Graphical comparison of cumulative hazards for two populations

Abstract

We consider graphs, confidence procedures and tests that can be used with censored survival data to compare the hazard experience of a treatment group with that of a control group. In particular, we consider the relative change Δ(t) in a cumulative rate function which is used to define a generalized proportional hazard model. This model is equivalent to the model obtained by introducing a gamma distributed frailty term in the proportional hazard model. We introduce an estimate (t) or δt based on the Kaplan-Meier and Nelson-Aalen estimates, and obtain the asymptotic distribution of the process (t) − δtappropriately scaled. This result is used to construct asymptotic pointwise confidence intervals for Δ(t). Under the proportional hazard model assumption, Doob's transformation of Gaussian processes is used to derive asymptotic simultaneous confidence bands for Δ(t) from a general class of test statistics defined as sup norms of weighted differences between Δ(t) and its estimate Δ^(t) We also consider procedures based on estimates (u) of the relative trend function r(u) between cumulative hazards. The methods are illustrated on a data set from a study of the effect of radiation on mice, and on a data set from a study of the effect of tumour size on the survival of cancer patients.