Abstract
Mixture cure models have become increasingly popular for modeling event time data when some individuals in a study may never experience an event of interest. While traditional linear assumptions are often made about the effects of continuous covariates, these assumptions can fail in practical situations where real-life effects are typically nonlinear. To address this issue, a linear spline Cox cure model is proposed, using a spline to approximate the unknown functional form for the effect of a continuous covariate and identify the nonlinear relationship. Laplace's approximation of the marginal log-likelihood function is used to justify and estimate the model parameters, leading to a penalized log-likelihood. The expectation-maximization algorithm is then employed for parameter estimation, and the methodology is used to evaluate the linearity of the continuous covariate effect through the likelihood ratio procedure. An extensive simulation study is conducted to investigate the proposed lack-of-fit test for linearity of the continuous covariate effect. Finally, the practical use of the methodology is illustrated with fibrous histiocytoma data from the Surveillance, Epidemiology, and End Results (SEER) program database.