Proportional Hazards Models - Introduction

Introduction

Survival models can be viewed as consisting of two parts: the underlying hazard function, often denoted, describing how the hazard (risk) changes over time at baseline levels of covariates; and the effect parameters, describing how the hazard varies in response to explanatory covariates. A typical medical example would include covariates such as treatment assignment, as well as patient characteristics such as age, gender, and the presence of other diseases in order to reduce variability and/or control for confounding.

The proportional hazards condition states that covariates are multiplicatively related to the hazard. In the simplest case of stationary coefficients, for example, a treatment with a drug may, say, halve a subject's hazard at any given time, while the baseline hazard may vary. Note however, that the covariate is not restricted to binary predictors; in the case of a continuous covariate, the hazard responds logarithmically; each unit increase in results in proportional scaling of the hazard. The Cox partial likelihood shown below, is obtained by using Breslow's estimate of the baseline hazard function, plugging it into the full likelihood and then observing that the result is a product of two factors. The first factor is the partial likelihood shown below, in which the baseline hazard has "canceled out". The second factor is free of the regression coefficients and depends on the data only through the censoring pattern. The effect of covariates estimated by any proportional hazards model can thus be reported as hazard ratios.

Sir David Cox observed that if the proportional hazards assumption holds (or, is assumed to hold) then it is possible to estimate the effect parameter(s) without any consideration of the hazard function. This approach to survival data is called application of the Cox proportional hazards model, sometimes abbreviated to Cox model or to proportional hazards model.