Partial Likelihood Function for the Cox Model


Let $t_{(1)} < t_{(2)} < \ldots <t_{(K)}$ denote the K distinct, ordered event times. Let $d_{k}$ denote the multiplicity of failures at $t_{(k)}$; that is, $d_{k}$ is the size of the set $\mc{D}_{k}$ of individuals that fail at $t_{(k)}$. Let $w_{hij}$ be the weight associated with the jth observation unit in the ith cluster in stratum h. Using this notation, the pseudo-likelihood functions used in PROC SURVEYPHREG to estimate $\bbeta _ N$ are described in the following sections.

Continuous Time Scale

Let $\mc{R}_ k$ denote the risk set just before the kth ordered event time $t_{(k)}$.

The Breslow likelihood is expressed as

\[  L_{\text {Breslow}}({\bbeta }) = \prod _{k=1}^{K} \frac{ \exp \left( {\bbeta ’\sum _{\mc{D}_ k}w_{hij}\bZ _{hij}(t)} \right)}{\left\{  {\displaystyle \sum _{\mc{R}_ k} w_{hij} \exp ( \bbeta ’\bZ _{hij}(t) ) } \right\} ^{\sum _{\mc{D}_ k} w_{hij}} }  \]

The Efron likelihood is expressed as

\[  L_{\text {Efron}}({\bbeta })=\prod _{k=1}^{K} \frac{ \exp \left( \bbeta '\sum _{\mc{D}_ k} w_{hij} \bZ _{hij}(t) \right) }{ \left\{  \phi (\bbeta , \mb Z, \mb w, k) \right\} ^{\frac{1}{d_ k}\sum _{\mc{D}_ k} w_{hij}} }  \]

where $\phi (\bbeta , \mb Z, \mb w, k)$ is

\[  \phi (\bbeta , \mb Z, \mb w, k) = \displaystyle \prod _{l=1}^{d_ k} \left\{  \sum _{\mc{R}_{k}} w_{hij} \exp \left( \bbeta ’ \bZ _{hij}(t) \right) - \frac{l-1}{d_ k} \sum _{\mc{D}_ k} w_{hij} \exp \left(\bbeta ’\bZ _{hij}(t) \right) \right\}   \]