# note08 - Chapter 3 K-Sample Methods 3 Multiple Comparisons...

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Unformatted text preview: Chapter 3: K-Sample Methods 3 Multiple Comparisons • Which treatments differ from the others among more than two treatments? – pairwise test: by doing so many pairwise comparisons, the probability of declaring at least two treatments to be different may be considerably greater than given significant level – multiple comparisons: determine which treatments differ from others in a way that will reduce the chance of spurious results • the experiment-wise error rate: the probability of declaring at least two treatments to be different when there are no differences among the k treat- ments • Three Rank-based procedures for controlling experiment-wise error rate (not ties in the data) – Bonferroni Adjustment * to have an experiment-wise error rate no greater than α * to do each of the k ( k- 1) / 2 comparisons at the level of significance α ′ = α k ( k- 1) / 2 * t-test, Wilcoxon rank-sum test or any other nonparametric test – Fisher’s Protected Least Significant Difference (LSD) * don’t try to determine which treatments differ from the others unless the ANOVA indicates that there are differences among the treatments * observations from normal distribution (a) F-test for equality of means as in the one-way ANOVA (b) if test is significant at a desired level α , do all pairwise t-test at level α | ¯ X i- ¯ X j | ≥ t α/ 2 ,df = S − k radicalBigg MSE parenleftbigg 1 n i + 1 n j parenrightbigg bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright least significant difference where S = n 1 + n 2 + ··· + n k . 1 * observations from non-normal distribution (a) Kruskal-Wallis test for equality of distributions (b) if test is significant at a desired level α , then declare the distri- bution of treatment i and j to be different if – Tukey’s Honest Significant Difference (HSD) * the test statistic Q Q = max ij parenleftbigg √ n | ¯ X i- ¯ X j | √ MSE parenrightbigg = √ n ( max i braceleftbig ¯ X i bracerightbig- min i braceleftbig ¯ X i bracerightbig) √ MSE * q ( α,k,df ): the upper-tail 100 α % point of the distribution of Q under H * sample sizes are equal ( n 1 = n 2 = ··· = n k = n ) · observations from normal distribution: declare treatments i and j to be different if | ¯ X i- ¯ X j | ≥ q ( α,k,df ) radicalbigg MSE n bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright...
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note08 - Chapter 3 K-Sample Methods 3 Multiple Comparisons...

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