The Multiplicative Hazards Model

The PHREG Procedure

The Multiplicative Hazards Model

Consider a set of n subjects such that the counting process $N_i \equiv \{N_i(t), t \geq 0\}$ 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 ${\beta}$ denote the vector of unknown regression coefficients. The multiplicative hazards function $\Lambda(t,Z_{i}(t))$ for N_i is given by

$Y_{i}(t)d\Lambda(t,Z_{i}(t)) = Y_{i}(t)\exp({\beta}'Z_{i}(t)) d\Lambda_{0}(t)$

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
$\Lambda_{0}(t)$ 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

${\cal L}({\beta}) = \prod_{i=1}^n \prod_{t \geq 0} \biggl\{ \frac{Y_{i}(t)\exp... ...}(t))} {\sum_{j=1}^nY_{j}(t)\exp({\beta}'Z_{j}(t))} \biggr\}^{\Delta N_{i}(t)}$

where $\Delta N_{i}(t) = 1$ if N_i(t) - N_i(t-) = 1, and $\Delta N_{i}(t) = 0$ otherwise.

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