Applied Statistics and Probability for Engineers

12-6 ASPECTS OF MULTIPLE REGRESSION MODELING 447

12-43. In Exercise 12-24 we fit a model to the response

PITCH in the heat treating data of Exercise 12-10 using new

regressors x 1 SOAKTIME SOAKPCT and x 2 DIFF-

TIME DIFFPCT.

(a) Calculate the R^2 for this model and compare it to the

value of R^2 from the original model in Exercise 12-10.

Does this provide some information about which model

is preferable?

(b) Plot the residuals from this model versus and on a

normal probability scale. Comment on model adequacy.

(c) Find the values of Cook’s distance measure. Are any ob-

servations unusually influential?

12-44. Consider the regression model for the NHL data

from Exercise 12-11.

(a) Fit a model using “pts” as the only regressor.

(b) How much variability is explained by this model?

yˆ

(c) Plot the residuals versus and comment on model

adequacy.

(d) Plot the residuals versus “PPG,” the points scored while in

power play. Does this indicate that the model would be

better if this variable were included?

12-45. The diagonal elements of the hat matrix are often

used to denote leverage—that is, a point that is unusual in its

location in the x-space and that may be influential. Generally,

the ith point is called a leverage pointif its hat diagonal

hiiexceeds 2p/n, which is twice the average size of all the hat

diagonals. Recall that pk1.

(a) Table 12-11 contains the hat diagonal for the wire bond

pull strength data used in Example 12-1. Find the average

size of these elements.

(b) Based on the criterion above, are there any observations

that are leverage points in the data set?

yˆ

12-6 ASPECTS OF MULTIPLE REGRESSION MODELING

In this section we briefly discuss several other aspects of building multiple regression models.

For more extensive presentations of these topics and additional examples refer to Montgomery,

Peck, and Vining (2001) and Myers (1990).

12-6.1 Polynomial Regression Models

The linear model is a general model that can be used to fit any relationship that

is linear in the unknown parameters .This includes the important class of polynomial

regression models.For example, the second-degree polynomial in one variable

(12-45)

and the second-degree polynomial in two variables

(12-46)

are linear regression models.

Polynomial regression models are widely used when the response is curvilinear, because

the general principles of multiple regression can be applied. The following example illustrates

some of the types of analyses that can be performed.

EXAMPLE 12-11 Sidewall panels for the interior of an airplane are formed in a 1500-ton press. The unit man-

ufacturing cost varies with the production lot size. The data shown below give the average

cost per unit (in hundreds of dollars) for this product (y) and the production lot size (x). The

scatter diagram, shown in Fig. 12-11, indicates that a second-order polynomial may be

appropriate.

y 1.81 1.70 1.65 1.55 1.48 1.40 1.30 1.26 1.24 1.21 1.20 1.18

x 20 25 30 35 40 50 60 65 70 75 80 90

Y 0  1 x 1  2 x 2  11 x^21  22 x^22  12 x 1 x 2 

Y 0  1 x 11 x^2 

yX

c 12 .qxd 5/20/02 9:34 M Page 447 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: