Diagnostics Based on Weighted Residuals

The PHREG Procedure

Diagnostics Based on Weighted Residuals

The vector of weighted Schoenfeld residuals, r_i, is computed as

$r_i = n_e \hat{V}U_i(t_i)$

where n_e is the total number of events, $\hat{V}=\hat{V}(\hat{{\beta}})$ is the estimated covariance matrix of $\hat{{\beta}}$ ,and U_i(t_i) is the vector of Schoenfeld residuals at the event time t_i. The components of 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

$\lambda(t,Z) = \lambda_0(t)\exp(\beta_1(t)Z_1+\beta_2(t)Z_2 + ... )$

where $\lambda(t,Z)$ and $\lambda_0(t)$ are hazards rates. Let $\hat{\beta}_j$ and r_ij be the jth component of $\hat{{\beta}}$ and r_i, respectively. Grambsch and Therneau (1993) 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 $\hat{{\beta}}_{(i)}$ be the estimate of ${\beta}$ when the ith subject is left out, and let ${\Delta}_i= \hat{{\beta}}- \hat{{\beta}}_{(i)}$ .The jth component of ${\Delta}_i$ can be used to assess any untoward effect of the ith subject on $\hat{\beta}_j$ . The exact computation of ${\Delta}_i$ involves refitting the model each time a subject is omitted. Cain and Lange (1984) derived the following approximation of ${\Delta}_i$ as weighted score residuals:

$\hat{{\Delta}}_i = \hat{V}L_i$

Here, $\hat{V}=\hat{V}(\hat{{\beta}})$ is the estimated covariance matrix of $\hat{{\beta}}$ , and L_i is the vector of the score residuals for the ith subject. Values of $\hat{{\Delta}}_i$ are output to the DFBETA= variables. Again, when the counting process MODEL specification is used, the DFBETA= variables contain the component $(\hat{V}L_i(t2) - \hat{V}L_i(t1))$ . The vector $\hat{{\Delta}}_i$ for a 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 49.8).

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