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There are also non-parametric tests which can be done to compare scale parameters...

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 – we’ll look at the Van der Waerden scores: For X 1 , ... , X N as our combined set of data (two groups m+n=N), the Van der Waerdan score is defined as

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There are also non-parametric tests which can be done to compare scale parameters...

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  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 – we’ll look at the Van der Waerden scores: For X1, ... , XN as our combined set of data (two groups m+n=N), the Van der Waerdan score is defined as V(i) = F-1(i/(N+1)) i.e., the standard normal quantile of i/(N+1) (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... • 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. There are also non-parametric tests which can be done to compare scale parameters... • 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 follow: 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 that and there are two nonparametric tests we'll consider: • the Siegel-Tukey test; and • 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 anymore…but 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 ( and apply to #15 on page 74)

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