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