Consider a set of n subjects such that the counting process for the ith subject represents the number of observed events experienced over time t. The sample paths of the process are step functions with jumps of size , with . Let denote the vector of unknown regression coefficients. The multiplicative hazards function for is given by
where
indicates whether the ith subject is at risk at time t (specifically, if at risk and otherwise)
is the vector of explanatory variables for the ith subject at time t
is an unspecified baseline hazard function
See Fleming and Harrington (1991) and Andersen et al. (1992). The Cox model is a special case of this multiplicative hazards model, where until the first event or censoring, and thereafter.
The partial likelihood for n independent triplets , has the form
where if , and otherwise.