Performance of two formal tests based on martingales residuals to check the proportional hazard assumption and the functional form of the prognostic factors in flexible parametric excess hazard models

Net survival, the one that would be observed if the disease under study was the only cause of death, is an important, useful, and increasingly used indicator in public health, especially in population-based studies. Estimates of net survival and effects of prognostic factor can be obtained by excess hazard regression modeling. Whereas various diagnostic tools were developed for overall survival analysis, few methods are available to check the assumptions of excess hazard models. We propose here two formal tests to check the proportional hazard assumption and the validity of the functional form of the covariate effects in the context of flexible parametric excess hazard modeling. These tests were adapted from martingale residual-based tests for parametric modeling of overall survival to allow adding to the model a necessary element for net survival analysis: the population mortality hazard. We studied the size and the power of these tests through an extensive simulation study based on complex but realistic data. The new tests showed sizes close to the nominal values and satisfactory powers. The power of the proportionality test was similar or greater than that of other tests already available in the field of net survival. We illustrate the use of these tests with real data from French cancer registries.