Applied Statistics and Probability for Engineers

518 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

assumptions are made. One possible assumption is to assume the interaction effect is negligi-

ble and use the interaction mean square as an error mean square. Thus, the analysis is equiva-

lent to the analysis used in the randomized block design. This no-interaction assumption can be

dangerous, and the experimenter should carefully examine the data and the residuals for indi-

cations as to whether or not interaction is present. For more details, see Montgomery (2001).

14-4.4 Factorial Experiments with Random Factors: Overview

In Section 13-3 we introduced the concept of a random factor.This is, of course, a situation

in which the factor of interest has a large number of possible levels and the experimenter

chooses a subset of these levels at random from this population. Conclusions are then drawn

about the population of factor levels.

Random factors can occur in factorial experiments. If all the factors are random, the analysis

of variance model is called a random-effects model.If some factors are fixed and other factors are

random, the analysis of variance model is called a mixed model.The statistical analysis of random

and mixed models is very similar to that of the standard fixed-effects models that are the primary

focus of this chapter. The primary differences are in the types of hypotheses that are tested, the con-

struction of test statistics for these hypotheses, and the estimation of model parameters. Some

additional details on these topics are presented in Section 14-6 on the CD. For a more in-depth pres-

entation, refer to Montgomery (2001) and Neter, Wasserman, Nachtsheim, and Kutner (1996).

• 0.5


• 2.0


• 0.3 –0.1 + 0.1 + 0.3


• 1.0

0.0

1.0

2.0

zj

eijk, residual

+0.5

0

• 0.5

3

eijk

12 Primer type

Figure 14-9 Normal probability plot of the Figure 14-10 Plot of residuals versus primer type.

residuals from Example 14-1.

+0.5

0

• 0.5

eijk

D

Application method

S

+0.5

0

• 0.5

eijk

4 5 6

y^ijk

Figure 14-11 Plot of residuals versus application

method.

Figure 14-12 Plot of residuals versus predicted

values yˆijk.

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