The PHREG Procedure

Diagnostics Based on Weighted Residuals

The vector of weighted Schoenfeld residuals, $\mb {r}_ i$, is computed as

\[  \mb {r}_ i = n_ e \mc {I}^{-1}(\hat{\bbeta }) \hat{\mb {U}}_ i(t_ i)  \]

where $n_ e$ is the total number of events and $\hat{\mb {U}}_ i(t_ i)$ is the vector of Schoenfeld residuals at the event time $t_ i$. The components of $\mb {r}_ i$ are output to the WTRESSCH= variables.

The weighted Schoenfeld residuals are useful in assessing the proportional hazards assumption. The idea is that most of the common alternatives to the proportional hazards can be cast in terms of a time-varying coefficient model

\[  \lambda (t,\bZ ) = \lambda _0(t)\exp (\beta _1(t)Z_1+\beta _2(t)Z_2 +\ldots )  \]

where $\lambda (t,\bZ )$ and $\lambda _0(t)$ are hazard rates. Let $\hat{\beta }_ j$ and $r_{ij}$ be the jth component of $\hat{\bbeta }$ and $\mb {r}_ i$, respectively. Grambsch and Therneau (1994) suggest using a smoothed plot of ($\hat{\beta }_ j + r_{ij}$) versus $t_ i$ to discover the functional form of the time-varying coefficient $\beta _ j(t)$. A zero slope indicates that the coefficient is not varying with time.

The weighted score residuals are used more often than their unscaled counterparts in assessing local influence. Let $\hat{\bbeta }_{(i)}$ be the estimate of $\bbeta $ when the ith subject is left out, and let $\delta \hat{\bbeta }_ i= \hat{\bbeta } - \hat{\bbeta }_{(i)}$. The jth component of $\delta \hat{\bbeta }_ i$ can be used to assess any untoward effect of the ith subject on $\hat{\beta }_ j$. The exact computation of $\delta \hat{\bbeta }_ i$ involves refitting the model each time a subject is omitted. Cain and Lange (1984) derived the following approximation of $\bDelta _ i$ as weighted score residuals:

\[  \delta \hat{\bbeta }_ i = \mc {I}^{-1}(\hat{\bbeta }) \hat{\bL }_ i  \]

Here, $\hat{\bL }_ i$ is the vector of the score residuals for the ith subject. Values of $\delta \hat{\bbeta }_ i$ are output to the DFBETA= variables. Again, when the counting process MODEL specification is used, the DFBETA= variables contain the component $ \mc {I}^{-1}(\hat{\bbeta })(\bL _ i(\hat{\bbeta },t_2) - \bL _ i(\hat{\bbeta },t_1))$, where the score process $\bL _ i(\bbeta ,t)$ is defined in the section Residuals. The vector $\delta \hat{\bbeta }_ i$ for the ith subject can be obtained by summing these components within the subject.

Note that these DFBETA statistics are a transform of the score residuals. In computing the robust sandwich variance estimators of Lin and Wei (1989) and Wei, Lin, and Weissfeld (1989), it is more convenient to use the DFBETA statistics than the score residuals (see Example 67.10).