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Table S13-1 Analysis of Variance for the Tensile Strength Data Source of Variation Hardwood concentration c (1, 4 vs. 2, 3) d (1, 2 vs. 3, 4) e (1, 3 vs. 2, 4) Error Total Sum of Squares 382.79 3.38 234.38 145.04 130.17 512.96 Degrees of Freedom 3 1 1 1 20 23 Mean Square 127.60 3.38 234.38 145.04 6.51 f0 19.61 0.52 36.00 22.28
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Notice that the contrast constants are orthogonal. Using the data from Table S13-1, we nd the numerical values of the contrasts and the sums of squares as follows: c d 60 60 94 94 102 102 127 127 9 75 SSc SSd SSe 1 92 2 6142 1752 2 6142 1592 2 6142 3.38 234.38
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These contrast sums of squares completely partition the treatment sum of squares; that is, SSTreatments SSc SSd SSe. These tests on the contrasts are usually incorporated in the analysis of variance, such as is shown in Table S13-1. From this analysis, we conclude that there are signi cant differences between hardwood concentration 3 and 4, and 1 and 2, but that the average of 1 and 4 does not differ from the average of 2 and 3. Also, the average of 1 and 3 differs from the average of 2 and 4. Tukey s Method The Tukey procedure for comparing pairs of means makes use of the studentized range statistic Q Ymax Ymin 1MSE n
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where Ymax and Ymin are the largest and smallest sample means, respectively, out of a group of p sample means. For equal sample sizes, the Tukey procedure would indicate that the two means i and j are different if the absolute value of the observed difference 0 yi. yj. 0 exceeds T g 1a, f 2 MSE B n
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where g (a, f ) is the upper percentage point of the studentized range statistic, a is the number of treatments, and f is the number of even degrees of freedom. Tables of g (a, f ) are
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Table S13-2 Minitab Output Illustrating Tukey s Method One-way ANOVA: Tensile Str versus Conc Analysis of Variance for Tensile Source DF SS Conc 3 382.79 Error 20 130.17 Total 23 512.96 MS 127.60 6.51 F 19.61 P 0.000
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StDev 2.828 2.805 1.789 2.639
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Individual 95% CIs For Mean Based on Pooled StDev -----+----------+----------+----------+(---*---) (---*----) (---*----) (---*----) -----+----------+----------+----------+10.0 15.0 20.0 25.0
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Tukey s pairwise comparisons Family error rate Individual error rate Critical value 3.96 (row level mean) 15 0.0500 0.0111
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Intervals for (column level mean) 5 10 10 9.791 1.542 15 20 11.124 2.876 15.291 7.042 5.458 2.791 9.624 1.376
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widely available; for example, see Montgomery (2001). Equivalently, we could construct a set of 100(1 )% con dence intervals for all pairs of mean using yi. For unequal sample sizes, use T yj. T
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The Tukey con dence intervals are a set of simultaneous con dence intervals that hold with probability 1 . Tukey s method is a very conservative procedure relative to Fisher s LSD because it requires a larger observed difference in treatment averages to declair the pair of means different. Minitab implements the Tukey procedure and reports the results in terms of the con dence interval. Table S13-2 is the Minitab output for the hardwood concentration experiment of Example S13-1. Notice that, like Fisher s LSD, Tukey s method indicates that all pairs of means are different except at 10% and 15% concentrations.
13-6 13-2.7 Technical Details about the Analysis of Variance (CD Only)
Derivation of the ANOVA Identity The proof of the fundamental ANOVA identity in Equation 13-5 is straightforward. Note that we may write a a 1 yij
y..2 2