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Bonferroni: If there are k groups, then there are

the F-statistic or KW statistic only tell you whether the populations are the same or not – not how they differ.

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Bonferroni: If there are k groups, then there are

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  1. the F-statistic or KW statistic only tell you whether the populations are the same or not – not how they differ. • the problem of trying to compare multiple populations pairwise is that for even a small number of treatments, there can be many comparisons, some of which are likely to be shown significantly different by chance even when they are not… so several techniques have been developed to try to control the so-called experiment-wise error rate – the probability of declaring at least two treatments different when there is no difference among all the treatments. • review the 3 multiple comparison procedures in section 3.3.1 for controlling experiment-wise error rates: • Bonferroni adjustment (divide the desired level of significance by the number of comparisons and do each comparison at that level). Do pairwise comparisons with the two-sample t-test if data is normal; otherwise use Wilcoxon rank-sum tests to do the comparisons.

  2. Fisher’s protected LSD (find the least significant difference for each pairwise comparison and declare two groups different if their means are greater than this LSD). The “protected” part of this comes from the initial use of the F-test (or Kruskal-Wallis) to declare at least one of the distributions is different. See page 93 for the LSD in both the usual parametric case and its non-parametric alternative... • Tukey’s honest significant difference (HSD) procedure (more complicated – see p.94-95 for a good discussion…). When there are unequal sample sizes, the Tukey-Kramer procedure is used. • Go over example 3.3.1 on page 95-96 to illustrate each of these three procedures in both the normal theory and in the nonparametric case… • Each of these multiple comparison procedures may be applied to midranks or to van der Waerden scores, etc.: replace the MSE with the sample variance of the scores and use df=infinity. • Permutation tests of each of the above tests are also possible – see section 3.3.3. Can you use Figure 3.3.1 on page 100 to write an R program to do permutation tests of the three multiple comparison procedures??

  3. Bonferroni: If there are k groups, then there are different pairwise comparisons that could be made. So divide the level of significance by this value and use this new level to test for differences in the groups. Use t-tests or Wilcoxon tests as appropriate. • Fisher’s (protected) LSD The rank-based equivalen is: Note the replacement of the MSE by its rank equivalent, the sample variance of the ranks (see bottom of p.90).

  4. Tukey-Kramer (unequal sample sizes), uses the Studentized Range statistic Q def. on p. 94: • Now the rank equivalent: • Try these on Example 3.3.1 on page 95. In R, use the Studentized Range distribution (see Tukey: ptukey or qtukey are the distribution and quantile functions.

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