Fifty Years of the Cox Model

Abstract

The Cox model is now 50 years old. The seminal paper of Sir David Cox has had an immeasurable impact on the analysis of censored survival data, with applications in many different disciplines. This work has also stimulated much additional research in diverse areas and led to important theoretical and practical advances. These include semiparametric models, nonparametric efficiency, and partial likelihood. In addition to quickly becoming the go-to method for estimating covariate effects, Cox regression has been extended to a vast number of complex data structures, to all of which the central idea of sampling from the set of individuals at risk at time t can be applied. In this article, we review the Cox paper and the evolution of the ideas surrounding it. We then highlight its extensions to competing risks, with attention to models based on cause-specific hazards, and to hazards associated with the subdistribution or cumulative incidence function. We discuss their relative merits and domains of application. The analysis of recurrent events is another major topic of discussion, including an introduction to martingales and complete intensity models as well as the more practical marginal rate models. We include several worked examples to illustrate the main ideas.