Applied Statistics and Probability for Engineers

498 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE

Figure 13-9 presents the results graphically. The underlined pairs of means are not different.

The LSD procedure indicates that chemical type 4 results in significantly different strengths

than the other three types do. Chemical types 2 and 3 do not differ, and types 1 and 3 do not

differ. There may be a small difference in strength between types 1 and 2.

13-4.3 Residual Analysis and Model Checking

In any designed experiment, it is always important to examine the residuals and to check for

violation of basic assumptions that could invalidate the results. As usual, the residuals for the

randomized complete block design are just the difference between the observed and estimated

(or fitted) values from the statistical model, say,

(13-33)

and the fitted values are

The fitted value represents the estimate of the mean response when the ith treatment is run in

the jth block. The residuals from the chemical type experiment are shown in Table 13-15.

Figures 13-10, 13-11, 13-12, and 13-13 present the important residual plots for the ex-

periment. These residual plots are usually constructed by computer software packages. There

is some indication that fabric sample (block) 3 has greater variability in strength when treated

with the four chemicals than the other samples. Chemical type 4, which provides the greatest

strength, also has somewhat more variability in strength. Followup experiments may be nec-

essary to confirm these findings, if they are potentially important.

13-4.4 Randomized Complete Block Design with Random Factors

(CD Only)

yˆijyi.y.jy..

eijyijyˆij

0 1234 6

1 3 24

5

Chemical type

Figure 13-9 Results of Fisher’s LSD method.

Table 13-15 Residuals from the Randomized Complete Block Design

Chemical Fabric Sample

Type 1 2 3 4 5

1 0.18 0.10 0.44 0.18 0.02

2 0.10 0.08 0.28 0.00 0.10

3 0.08 0.24 0.30 0.12 0.02

4 0.00 0.28 0.48 0.30 0.10

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