Flexible modeling of the effects of continuous prognostic factors in relative survival

Relative survival methods permit separating the effects of prognostic factors on disease-related 'excess mortality' from their effects on other-causes 'natural mortality', even when individual causes of death are unknown. As in conventional 'crude' survival, accurate assessment of prognostic factors requires testing and possibly modeling of non-proportional effects and, for continuous covariates, of non-linear relationships with the hazard. We propose a flexible extension of the additive-hazards relative survival model, in which the observed all-causes mortality hazard is represented by a sum of disease-related 'excess' and natural mortality hazards. In our flexible model, the three functions representing (i) the baseline hazard for 'excess' mortality, (ii) the time-dependent effects, and (iii) for continuous covariates, non-linear effects, on the logarithm of this hazard, are all modeled by low-dimension cubic regression splines. Non-parametric likelihood ratio tests are proposed to test the time-dependent and non-linear effects. The accuracy of the estimated functions is evaluated in multivariable simulations. To illustrate the new insights offered by the proposed model, we apply it to re-assess the effects of patient age and of secular trends on disease-related mortality in colon cancer.