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 δ t based on the Kaplan-Meier and Nelson-Aalen estimates, and obtain the asymptotic distribution of the process ( t ) − δ t appropriately 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.