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Table 13-12 Fabric Strength Data Randomized Complete Block Design Fabric Sample Chemical Type 1 2 3 4 Block totals y.j Block averages y.j 1 1.3 2.2 1.8 3.9 9.2 2.30 2 1.6 2.4 1.7 4.4 10.1 2.53 3 0.5 0.4 0.6 2.0 3.5 0.88 4 1.2 2.0 1.5 4.1 8.8 2.20 5 1.1 1.8 1.3 3.4 7.6 1.90 Treatment Totals yi. 5.7 8.8 6.9 17.8 39.2(y..) 1.96( y..) Treatment Averages yi. 1.14 1.76 1.38 3.56
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Table 13-13 Analysis of Variance for the Randomized Complete Block Experiment Source of Variation Chemical types (treatments) Fabric samples (blocks) Error Total
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The ANOVA is summarized in Table 13-13. Since f0 75.13 f0.01,3,12 5.95 (the P-value is 4.79 10 8), we conclude that there is a signi cant difference in the chemical types so far as their effect on strength is concerned. When Is Blocking Necessary Suppose an experiment is conducted as a randomized block design, and blocking was not really necessary. There are ab observations and (a 1)(b 1) degrees of freedom for error. If the experiment had been run as a completely randomized single-factor design with b replicates, we would have had a(b 1) degrees of freedom for error. Therefore, blocking has cost a(b 1) (a 1)(b 1) b 1 degrees of freedom for error. Thus, since the loss in error degrees of freedom is usually small, if there is a reasonable chance that block effects may be important, the experimenter should use the randomized block design. For example, consider the experiment described in Example 13-5 as a single-factor experiment with no blocking. We would then have 16 degrees of freedom for error. In the randomized block design, there are 12 degrees of freedom for error. Therefore, blocking has cost only 4 degrees of freedom, which is a very small loss considering the possible gain in information that would be achieved if block effects are really important. The block effect in Example 13-5 is large, and if we had not blocked, SSBlocks would have been included in the error sum of squares for the completely randomized analysis. This would have resulted in a much larger MSE, making it more dif cult to detect treatment differences. As a general rule, when in doubt as to the importance of block effects, the experimenter should block and gamble that the block effect does exist. If the experimenter is wrong, the slight loss in the degrees of freedom for error will have a negligible effect, unless the number of degrees of freedom is very small. Computer Solution Table 13-14 presents the computer output from Minitab for the randomized complete block design in Example 13-5. We used the analysis of variance menu for balanced designs to solve this problem. The results agree closely with the hand calculations from Table 13-13. Notice that Minitab computes an F-statistic for the blocks (the fabric samples). The validity of this ratio as a test statistic for the null hypothesis of no block effects is doubtful because the blocks represent a restriction on randomization; that is, we have only randomized within the blocks. If the blocks are not chosen at random, or if they are not run in random order, the
Table 13-14 Minitab Analysis of Variance for the Randomized Complete Block Design in Example 13-5 Analysis of Variance (Balanced Designs) Factor Chemical Fabric S Type xed xed Levels 4 5 Values 1 1 2 2 3 3 4 4
Analysis of Variance for strength Source Chemical Fabric S Error Total DF 3 4 12 19 SS 18.0440 6.6930 0.9510 25.6880 MS 6.0147 1.6733 0.0792 F 75.89 21.11 P 0.000 0.000
F-test with denominator: Error Denominator MS 0.079250 with 12 degrees of freedom Numerator Chemical Fabric S DF 3 4 MS 6.015 1.673 F 75.89 21.11 P 0.000 0.000
F-ratio for blocks may not provide reliable information about block effects. For more discussion see Montgomery (2001, 4).