Proportional hazards regression models with unknown link function and time-dependent covariates

Proportional hazards regression models assume that the covariates affect the survival time through a link function and an index which is a linear function of the covariates. We study the situation when the link is unspecified and some covariates are time-dependent. Due to the nature of irregular designs, oftentimes the history of the time-dependent covariates is not observable. We propose a two-stage approach to account for the missingness. In the first stage, we impute the missing time-dependent covariates using functional data analysis methods. In the second stage, we perform a two-step iterative algorithm to estimate the unknown link function. Asymptotic properties are derived for the non-parametric estimated link function when time-dependent covariates history is observable. The approach is illustrated through several simulations and a data set of a prostate cancer clinical trial.