10/25/2023

Multiple Comparisons Procedures

• For general alternatives (two-sided, all treatments), after Kruskal-Wallis (Section 5)
• For ordered alternatives (one-sided, all treatments), after Jonckheere-Terpstra (Section 6)
• For treatment versus control, after Fligner-Wolfe (Section 7)

General Alternatives

After rejecting the null hypothesis of no difference in the treatments with the Kruskal-Wallis procedure, we identify the significant differences by:

1. Computing the Wilcoxon rank sum for pairwise comparison of the ith and jth samples.
2. Standardizing the Wilcoxon rank sums.
3. Comparing the absolute value of the observed statistic with a critical value from cSDCFlig(alpha, n)

Exercise 42 on p. 265

• Return to Google Drive\Stat216-Nonparametrics-F2023\!Class examples and labs\pine.R

Multiple Comparisons Procedures

• For ordered alternatives - Section 6.6 - use cHaySton and pHaySton (or cHayStonLSA) in R
• For treatments vs. control - Section 6.7 - use cMaxCorrNorm and cNDWol in R
  • Exercises 63 and 64 on page 277

One Sided All Treatment Multiple Comparisons

• After rejecting the null hypothesis in favor of the ordered alternative with the Jonckheere-Terpstra procedure, we identify the significant differences by
  • computing the pairwise Wilcoxon rank sum statistics and standardized versions as in Section 6.5
  • deciding tau (v)>tau(u) if the standardized rank sum is > the C.V.
• L.S.A. - we decide tau(v)>tau(u) if the standardized rank sum is > the C.V.
• Ties - Use average ranks and the same correction as in Section 6.5 in the denominator of the standardized rank sums

One Sided Treatment Versus Control Multiple Comparisons

• After rejecting the null hypothesis in favor of the alternative that the treatments are at least as large as the control with the Fligner-Wolfe procedure, we identify the significant differences by
  • jointly ranking all the sample observations from 1, …, N
  • computing the averages of the ranks assigned to each treatment
  • calculating the differences between the treatment averages and the control average
  • deciding tau(u)>tau(1) (at an experimentwise error rate of alpha) if N*(R.u-R.1) > the C.V., where N* is the least common multiple of n1, n2, …, nk. Note: N* is defined differently in Section 6.4.

Final Comments on Treatment versus Control Comparisons

• If the null hypothesis is rejected in favor of the alternative that the treatments are at least as small as the control with the Fligner-Wolfe procedure, then we apply the procedures above with (R.u-R.1) replaced by (R.1-R.u).
  • Use average ranks for ties
  • L.S.A.

Please complete your reading of Chapter 6 for class on Friday.