542 Chapter 15 Multiple Regression
If the data met the underlying assumptions, we would expect the values of Yto be
normally distributed about the regression line. In other words, with a very large data
set all of the Yvalues corresponding to a specific value of Xwould have a normal dis-
tribution. Five percent of these values would lie more than 1.96 adjusted standard er-
rors from the regression line. (I use the word “adjusted” because the size of the
standard error will depend in part on the degree to which Xdeparts from the mean of
X, as measured by .) Within this context, it may be meaningful to ask if a point lies
significantly far from the regression line. If so, we should be concerned about it. A t test
on the magnitude of the residuals is given by the statistic RStudent, sometimes
called the Studentized residual.This can be interpreted as a standard t statistic on
(N 2 p 2 1) degrees of freedom. Here we see that for case 11 RStudent 52 3.54. This
should give us pause because that is a substantial, and significant, deviation. It is often
useful to think of RStudent less as a hypothesis-testing statistic and more as just an indi-
cator of the magnitude of the residual. (Remember that here we are computing Nt-tests,
with a resulting very large increase in the familywise error rate.) But significant or not,
something that is 3.54 standard errors from the line is unusual and therefore noteworthy.
We are not predicting that case well.
We now turn to leverage ( ), shown in the column headed Hat Diag. Here we see that
most observations have leverage values that fall between about 0.00 and .20. The mean
leverage is (p 1 1)N 5 2 12 5 .167, and that is about what we would expect. Notice,
however, that two cases have larger leverage; namely, cases 11 and 12, which exceeds
Steven’s rule of thumb of 3(p 1 1)n 5 3(2) 12 5 .50. We have already seen that 11 has a
large residual, so its modest leverage may make it an influential point. Case 12 has a lever-
age value nearly twice as large. However, it falls quite close to the regression line with a
fairly small residual, and it is likely to be less influential.
Cook’s D, which is a measure of influence, varies as a function of distance (residual),
leverage ( ), and. Most of the values in the last column are quite small, but cases
11 and 12 are exceptions. In particular, observation 11 has a Dexceeding 1.00. The sam-
pling distribution of Cook’s Dis in dispute, and there is no general rule for what consti-
tutes a large value, but values over 1.00 are unusual.
We can summarize the results shown in Exhibit 15.2 by stating that each of the three
points labeled in Figure 15.5 is reflected in that table. Point Ahas a fairly, though not sig-
nificantly, large residual but has small values for both leverage and influence. Point Bhas a
large leverage, but Cook’s Dis not high and its removal would not substantially reduce. Point Chas a large residual, a fairly large leverage, and a substantial Cook’s D;
its removal would provide a substantial reduction in. This is the kind of observa-
tion that we should consider seriously. Although data should not be deleted merely because
they are inconvenient and their removal would make the results look better, it is important
to pay attention to observations such as case 11. There may be legitimate reasons to set that
case aside and to treat it differently. Or it may in fact be erroneous. Because this is not a
real data set, we cannot do anything further with it.
It may seem like overkill to compute regression diagnostics simply to confirm what
anyone can see simply by looking at a plot of the data. However, we have looked only at a
situation with one predictor variable. With multiple predictors there is no reasonable way
to plot the data and visually identify influential points. In that situation you should at least
create univariate displays, perhaps bivariate plots of each predictor against the criterion
looking for peculiar distributions of points, and compute diagnostic statistics. From those
statistics you can then target particular cases for closer study.
Returning briefly to the data on course evaluations, we can illustrate some additional
points concerning diagnostic statistics. Exhibit 15.3 contains additional statistics that were
MSresidualMSresidualhi MSresidual> >
> >
hihiStudentized
residual