5. Regression Analysis Using the Proportional Hazards Model

In the previous chapter we saw how to compare two survival distributions without assuming a particular parametric form for the survival distributions, and we also introduced a parameter ψ that indexes the difference between the two survival distributions via the Lehmann alternative, \(S_{1}(t) = \left [S_{0}(t)\right ]^{\psi }\). Using Eq. 2.2.1 we can see that we can re-express this relationship in terms of the hazard functions, yielding the proportional hazards assumption , This equation is the key to quantifying the difference between two hazard functions, and the proportional hazards model is widely used. (Later we will see how to assess the validity of this assumption, and ways to relax it when necessary.) Furthermore, we can extend the model to include covariate information in a vector z as follows: While other functional relationships between the proportional hazards constant ψ and covariates z are possible, this is by far the most common in practice. This proportional hazards model will allow us to fit regression models to censored survival data, much as one can do in linear and logistic regression. However, not assuming a particular parametric form for h ₀(t), along with the presence of censoring, makes survival modeling particularly complicated. In this chapter we shall see how to do this using what we shall call a partial likelihood . This modification of the standard likelihood was developed initially by D.R. Cox [12], and hence is often referred to as the Cox proportional hazards model.