Functional Form

Abstract

In the Cox model, we assume that the hazard function satisfies

$${\lambda _i}(t) = {\lambda _0}(t)\exp ({X_i}\beta ),$$

that is, a proportional hazards structure with a loglinear model for the covariates. For a continuous variable, age, for instance, this implicitly assumes the ratio of risks between a 45- and a 50-year-old is the same as that between an 80- and an 85-year-old. Yet experience with biological data shows that threshold effects, both upper and lower, are common. Perhaps the risk does not begin to rise until after age 65, or even rises and then falls again at a later age.