The PHREG Procedure

Partial Likelihood Function for the Cox Model

Subsections:

Let $\bZ _ l(t)$ denote the vector explanatory variables for the lth individual at time t. Let $t_{1} < t_{2} < \ldots <t_{k}$ denote the k distinct, ordered event times. Let $ d_ i$ denote the multiplicity of failures at $t_ i$; that is, $d_ i$ is the size of the set $\mc{D}_ i$ of individuals that fail at $t_ i$. Let $w_ l$ be the weight associated with the lth individual. Using this notation, the likelihood functions used in PROC PHREG to estimate $\bbeta $ are described in the following sections.

Continuous Time Scale

Let $\mc{R}_ i$ denote the risk set just before the ith ordered event time $t_ i$. Let $\mc{R}_{i}^{\ast }$ denote the set of individuals whose event or censored times exceed $t_ i$ or whose censored times are equal to $t_ i$.

Exact Likelihood
\[ L_{1}({\bbeta })=\prod _{i=1}^{k} \left\{ \int _{0}^{\infty } \prod _{j\in \mc{D}_ i} \left[ 1-\mr{exp} \left( \raisebox{1.5ex}{\mbox{ $ -\frac{ \raisebox{1ex}{\mbox{$ \mr{e}^{\bbeta '\bZ _ j(t_ i)} $} } }{\displaystyle \sum _{l \in \mc{R}_{i}^{\ast }}\mr{e}^{\bbeta '\bZ _ l(t_ i)}}t $} } \right) \right] \mr{exp}(-t)dt \right\} \]

Breslow Likelihood
\[ L_{2}({\bbeta })=\prod _{i=1}^{k} \frac{ \mr{e}^{\bbeta '\sum _{j\in \mc{D}_ i}\bZ _ j(t_ i)}}{\left[ \raisebox{.8ex}{\mbox{$\displaystyle \sum _{l \in \mc{R}_{i}} \mr{e}^{\bbeta '\bZ _ l(t_ i)} $} } \right]^{d_ i} } \]

Incorporating weights, the Breslow likelihood becomes

\[ L_{2}({\bbeta })=\prod _{i=1}^{k} \frac{ \mr{e}^{\bbeta '\sum _{j\in \mc{D}_ i}w_ j\bZ _ j(t_ i)}}{\left[ \raisebox{.8ex}{\mbox{$\displaystyle \sum _{l \in \mc{R}_{i}} w_ l \mr{e}^{\bbeta '\bZ _ l(t_ i)} $} } \right]^{\sum _{j\in \mc{D}_ i}w_ i} } \]
Efron Likelihood
\[ L_{3}({\bbeta })=\prod _{i=1}^{k}\frac{ \mr{e}^{\bbeta ' \sum _{j\in \mc{D}_ i}\bZ _ j(t_ i)} }{\displaystyle \prod _{j=1}^{d_ i} \left( \sum _{l \in \mc{R}_{i}} \mr{e}^{\bbeta '\bZ _ l(t_ i)} - \frac{j-1}{d_ i}\sum _{l\in \mc{D}_ i} \mr{e}^{\bbeta '\bZ _ l(t_ i)} \right) } \]

Incorporating weights, the Efron likelihood becomes

\[ L_{3}({\bbeta })=\prod _{i=1}^{k}\frac{ \mr{e}^{\bbeta ' \sum _{j\in \mc{D}_ i}w_ j\bZ _ j(t_ i)} }{\left[ \displaystyle \prod _{j=1}^{d_ i} \left( \sum _{l \in \mc{R}_{i}} w_ l \mr{e}^{\bbeta '\bZ _ l(t_ i)} - \frac{j-1}{d_ i}\sum _{l\in \mc{D}_ i} w_ l \mr{e}^{\bbeta '\bZ _ l(t_ i)} \right) \right]^{\frac{1}{d_ i}\sum _{j\in \mc{ D}_ i} w_ j}} \]

Discrete Time Scale

Let $\mc{Q}_ i$ denote the set of all subsets of $d_ i$ individuals from the risk set $\mc{R}_ i$. For each $\mb{q} \in \mc{Q}_{i}$, $\mb{q}$ is a $d_ i$-tuple $(q_{1},q_{2},\ldots ,q_{d_{i}})$ of individuals who might have failed at $t_ i$.

Discrete Logistic Likelihood

\[ L_{4}({\bbeta })=\prod _{i=1}^{k} \frac{ \mr{e}^{\bbeta '\sum _{j\in \mc{ D}_ i} \bZ _ j(t_ i)} }{\displaystyle \sum _{\mb{q} \in \mc{Q}_ i} \mr{e}^{\bbeta '\sum _{l=1}^{d_ i}\bZ _{q_ l}(t_ i)} } \]

The computation of $L_4(\bbeta )$ and its derivatives is based on an adaptation of the recurrence algorithm of Gail, Lubin, and Rubinstein (1981) to the logarithmic scale. When there are no ties on the event times (that is, $d_{i} \equiv 1$), all four likelihood functions $L_{1}({\bbeta })$, $L_{2}({\bbeta })$, $L_{3}({\bbeta })$, and $L_{4}({\bbeta })$ reduce to the same expression. In a stratified analysis, the partial likelihood is the product of the partial likelihood functions for the individual strata.