Abstract Partial likelihood, introduced in Cox (1975, Partial likelihood. Biometrika, 62(2),269–276), formalizes the construction of the inference function developed in Cox (1972, Regression models and life-tables (with discussion). Journal of the Royal Statistical Society Series B, 34(2),187–220) and referred there to as a conditional likelihood. Partial likelihood can also be viewed as a version of composite likelihood, a different example of which was studied in Cox, and Reid (2004, A note on pseudolikelihood constructed from marginal densities. Biometrika, 91(3),729–737). In this note, I describe the links between partial and composite likelihood, and the connections to profile, marginal, and conditional likelihood. Somewhat tangentially, two recent applications of the Cox proportional hazards model from the medical literature are briefly discussed, as they highlight the model’s ongoing relevance while also raising some more general questions about inference.