Testing Kaplan-Meier curves against an alternative of concave or convex non-proportional hazards

We show the optimality of the log-rank test for comparing two groups when the hazards are proportional. When they are not proportional we can see that this optimality is lost. Many authors have considered alternatives to the log-rank in situations of non-proportionality. We recall this work before considering two sub-classes of non-proportional hazards - concave and convex non-proportional hazards. Although less broad than just non-proportional hazards, the two classes represent a very large class of possibilities. We will see that situations such as delayed effect (immunotherapy trials), situations of non-responders and situations of group attribution errors all belong to these classes. For these two classes, optimal tests can be identified theoretically although, in practice, are not typically available. Nonetheless we describe tests that are close to optimal for the concave and convex classes. Some practical concerns are described. These include efficiency losses and sample size calculation for concave and convex alternatives to the null.