Competing risks models with two time scales.

Competing risks models can involve more than one time scale. A relevant example is the study of mortality after a cancer diagnosis, where time since diagnosis but also age may jointly determine the hazards of death due to different causes. Multiple time scales have rarely been explored in the context of competing events. Here, we propose a model in which the cause-specific hazards vary smoothly over two times scales. It is estimated by two-dimensional $P$-splines, exploiting the equivalence between hazard smoothing and Poisson regression. The data are arranged on a grid so that we can make use of generalised linear array models for efficient computations. The R-package TwoTimeScales implements the model. As a motivating example we analyse mortality after diagnosis of breast cancer and we distinguish between death due to breast cancer and all other causes of death. The time scales are age and time since diagnosis. We use data from the Surveillance, Epidemiology and End Results (SEER) program. In the SEER data, age at diagnosis is provided with a last open-ended category, leading to coarsely grouped data. We use the two-dimensional penalised composite link model to ungroup the data before applying the competing risks model with two time scales.