4. Proportional hazards models

We consider several models that describe survival in the presence of observable covariates, these covariates measuring subject heterogeneity. The most general situation can be described by a model with a parameter of high, possibly unbounded, dimension. We refer to this as the general or non-proportional hazards model since dependence is expressed via a parameter, \(\beta (t),\) that is not constrained or restricted. Proportional hazards models have the same form but constrain \(\beta (t)\) to be a constant. We write the constant as \(\beta ,\) sometimes \(\beta _0\), since it does not change with time. When the covariate itself is constant, the dependence structure corresponds to the Cox regression model. We describe this model, its connection to the well-known log-rank test, and its use in many applications. We recall the founding paper of Cox ((Cox, 1972)) and the many discussions that surrounded that paper. Some of the historical backgrounds that lay behind Cox’s proposal is also recalled in order to for the new reader to quickly appreciate that, brilliant though Professor Cox’s insights were, they leant on more than just his imagination. They did not emerge from a vacuum. Some discussion on how the model should be used in practice is given.