On a connection between the Bradley–Terry model and the Cox proportional hazards model

This article addresses a connection between the Bradley and Terry [1952a. The rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika 39, 324–345] model and Cox [1972. Regression models and life-tables. J. Roy. Statist. Soc. B 24, 187–220 (with discussion)] proportional hazards model. We show that the partial likelihood of random variables that satisfy the stratified proportional hazards assumption of Cox [1972. Regression models and life-tables. J. Roy. Statist. Soc. B 24, 187–220 (with discussion)] coincides with the likelihood given by the Bradley–Terry (BT) model for rank order events. Such a connection not only allows many available software for the Cox model to be used for regression analysis based on the BT model, but also enables the new developments to fit certain rank order data by using ideas that stem from the Cox model and its extensions available in many recent literatures from survival analysis.