Analysis of time-dependent covariates in a regressive relative survival model
Statistics in Medicine
Relative survival is a method for assessing prognostic factors for disease-specific mortality. However, most relative survival models assume that the effect of covariate on diseasespecific mortality is fixed-in-time, which may not hold in some studies and requires adapted modelling. We propose an extension of the Esteve et al. regressive relative survival model that uses the counting process approach to accommodate time-dependent effect of a predictor's on disease-specific mortality. This
... ch had shown its robustness, and the properties of the counting process give a simple and attractive computational solution to model time-dependent covariates. Our approach is illustrated with the data from the Stanford Heart Transplant Study and with data from a hospital-based study on invasive Published in: Statistics in Medicine 2005; 24:3863-3870. breast cancer. Advantages of modelling time-dependent covariates in relative survival analysis are discussed. INTRODUCTION Analysis of net survival provides an estimate of the effect on survival corrected for the effect of other independent causes of death, and therefore emphasises the mortality excess to which the studied group is subject. Such analysis may be carried out considering nondisease related deaths as censored observations in the Kaplan-Meier estimates of survival . However, specific causes of death are often unavailable or unreliable  and even if accurate information is available, as in randomized trials for example, it is often difficult to conclude whether or not a death should be classified as being due to disease of interest . A way around this difficulty consists of applying so-called relative survival analysis which provides an estimate of the patients' survival corrected for the effect of other independent causes of death, using the natural mortality in the general population . Although relative survival is generally used in survival population-based studies of cancer survival, it also applies in all survival clinical studies. Moreover, it has been shown that this method allows to differentiate whether covariates such as age and sex, are strictly related to the disease specific mortality, the natural mortality in the source population, or to both [4,5]. Several relative survival regression models have been proposed [1,6-8]. In these models (i) the effect of a prognostic factor on survival is assumed to be constant over time, according to the proportional hazards (PH) assumption which constrains the hazards ratio to be constant over time; and (ii) covariate are modelled with a fixed-in-time effect. Software used to obtain estimations of the Hakulinen and Tenkanen model  allow to take into account non-proportional hazards through introducing an interaction between the prognostic factor of interest and the follow-up time interval. Nevertheless, this method implies an a priori specification of the parametric form of the interaction function. Extensions of the Esteve et al. PH model , which has shown to produce accurate Published in: Statistics in Medicine 2005; 24:3863-3870. covariates. This method is an adaptation of the method used for the Cox PH model  to the Esteve et al. PH relative survival model . Our objective is to provide a very simple and attractive computational solution to model time-dependent covariates in relative survival analysis. Our approach is exemplified with the well-known data from the Stanford Heart Transplant Study  and with data from a hospital-based study on invasive breast cancer. RELATIVE SURVIVAL WITH FIXED-IN-TIME COVARIATES In the relative survival regression model proposed by Esteve et al.  , the observed hazard for total mortality, t λ , at time t after diagnosis of an individual with age at diagnosis and with a vector of covariates , which could contain age, is defined as: a z Published in: Statistics in Medicine 2005; 24:3863-3870.