## Diagnostics Based on Weighted Residuals

The vector of weighted Schoenfeld residuals, **r**_{i},
is computed as

where *n*_{e} is the total number of events, is the
estimated covariance matrix of ,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

where and are hazards rates.
Let and *r*_{ij} be the *j*th component of and
**r**_{i}, respectively.
Grambsch and Therneau (1993) suggest using a smoothed plot
of () versus *t*_{i} to discover
the functional form
of the time-varying coefficient .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 be the estimate of when the *i*th
subject is left out, and let .The *j*th component of can be used to assess any untoward effect of the *i*th subject
on . The exact computation of involves refitting the model each time a subject is omitted.
Cain and Lange (1984) derived the following approximation of
as weighted score residuals:

Here, is the
estimated covariance matrix of , and
**L**_{i} is the vector of the score residuals for the
*i*th subject. Values of are output
to the DFBETA= variables.
Again, when the counting process MODEL specification
is used, the DFBETA= variables contain the component
. The vector 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).

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.