A partly parametric additive risk model

Abstract

SUMMARY Aalen's additive risk model allows the influence of each covariate to vary separately over time. Although allowing greater flexibility of temporal structure than a Cox model, Aalen's model is more limited in the number of covariates it can handle. We introduce a partly parametric version of Aalen's model in which the influence of only a few covariates varies nonparametrically over time, and that of the remaining covariates is constant. Efficient procedures for fitting this new model are developed and studied. The approach is applied to data from the Medical Research Council's myelomatosis trials. it is the first step of a Taylor series expansion of a general hazard function about the zero of the covariate vector. However, in estimating the unknown functions in such a general model there is a variance-bias trade-off that may be critical in small and medium samples. Also, after fitting the model one does not have parameters or formulae that are easily reported. We propose a model that takes the additive structure of Aalen's model and imposes parametric constraints to obtain a semiparametric submodel, which may be more appropriate in some applications. The model will be illustrated with data from clinical trials on myelomatosis. Covariates include treatment, sex and four age strata, which will be treated parametrically, together with serum levels of haemoglobin and f32-microglobulin, whose effects will be investigated nonparametrically. The additive form can be interpreted loosely in terms of unobserved competing risks since the hazard function for the minimum of independent random vari- ables is the sum of the hazard functions for the individual variables. Microglobulin levels are related to kidney function and tumour mass, whereas haemoglobin is unaffected by kidney function. Hence one might anticipate that the hazard function associated with each covariate represents a different cause of death.