Applied Statistics and Probability for Engineers

14-4 TWO-FACTOR FACTORIAL EXPERIMENTS 517

14-4.2 Model Adequacy Checking

Just as in the single-factor experiments discussed in Chapter 13, the residualsfrom a factorial

experiment play an important role in assessing model adequacy.The residuals from a

two-factor factorial are

That is, the residuals are just the difference between the observations and the corresponding

cell averages.

Table 14-8 presents the residuals for the aircraft primer paint data in Example 14-1. The

normal probability plot of these residuals is shown in Fig. 14-9. This plot has tails that do not

fall exactly along a straight line passing through the center of the plot, indicating some poten-

tial problems with the normality assumption, but the deviation from normality does not appear

severe. Figures 14-10 and, 14-11 plot the residuals versus the levels of primer types and ap-

plication methods, respectively. There is some indication that primer type 3 results in slightly

lower variability in adhesion force than the other two primers. The graph of residuals versus

fitted values in Fig. 14-12 does not reveal any unusual or diagnostic pattern.

14-4.3 One Observation per Cell

In some cases involving a two-factor factorial experiment, we may have only one replicate—

that is, only one observation per cell. In this situation, there are exactly as many parameters in

the analysis of variance model as observations, and the error degrees of freedom are zero. Thus,

we cannot test hypotheses about the main effects and interactions unless some additional

eijkyijkyij.

Table 14-7 Analysis of Variance From Minitab for Example 14-1

ANOVA (Balanced Designs)

Factor Type Levels Values

Primer fixed 3 1 2 3

Method fixed 2 Dip Spray

Analysis of Variance for Adhesion

Source DF SS MS F P

Primer 2 4.5811 2.2906 27.86 0.000

Method 1 4.9089 4.9089 59.70 0.000

Primer *Method 2 0.2411 0.1206 1.47 0.269

Error 12 0.9867 0.0822

Total 17 10.7178

Table 14-8 Residuals for the Aircraft Primer Experiment in Example 14-1

Application Method

Primer Type Dipping Spraying

1 0.27, 0.23, 0.03 0.10,0.40, 0.30

2 0.30,0.40, 0.10 0.27, 0.03, 0.23

3 0.03,0.13, 0.17 0.33,0.17,0.17

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