Statistical Methods in Medical Research 4ed

ti ˆ
ei
s… i† …
p 1 hi†
gives the externally studentized residual or jackknife residual. From (11.67),
ti ˆ ri
s
s … i†
11.9 Checking the model 367
…11:68†
…11:69†
and it can be shown mathematically that
n p 2
ti ˆ ri
n p 1 r2 r
: …11:70†
i
so that the jackknife residuals are easily computed for every point without
having to fit the n separate regressions, each excluding one point. The numerator
and denominator of (11.68) are uncorrelated and the jackknife residuals are
distributed exactly as the t distribution on n p 2DF.
The definition of ti might seem to be no more than an adjustment using a
more appropriate estimate of standard deviation, but it is in fact much more than
this. Suppose the residual were redefined as the difference between the observed
yi and the value, Y… i†, that would be predicted for the ith point using the
regression based on all the points except that point. The standard error of this
residual could be calculated, using the multivariate extension of (7.22). Then
define as a standardized residual
ti ˆ yi Y… i†
: …11:71†
SE…yi Y… i††
The residual defined in this way, as an externally studentized residual, is mathematically
equivalent to the jackknife residual as defined in (11.70). Therefore
each jackknife residual is a standardized residual about the regression fitted
using all the data except the point under consideration. This rather remarkable
mathematical property hinges on the leverage. From (11.70) the studentized
residual, ri, cannot exceed …
p n p 1†, and this limit is only reached when
deletion of the ith point results in a regression that is a perfect fit to all the other
points (Gray & Woodall, 1994). In this extreme case the jackknife residual has an
infinitely high value. Mathematically inclined readers are referred to the books
by Cook and Weisberg (1982) and Belsley (1991) for a full mathematical treatment.
These residuals are given in Table 11.7 for the data shown in Fig. 11.10. The
column headed D10 will be referred to later. Since all the residuals are standardized
and independent of the scale on which y and x are measured, these scales
have been chosen so that the slope of the line and the standard deviation in
Fig. 11.10(a) are both unity. For Fig. 11.10(b to e) the residual diagnostic statistics
are given for the extra 10th point. When judged against the t distribution with 8