  The classical approach to analyze time-to-event data, e.g. in clinical
trials, is to fit Kaplan-Meier curves yielding the treatment effect as the
hazard ratio between treatment groups. Afterwards commonly a log-rank test is
performed in order to investigate whether there is a difference in survival,
or, depending on additional covariates, a Cox proportional hazard model is
used. However, in numerous trials these approaches fail due to the presence of
non-proportional hazards, resulting in difficulties of interpreting the hazard
ratio and a loss of power. When considering equivalence or non-inferiority
trials, the commonly performed log-rank based tests are similarly affected by a
violation of this assumption. Here we propose a parametric framework to assess
equivalence or non-inferiority for survival data. We derive pointwise
confidence bands for both, the hazard ratio and the difference of the survival
curves. Further we propose a test procedure addressing non-inferiority and
equivalence by directly comparing the survival functions at certain time points
or over an entire range of time. We demonstrate the validity of the methods by
a clinical trial example and by numerous simulation results.
