|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.
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.