A note on some recent likelihoods leading to the log rank test

Abstract

A discussion is given of the derivation by Peto (1972) of the log rank test as the locally most powerful rank invariant test against certain Lehmann-type alternatives in the two-sample problem with censored data. The random censorship model is used to show that the test with this local optimality property may depend on the [censoring mechanism, even if censoring is applied equally to both groups. An explanation is offered of Peto's likelihood and of a special case of the conditional likelihood of Cox (1972). Reference is made to asymptotic relative efficiency results which demonstrate that the log rank test may not be fully efficient when censoring is not equal.