Equivalence tests under the non mixture and mixture cure fraction models

In this article, we propose equivalence tests for the difference of two survival functions in the presence of cure fraction. We consider both the non mixture cure model (NMCM) and mixture cure model (MCM). We first demonstrate that the existing tests can be used to test equivalence of two improper survival functions under the class of log transformation (LT) and Box-Cox transformation (BCT) NMCM. For the MCM, a logistic regression model is assumed for the probability of being uncured and the class of LT or BCT models are used to describe the distribution of event time for the uncured subjects. Under the MCM, we consider testing separately the null hypothesis of identical cure rate and that of equivalence of two survival functions of the uncured subjects. We also develop a Wald statistic for simultaneously testing whether or not both the maximal difference between cure rates and that between two short-term survival functions fall within some equivalence margins. Simulation results indicate that the proposed tests have a satisfactory size and an adequate power in finite sample.