Proportional hazards regression model with unknown link function and applications to longitudinal time-to-event data

Proportional hazards regression model has played a pivotal role in survival analysis since Cox proposed it in 1972. This model assumes that the covariates affect the survival time through a link function and an index which is a linear function of the covariates. The link function is assumed to be known in the literature, and the most popular choice is the identity link function. In reality, the link function is often unknown and thus needs to be either estimated from the data, or to be validated before a specific form of the link function is applied. The goal of this thesis is to fill this gap by estimating both the unknown link function, and the unknown parameters of the index simultaneously.; A two-step iterative algorithm to estimate the link function and the covariate effects is proposed which does not involve the baseline hazard estimate. The link function is estimated by a smoothing method based on a local version of partial likelihood and the index function is then estimated using a full version of partial likelihood. Asymptotic properties of the estimators are derived for both the parametric covariate effects and the non-parametric estimated link function. The approach is illustrated through a liver disease data and simulations. Both the theory and methods are applicable to censored survival data.; This procedure applies to time-independent covariates and also to time-dependent covariates whose entire history is observed. Asymptotic results involving the link function and root-n consistency of the covariate coefficient also hold for such time-dependent covariates.; In practice, the entire history of the time-dependent covariates is not observable as patients are observed at certain scheduled times. This induced missing covariates values and a new functional principal components analysis method is proposed to impute the missing covariate values. Simulation studies show that the new estimates outperform some of the existing methods.