Regression models – parametric and Cox proportional hazards models

You may recall that the survival data consists of complete as well as censored observations, and we saw that the lifetimes look like 400, 4500+, 1012, 1925, 1504+, … for the pbc dataset. Although the lifetimes are continuous random variables, a regression model of the form will not be appropriate here. In fact, there were many attempts to correct and improvise on models of this form in the 1970s, and most often the results were detrimental. We will define a generic hazards regression model as follows:

Here, t is the lifetime, is the lifetime indicator, is the covariate vector, is the vector of regression coefficients, and is the baseline hazard rate. A relative risks model that is of specific interest is the following:

We will focus solely on this class of model. First, the parametric hazards regression is considered. This means that we will specify the hazard rate through a parametric model, for example, through exponential...