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 N_{i} are step functions with jumps of size
+1, with N_{i}(0)=0. Let denote the vector of
unknown regression coefficients. The multiplicative hazards function
for N_{i} is given by
where
- Y_{i}(t) indicates whether the ith subject is at risk
at time t (specifically, Y_{i}(t)=1 if at risk and Y_{i}(t)=0 otherwise)
- Z_{i}(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 Y_{i}(t)=1 until the first
event or censoring, and Y_{i}(t)=0 thereafter.
The partial likelihood for n independent triplets
(N_{i},Y_{i},Z_{i}), i = 1, ... , n, has the form
where
if N_{i}(t) - N_{i}(t-) = 1, and
otherwise.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.