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 $\bbeta $ denote the vector of unknown regression coefficients. The multiplicative hazards function $\Lambda (t,{\bZ }_{i}(t))$ for $N_{i}$ is given by

\[  Y_{i}(t)d\Lambda (t,{\bZ }_{i}(t)) = Y_{i}(t)\exp (\bbeta ’\bZ _{i}(t)) d\Lambda _{0}(t)  \]


  • $Y_ i(t)$ 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)

  • $\bZ _ i(t)$ is the vector of explanatory variables for the ith subject at time t

  • $\Lambda _{0}(t)$ 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 $Y_ i(t)=1$ until the first event or censoring, and $Y_ i(t)=0$ thereafter.

The partial likelihood for n independent triplets $(N_{i},Y_{i},{\bZ }_{i}), i=1, \ldots , n$, has the form

\[  \mc{L}(\bbeta ) = \prod _{i=1}^{n} \prod _{t \geq 0} \biggl \{  \frac{Y_{i}(t)\exp (\bbeta '\bZ _{i}(t))}{\sum _{j=1}^{n}Y_{j}(t)\exp (\bbeta '\bZ _{j}(t))} \biggr \} ^{\Delta N_{i}(t)}  \]

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