STT 430/530, Nonparametric Statistics

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

• Fishers 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...
• Tukeys 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 dfinfinity.
• 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.
• Fishers (protected) LSD
• The rank-based equivalen is

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.