Promotion Time Cure Model with Local Polynomial Estimation

In modeling survival data with a cure fraction, flexible modeling of covariate effects on the probability of cure has important medical implications, which aids investigators in identifying better treatments to cure. This paper studies a semiparametric form of the Yakovlev promotion time cure model that allows for nonlinear effects of a continuous covariate. We adopt the local polynomial approach and use the local likelihood criterion to derive nonlinear estimates of covariate effects on cure rates, assuming that the baseline distribution function follows a parametric form. This approach ensures that the model is identifiable and we adopt a flexible method to estimate the cure rate locally, the important part in cure models, and a convenient way to estimate the baseline function globally. An algorithm is proposed to implement estimation at both the local and global scales. Asymptotic properties of local polynomial estimates, the nonparametric part, are investigated in the presence of both censored and cured data, and the parametric part is shown to be root-n consistent. The proposed methods are illustrated by simulated and real data with discussions on the practical applications of the proposed method, including the selections of the bandwidths in the local polynomial approach and the parametric baseline distribution family. Extension of the proposed method to multiple covariates is also discussed.