Cox Proportional Hazards Regression Model

Abstract

The Cox proportional hazards model 132 is the most popular model for the analysis of survival data. It is a semiparametric model; it makes a parametric 1 assumption concerning the effect of the predictors on the hazard function, but makes no assumption regarding the nature of the hazard function λ(t) itself. The Cox PH model assumes that predictors act multiplicatively on the hazard function but does not assume that the hazard function is constant (i.e., exponential model), Weibull, or any other particular form. The regression portion of the model is fully parametric; that is, the regressors are linearly related to log hazard or log cumulative hazard. In many situations, either the form of the true hazard function is unknown or it is complex, so the Cox model has definite advantages. Also, one is usually more interested in the effects of the predictors than in the shape of λ(t), and the Cox approach allows the analyst to essentially ignore λ(t), which is often not of primary interest. The Cox PH model uses only the rank ordering of the failure and censoring times and thus is less affected by outliers in the failure times than fully parametric methods. The model contains as a special case the popular log-rank test for comparing survival of two groups. For estimating and testing regression coefficients, the Cox model is as efficient as parametric models (e.g., Weibull model with PH) even when all assumptions of the parametric model are satisfied. 171 When a parametric model's assumptions are not true (e.g., when a Weibull model is used and the population is not from a Weibull survival distribution so that the choice of model is incorrect), the Cox analysis is more efficient