Statistical Methods in Medical Research 4ed

368 Modelling continuous data
Table 11.7 Residual diagnostic statistics for Fig. 11.10.
Fig. 11.10 Fitted slope s r10 t10 Leverage h10 Cook's D10
(a) 1 00 1 00
(b) 1 03 0 94 0 14 0 13 0 54 0 01
(c) 0 64 1 11 1 51 1 68 0 54 1 37
(d) 1 61 1 37 2 07 2 84 0 54 2 55
(e) 1 21 2 15 2 55 5 49 0 11 0 41
and 7 DF, respectively, the studentized and jackknife residuals are clearly not
remarkable for either Fig. 11.10(b) or (c). The greater sensitivity of the
jackknife residual is shown for Fig. 11.10(d and e); the lower values of the
studentized residual, r10, in these two cases is a consequence of the increase in
standard deviation due to the outlier under test. For Fig. 11.10(d and e), the
extra point is suggested as a possible outlier by the values of the jackknife
residual. We now consider the formulation of a significance test of the residuals.
For Fig. 11.10(d), t10 ˆ 2 84 and if this is assessed as a t statistic with 7 DF
then P ˆ 0 025. However, this takes no account of the fact that this residual has
been chosen as the largest, in absolute magnitude, and the effect of this is quite
considerable and increases with increasing n. The required significance level is
then the probability that the largest residual in a set of n residuals will be as large
as or larger than the observed value. This probability is the probability that not
all 10 residuals are in the central part of the distribution and is given approximately
by P ˆ 1 …1 0 025† 10 ˆ 0 22, since the jackknife residuals are almost
independent. This is the Bonferroni correction (see §13.3). An equivalent means
of assessment is to work with a lower critical significance probability, and the
largest residual would be significant at level a only if the t statistic gave P < a=n.
For Fig. 11.10(d), and a significance test at the 0 05 level, we would require
P < 0 005 so that the largest residual with a nominal P of 0 025 is not significant.
In Fig. 11.10(e) the largest residual has a nominal P of just less than 0 001 and so
there is significant evidence …P < 0 01…ˆ 10 0 001†† that the 10th point is an
outlier. Kleinbaum et al. (1998) give a table (Table A±8A) to facilitate the
assessment of the significance of jackknife residuals and from this table the
significance level is slightly less than 0 01 (critical value 5 41).
Checking for influential points
A point may be regarded as influential if it exerts more than its fair share in
determining the values of the regression coefficients. Conceptually the influence
of the ith point may be envisaged in terms of the changes in the estimates of the