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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 * to do each of the k ( k- 1) / 2 comparisons at the level of significance = k ( k- 1) / 2 * t-test, Wilcoxon rank-sum test or any other nonparametric test Fishers Protected Least Significant Difference (LSD) * dont 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 , do all pairwise t-test at level | X i- X j | t / 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 , then declare the distri- bution of treatment i and j to be different if Tukeys 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 ( ,k,df ): the upper-tail 100 % 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 ( ,k,df ) radicalbigg MSE n bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright...
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- Spring '08