The PHREG Procedure

## The Multiplicative Hazards Model

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 Ni are step functions with jumps of size +1, with Ni(0)=0. Let denote the vector of unknown regression coefficients. The multiplicative hazards function for Ni is given by where
• Yi(t) indicates whether the ith subject is at risk at time t (specifically, Yi(t)=1 if at risk and Yi(t)=0 otherwise)
• Zi(t) is the vector of explanatory variables for the ith subject at time t
• is an unspecified baseline hazard function
Refer to Fleming and Harrington (1991) and Andersen and others (1992). The Cox model is a special case of this multiplicative hazards model, where Yi(t)=1 until the first event or censoring, and Yi(t)=0 thereafter.

The partial likelihood for n independent triplets (Ni,Yi,Zi), i = 1, ... , n, has the form where if Ni(t) - Ni(t-) = 1, and otherwise.