An Asymptotic Analysis of the Logrank Test

Asymptotic expansions for the null distribution of thelogrank statistic and its distribution under local proportionalhazards alternatives are developed in the case of iid observations.The results, which are derived from the work of Gu (1992) andTaniguchi (1992), are easy to interpret, and provide some theoreticaljustification for many behavioral characteristics of the logranktest that have been previously observed in simulation studies.We focus primarily upon (i) the inadequacy of the usual normalapproximation under treatment group imbalance; and, (ii) theeffects of treatment group imbalance on power and sample sizecalculations. A simple transformation of the logrank statisticis also derived based on results in Konishi (1991) and is foundto substantially improve the standard normal approximation toits distribution under the null hypothesis of no survival differencewhen there is treatment group imbalance.