Applied Statistics and Probability for Engineers

12-5 MODEL ADEQUACY CHECKING 445

We would like to examine the influential points to determine whether they control many

model properties. If these influential points are “bad” points, or erroneous in any way, they should

be eliminated. On the other hand, there may be nothing wrong with these points, but at least we

would like to determine whether or not they produce results consistent with the rest of the data. In

any event, even if an influential point is a valid one, if it controls important model properties, we

would like to know this, since it could have an impact on the use of the model.

Montgomery, Peck, and Vining (2001) and Myers (1990) describe several methods for

detecting influential observations. An excellent diagnostic is the distance measuredeveloped

by Dennis R. Cook. This is a measure of the squared distance between the usual least squares

estimate of based on all nobservations and the estimate obtained when the ith point is re-

moved, say, ˆ 1 i 2. The Cook distancemeasureis

Clearly, if the ith point is influential, its removal will result in changing considerably from

the value. Thus, a large value of Diimplies that the ith point is influential. The statistic Diis

actually computed using

ˆ

ˆ 1 i 2

Di

1 ˆ (^1) i 2 ˆ^2 ¿X¿X^1 ˆ (^1) i 2 ˆ^2
p ˆ^2
(^) i1, 2,p, n
Di (12-44)
ri^2
p^
hii
11 hii 2

i1, 2,p, n

From Equation 12-44 we see that Diconsists of the squared studentized residual, which

reflects how well the model fits the ith observation yi[recall that and a

component that measures how far that point is from the rest of the data is a

measure of the distance of the ith point from the centroid of the remaining n1 points]. A

value of Di1 would indicate that the point is influential. Either component of Di(or both)

may contribute to a large value.

EXAMPLE 12-10 Table 12-11 lists the values of the hat matrix diagonals hiiand Cook’s distance measure Difor

the wire bond pull strength data in Example 12-1. To illustrate the calculations, consider the

first observation:

The Cook distance measure Didoes not identify any potentially influential observations in the

data, for no value of Diexceeds unity.

0.035



3 1.57 2 5.2352 11 0.1573 242

3

̨ ̨

0.1573

11 0.1573 2



3 e 1  2 MSE 11 h 11242

p ̨^

h 11

11 h 112

D 1 

r 12

p ̨^ ̨

h 11

11 h 112

3 hii 11 hii 2

riei 2 ˆ^211 hii 24

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