Tests for equality of scale

1

• Computing the ranks of data is only one of
  several possible so-called scoring methods that
  are in use... Section 2.7 reviews three of them
  well look at the Van der Waerden scores For
  X1, ... , XN as our combined set of data (two
  groups mnN), the Van der Waerdan score is
  defined as V(i) F-1(i/(N1))
• i.e., the standard normal quantile of i/(N1)
• (in R, this is done with the qnorm function and
  the values for various N are given in Table A.5)
• Once these scores are computed, we test the
  usual two-sample hypothesis by doing a
  permutation test on the sum of the Van der
  Waerden scores on one of the groups if there
  are ties in the data, we average the V. d. W.
  scores as we did with the ranks...
• See Table 2.7.1, p. 51 for a comparison. Try
  in R
• sum(qnorm(rank(c(granite,basalt))/13)16)
• Try this test on problem 5, page 73... How
  does the p-value relate to the one you computed
  with the Wilcoxon rank-sum statistic?

2

• FIGURE 2.8.1, page 52 two distributions with
  the same location parameters, but different scale
  parameters.

3
Tests for equality of scale

• Assume Xi from treatment 1 and Yj from
  treatment 2 as follows Xi ??????x and Yj
  ??????y where the epsilons are iid with median
  0. Both the X's and the Y's have the same
  location m but different scale parameters. The
  null hypothesis is H0 ?????????
• There are two nonparametric tests we'll
  consider
• The Siegel-Tukey test
• The Ansari-Bradley test
• Both of these tests require a different way of
  ranking the data
• arrange the data from smallest to largest
• assign rank 1 to the smallest obs, rank 2 to
  the largest, rank 3 to the next largest, rank 4
  to the next smallest, etc.
• to get the Siegel-Tukey statistic, apply the
  Wilcoxon rank sum test - the smaller ranks and
  smaller rank sum come from the group with higher
  variability (see Fig. 2.8.1). Use the critical
  values of the Wilcoxon statistic (Table A3 in our
  book) or in R

4

• to get the Ansari-Bradley statistic rank in the
  following way rank 1 goes to both the smallest
  and largest observations rank 2 to both the
  next smallest and the next largest etc. Then
  compute the sum of the ranks of treatment 1 - the
  problem is that p-values can't be obtained from
  Table 3 anymorebut both SAS and R have exact
  p-values available in their software.
• Let's look at Example 2.8.1 on page 53
• in SAS use PROC NPAR1WAY ST AB CLASS TREATMENT
  EXACT ST AB VAR OUNCES
• here, TREATMENT is the grouping variable and
  OUNCES is the response. ST stands for
  Siegel-Tukey and AB stands for Ansari-Bradley
• there is a function in R called ansari.test (the
  package is ctest - you might have to load it?).
  Check out the help file on this function I
  haven't found one for doing the Siegel-Tukey
  test can you write one?
• another possibility is to use a permutation test
  based on the ratio of the sample deviances see
  p. 54 and Figure 2.8.2 for the RMD test. Use R
  do this for HW. That is, replicate the results
  of Example 2.8.2 and show your empirical p-value
  is around .07