Proportional Hazards Regression with Unknown Link Function

Proportional hazards regression model assumes that the covariates affect the hazard function through a link function and an index which is a linear function of the covariates. Traditional approaches, such as the Cox proportional hazards model, focus on estimating the unknown index by assuming a known link function between the log-hazard function and covariates. A linear link function is often employed for convenience without any validation. This paper provides an approach to estimate the link function, which can then be used to guide the choice of a proper parametric link function. This is accomplished through a two-step algorithm to estimate the link function and the effects of the covariates iteratively without involving 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 nonparametric link function estimate are derived, which facilitates model checking of the adequacy of the Cox Proportional hazards model. The approach is illustrated through a survival data and simulations.