Smooth Hazards With Multiple Time Scales.

Hazard models are the most commonly used tool to analyze time-to-event data. If more than one time scale is relevant for the event under study, models are required that can incorporate the dependence of a hazard along two (or more) time scales. Such models should be flexible to capture the joint influence of several time scales, and nonparametric smoothing techniques are obvious candidates. $P$ -splines offer a flexible way to specify such hazard surfaces, and estimation is achieved by maximizing a penalized Poisson likelihood. Standard observation schemes, such as right-censoring and left-truncation, can be accommodated in a straightforward manner. Proportional hazards regression with a baseline hazard varying over two time scales is presented. Efficient computation is possible by generalized linear array model (GLAM) algorithms or by exploiting a sparse mixed model formulation. A companion R-package is provided.