A method for sequential analysis of survival data with nonproportional hazards

Two tests are proposed for comparing the survival curves of patients randomised between an experimental treatment and a control treatment when it is anticipated that the two survival curves may not satisfy the assumption of proportional hazards. The tests are particularly useful for the situation in which the survival curves are coincident or cross over early in the follow-up period and then diverge. The tests compare the probabilities of survival for longer than some fixed time since randomisation for the two groups of patients. Both methods take account of the right-censored observations, and both are associated with methods for estimating and setting confidence limits for treatment differences. The first method is a mathematically direct approach based on the derivation of the efficient score statistic and Fisher's information. The second method is simpler, being based on Kaplan-Meier estimates and their variances. Conventional methods of sample size determination require the assumption of proportional hazards. Here a sequential approach is used, as it is difficult to set the sample size in advance without strong assumptions about the relationship between the two survival curves. Simulation results giving information on the size and power of the proposed tests are provided and the tests are applied to data from a clinical trial in breast cancer.