The PHREG Procedure

Robust Sandwich Variance Estimate

For the ith subject, $i=1,\ldots ,n$, let $X_ i$, $w_ i$, and $\bZ _ i(t)$ be the observed time, weight, and the covariate vector at time t, respectively. Let $\Delta _ i$ be the event indicator and let $Y_ i(t)=I(X_ i\geq t)$. Let

\[ S^{(r)}(\bbeta ,t) = \sum _{j=1}^ n w_ jY_ j(t) \mr{e}^{\bbeta '\bZ _ j(t)} \bZ _ j^{\bigotimes r}(t), r=0,1 \]

Let $ \bar{\bZ }(\bbeta ,t) = \frac{S^{(1)}(\bbeta ,t)}{S^{(0)}(\bbeta ,t)} $. The score residual for the ith subject is

\[ \bL _ i(\bbeta ) = \Delta _ i\biggl \{ \bZ _ i(X_ i) - \bar{\bZ }(\beta ,X_ i)\biggr \} - \sum _{j=1}^ n \Delta _ j \frac{w_ jY_ i(X_ j)\mr{e}^{\bbeta ' \bZ _ i(X_ j)}}{S^{(0)}(\bbeta ,X_ j)}\biggl \{ \bZ _ i(X_ j) - \bar{\bZ }(\bbeta ,X_ j)\biggr \} \]

For TIES=EFRON, the computation of the score residuals is modified to comply with the Efron partial likelihood. For more information, see the section Residuals.

The robust sandwich variance estimate of $\hat{\bbeta }$ derived by Binder (1992), who incorporated weights into the analysis, is

\[ \hat{\bV }_ s(\hat{\bbeta }) = \mc{I}^{-1}(\hat{\bbeta }) \biggl [\sum _{j=1}^ n (w_ j\bL _ j(\hat{\bbeta }))^{\bigotimes 2} \biggr ] \mc{I}^{-1}(\hat{\bbeta }) \]

where $\mc{I}(\hat{\bbeta })$ is the observed information matrix, and $\mb{a}^{\bigotimes 2}=\mb{a} \mb{a}’$. Note that when $w_ i \equiv 1$,

\[ \hat{\bV }_ s(\hat{\bbeta }) = \bD ’\bD \]

where $\bD $ is the matrix of DFBETA residuals. This robust variance estimate was proposed by Lin and Wei (1989) and Reid and Crépeau (1985).