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When the ANOVA indicates that a difference exists between the treatment means, we may need to perform some follow-up tests to isolate the speci c differences. Any multiple comparison method, such as Fisher s LSD method, could be used for this purpose. We will illustrate Fisher s LSD method. The four chemical type averages from Example 13-5 are: y1. 1.14 y2. 1.76 y3. 1.38 210.082 B 5 y4. 3.56
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Each treatment average uses b 5 observations (one from each block). We will use 0.05, so t0.025,12 2.179. Therefore the value of the LSD is LSD t0.025,12 2MSE B b 2.179 0.39
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Any pair of treatment averages that differ by 0.39 or more indicates that this pair of treatment means is signi cantly different. The comparisons are shown below: 4 vs. 1 4 vs. 3 4 vs. 2 2 vs. 1 2 vs. 3 3 vs. 1 y4. y4. y4. y2. y2. y3. y1. y3. y2. y1. y3. y1. 3.56 3.56 3.56 1.76 1.76 1.38 1.14 1.38 1.76 1.14 1.38 1.14 2.42 2.18 1.80 0.62 0.38 0.24 0.39 0.39 0.39 0.39 0.39 0.39
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Figure 13-9 Results of Fisher s LSD method.
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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 signi cantly 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.
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Residual Analysis and Model Checking
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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 tted) values from the statistical model, say, eij and the tted values are
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The tted 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 experiment. 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 necessary to con rm these ndings, if they are potentially important.
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13-4.4 Randomized Complete Block Design with Random Factors (CD Only)
Table 13-15 Residuals from the Randomized Complete Block Design Chemical Type 1 2 3 4 Fabric Sample 1 0.18 0.10 0.08 0.00 2 0.10 0.08 0.24 0.28 3 0.44 0.28 0.30 0.48 4 0.18 0.00 0.12 0.30 5 0.02 0.10 0.02 0.10
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1 0.5 0
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2 0.50 0.25 0 0.25 0.50 0.5 Residual value
Figure 13-10 Normal probability plot of residuals from the randomized complete block design.
0.5 0.5
Figure 13-11
Residuals by treatment.
eij 0 1 2 3 4 5
ei j 0 2 4 6
Figure 13-12
Residuals by block.
Figure 13-13
Residuals versus yij.
13-25. In The Effect of Nozzle Design on the Stability and Performance of Turbulent Water Jets (Fire Safety Journal, Vol. 4, August 1981), C. Theobald describes an experiment in which a shape measurement was determined for several different nozzle types at different levels of jet ef ux velocity. Interest in this experiment focuses primarily on nozzle type, and velocity is a nuisance factor. The data are as follows: Jet Ef ux Velocity (m/s) 11.73 0.78 0.85 0.93 1.14 0.97 14.37 0.80 0.85 0.92 0.97 0.86 16.59 0.81 0.92 0.95 0.98 0.78 20.43 0.75 0.86 0.89 0.88 0.76 23.46 0.77 0.81 0.89 0.86 0.76 28.74 0.78 0.83 0.83 0.83 0.75 (a) Does nozzle type affect shape measurement Compare the nozzles with box plots and the analysis of variance. (b) Use Fisher s LSD method to determine speci c differences between the nozzles. Does a graph of the average (or standard deviation) of the shape measurements versus nozzle type assist with the conclusions (c) Analyze the residuals from this experiment. 13-26. In Design and Analysis of Experiments, 5th edition (John Wiley & Sons, 2001), D. C. Montgomery describes an experiment that determined the effect of four different types of tips in a hardness tester on the observed hardness of a metal alloy. Four specimens of the alloy were obtained, and each tip was tested once on each specimen, producing the following data: (a) Is there any difference in hardness measurements between the tips