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# anova-b - 13 Additional ANOVA Topics Post hoc Comparisons |...

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Page 13.1 (C:\data\StatPrimer\anova-b.wpd 8/9/06) 13: Additional ANOVA Topics Post hoc Comparisons | ANOVA Assumptions | Assessing Group Variances When Distributional Assumptions are Severely Violated | Kruskal-Wallis Test Post hoc Comparisons 0 In the prior chapter we used ANOVA to compare means from k independent groups. The null hypothesis was H : all i : are equal. Moderate P -values reflect little evidence against the null hypothesis whereas small P -values indicate that either the null hypothesis is not true or a rare event had occurred. In rejecting the null declared, we would 0 1 2 declare that at least one population mean differed but did not specify how so. For example, in rejecting H : : = : = 3 4 : = : we were uncertain whether all four means differed or if there was one “odd man out.” This chapter shows how to proceed from there. Illustrative data (Pigmentation study) . Data from a study on skin pigmentation is used to illustrate methods and concepts in this chapter. Data are from four families from the same “racial group.” The dependent variable is a measure of skin pigmentation. Data are: Family 1: 36 39 43 38 37 = 38.6 Family 2: 46 47 47 47 43 = 46.0 Family 3: 40 50 44 48 50 = 46.4 Family 4: 45 53 56 52 56 = 52.4 1 2 3 4 There are k = 4 groups. Each group has 5 observations( n = n = n = n = n = 5), so there are N = 20 subjects total. A one-way ANOVA table (below) shows the means to differ significantly ( P < 0.0005): Sum of Squares SS df Mean Square F Sig. Between 478.95 3 159.65 12.93 .000 Within 197.60 16 12.35 Total 676.55 19 Side-by-side boxplots (below) reveal a large difference between group 1 and group 4, with intermediate resultsi n group 2 and group 3.

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Tukey, J. W. (1991). The Philosophy of Multiple Comparisons. Statistical Science, 6 (1), 100-116. * Rothman, K. J. (1990). No adjustments are needed for multiple comparisons. Epidemiology, 1 , 43-46. Page 13.2 (C:\data\StatPrimer\anova-b.wpd 8/9/06) The overall one-way ANOVA results are significant, so we concluded the not all the population means are equal. We now compare means two at a time in the form of post hoc (after-the-fact) comparisons . We conduct the following six tests: 0 1 2 1 1 2 0 1 3 1 1 3 Test 1: H : : = : vs. H : : : Test 2: H : : = : vs. H : : : 0 1 4 1 1 4 0 2 3 1 2 3 Test 3: H : : = : vs. H : : : Test 4: H : : = : vs. H : : : 0 2 4 1 2 4 0 3 4 1 3 4 Test 5: H : : = : vs. H : : : Test 6: H : : = : vs. H : : : Conducting multiple post hoc comparisons (like these) leads to a problem in interpretation called “The Problem of Multiple Comparisons.” This boils down to identifying too many random differences when many “looks” are taken: A man or woman who sits and deals out a deck of cards repeatedly will eventually get a very unusual set of hands. A report of unusualness would be taken quite differently if we knew it was the only deal ever made, or one of a thousand deals, or one of a million deals. * Consider testing 3 true null hypothesis. In using a = 0.05 for each test, the probability of making a correct retention is 0.95. The probability of making three consecutive correct retentions = 0.95 × 0.95 × 0.95 .
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