Partial Likelihood Function for the Cox Model

Let denote the vector explanatory variables for the th individual at time . Let denote the distinct, ordered event times. Let denote the multiplicity of failures at ; that is, is the size of the set of individuals that fail at . Let be the weight associated with the th individual. Using this notation, the likelihood functions used in PROC PHREG to estimate are described in the following sections.

Continuous Time Scale

Let denote the risk set just before the ith ordered event time . Let denote the set of individuals whose event or censored times exceed or whose censored times are equal to .

Exact Likelihood


Breslow Likelihood


Incorporating weights, the Breslow likelihood becomes


Efron Likelihood


Incorporating weights, the Efron likelihood becomes


Discrete Time Scale

Let denote the set of all subsets of individuals from the risk set . For each , is a -tuple of individuals who might have failed at .

Discrete Logistic Likelihood


The computation of and its derivatives is based on an adaptation of the recurrence algorithm of Gail, Lubin, and Rubinstein (1981) to the logarithmic scale. When there are no ties on the event times (that is, ), all four likelihood functions , , , and reduce to the same expression. In a stratified analysis, the partial likelihood is the product of the partial likelihood functions for the individual strata.