Time-dependent effects of fixed covariates in Cox regression

A nonparametric modification is proposed for Cox's proportional hazards model (Cox, 1972, Journal of the Royal Statistical Society, Series B 34, 187-220), where the covariates are fixed, but their effects are allowed to vary in time. Parameters are introduced for the covariate effects at the (uncensored) survival times. Estimates are obtained by maximizing the penalized partial log-likelihood that arises if a penalty function of first order differences of consecutive parameters is subtracted from the partial log-likelihood. The choice of the smoothing parameter, which determines the weight of the penalty, is based on Akaike's Information Criterion. The methods are illustrated in a set of ovarian cancer data and in a set of kidney transplantation data.