than a proper Fleming-Harrington weighted log-rank test.