The PHREG Procedure

## Partial Likelihood Function for the Cox Model

Let zl denote the vector of (possibly time-dependent) explanatory variables for the lth individual. Let t1 < t2 < ... <tk denote the k distinct, ordered event times. Let di denote the multiplicity of failures at ti; that is, di is the size of the set Di of individuals that fail at ti. Let si be the sum of the vectors zl over the individuals who fail at ti; that is, .Using this notation, the likelihood functions used in PROC PHREG to estimate are described in the following sections.

### Continuous Time Scale

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

The exact likelihood is The Breslow likelihood is The Efron likelihood is ### Discrete Time Scale

Let Qi denote the set of all subsets of di individuals from the risk set Ri. For each , q is a di-tuple (q1 , q2 , ... , qdi) of individuals who might have failed at ti. Let .

The discrete logistic likelihood is 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.