Nontransitivity in a class of weighted logrank statistics under nonproportional hazards

Transitivity is an important property of any statistic applied in the setting of multi-arm clinical trials and non-inferiority trials where active-controls are used. The G[rho],[gamma] class of weighted logrank statistics for right-censored survival data as proposed by Fleming and Harrington [1991. Counting Processes and Survival Analysis. Wiley, New York] is often used to improve efficiency in the setting of nonproportional hazards. These statistics utilize a weighting scheme based upon the combined Kaplan-Meier estimate of survival for all comparison groups. Members of this class include the usual logrank statistic as well as the generalized Wilcoxon statistic. It is demonstrated that all useful members of this class exhibit nontransitivity. We propose a general modification of the G[rho],[gamma] statistic which asymptotically achieves transitivity.