## Partial Likelihood Function for the Cox Model

Let **z**_{l} denote the vector of (possibly time-dependent)
explanatory variables for the *l*th individual.
Let *t*_{1} < *t*_{2} < ... <*t*_{k}
denote the *k* distinct, ordered event times.
Let *d*_{i} denote the multiplicity of failures at
*t*_{i}; that is,
*d*_{i} is the size of the set
*D*_{i} of individuals that fail at
*t*_{i}.
Let **s**_{i} be the sum of the vectors
**z**_{l} over the individuals who fail at
*t*_{i};
that is,
.Using this notation, the likelihood functions used in PROC PHREG
to estimate are described in the following sections.

*Continuous Time Scale*

Let *R*_{i} denote the risk set
just before the *i*th ordered event time
*t*_{i} .
Let denote the set of individuals whose event or censored times exceed
*t*_{i} or whose
censored times are equal to *t*_{i}.
The exact likelihood is

The Breslow likelihood is

The Efron likelihood is

*Discrete Time Scale*

Let *Q*_{i} denote the set of all subsets of
*d*_{i} individuals from
the risk set *R*_{i}.
For each , **q** is a
*d*_{i}-tuple
**(***q*_{1} , *q*_{2} , ... , *q*_{di})
of individuals who might have failed at *t*_{i}. 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.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.