Assessing time-by-covariate interactions in relative survival models using restrictive cubic spline functions.

Background: The Cox model is widely used in the evaluation of prognostic factors in clinical research. However, in population-based studies, which assess long-term survival of unselected populations, relative-survival models are often considered more appropriate. In both approaches, the validity of proportional hazards hypothesis should be evaluated.

Methods: We propose a new method in which restricted cubic spline functions are employed to model time-by-covariate interactions in relative survival analyses. The method allows investigation of the shape of possible dependence of the covariate effect on time without having to specify a particular functional form. Restricted cubic spline functions allow graphing of such time-by-covariate interactions, to test formally the proportional hazards assumption, and also to test the linearity of the time-by-covariate interaction.

Results: Application of our new method to assess mortality in colon cancer provides strong evidence against the proportional hazards hypothesis, which is rejected for all prognostic factors. The results corroborate previous analyses of similar data-sets, suggesting the importance of both modelling of non-proportional hazards and relative survival approach. We also demonstrate the advantages of using restricted cubic spline functions for modelling non-proportional hazards in relative-survival analysis. The results provide new insights in the estimated impact of older age and of period of diagnosis.

Discussion: Using restricted cubic splines in a relative survival model allows the representation of both simple and complex patterns of changes in relative risks over time, with a single parsimonious model without a priori assumptions about the functional form of these changes.