# Chapter 21 Section D

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1 Chapter 21 Section D Statistical Tests for Ordinal Data The rank-sum test. You can perform the rank-sum test in SPSS by selecting 2 Independent Samples from the Analyze/ Nonparametric Tests menu. The first (and default) choice in the dialog box under Test Type is the one that produces the rank-sum test, and it is labeled Mann- Whitney U. Just as in the Independent-Samples t test dialog box, you must specify the two levels of the Grouping Variable that you wish to compare (even if there are only two levels), as well as the list of Test Variables (DV s) on which you would like the two groups to be compared. Let s compare the phobia scores between the two genders. The dialog box will look like this: This test produces two boxes of output. The first is descriptive, and contains the N s, sums, and means of the ranks for each group. The second box contains three Test Statistics: the Mann-Whitney U (as mentioned in the text), the Wilcoxon W (the lower of the two sums of ranks), and Z, which is the z score calculated for this test using the normal approximation formula in the text. The p value associated with Z (SPSS uses an upper-case Z, to represent the same statistic for which I prefer to use the lower-case z) comes from the normal distribution, and is labeled Asymp. Sig. (2-tailed). The U and W statistics are provided probably so that you can look up their critical values from an appropriate statistical table. Given the p value for this test (.093), we can identify a trend (p <.1) for females to report a higher level of phobia, but the null hypothesis cannot be rejected.

2 Ranks gender N Mean Rank Sum of Ranks phobia Female Male Total 100 Test Statistics a phobia Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed).093 a. Grouping Variable: gender When the total N is below about 40, a row is added automatically to the bottom of the Test Statistics output box, labeled Exact Sig. The Exact Sig. is not based on an approximation, but rather on exact probabilities, so it is more accurate; the smaller your samples, the more Exact Sig. is preferred over Asymp. Sig. For total N s larger than about 40, the Exact Sig. is available by clicking on the Exact button in the upper-right corner of the Two- Independent-Samples dialog box (see previous figure), and then selecting Exact from the Exact tests box. That is what I did to produce the results box below. Test Statistics a phobia Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed).093 Exact Sig. (2-tailed).094 Exact Sig. (1-tailed).047 Point Probability.000 a. Grouping Variable: gender Note that for very large samples the Exact test can take a significant amount of computing time (and will barely differ at all from the normal approximation), which is why you are asked to set a time limit for the Exact test (the default is 5 minutes). However, producing the output above (total N = 100) required less than one second of computing time. A final note: If

3 you select the Exact test when the total N is less than 40, you not only get the Exact Sig. that is given automatically when N < 40, but also an Exact Sig. that is based on a correction for tied scores. The Kruskal-Wallis H test. To compute the Kruskal-Wallis H test, select K Independent Samples from the Analyze/ Nonparametric Tests menu. You will see that Kruskal-Wallis H is already selected as the default under Test Type (as in the following figure). The main difference between the dialog box shown above, and the one for two independent samples, is that you are asked to Define Range instead of Define Groups. You must enter integers to define the minimum and maximum values of interest for your Grouping Variable. Those integers entered, and all integer values between them, will each define a different group in the analysis. In the example depicted in the preceding figure, I am using undergrad major as the grouping variable (with number of previous math courses as the DV, or Test Variable), but by setting the range from 1 to 4, I am dropping the fifth ug major (Econ) from the analysis. [Note: If I had wanted to include only groups 2, 3, and 5, I would have entered 2 as the minimum of the range, and 5 as the maximum, but I would also have had to use Data/ Select Cases to exclude cases which have a value of 4 on the grouping variable (e.g., group ~= 4).] Like the Mann-Whitney rank-sum test, the Kruskal-Wallis procedure creates two boxes of output, the first of which contains the size and mean rank for each group in the specified range of the Grouping Variable. If you did not add to or change the default selection for Test Type, the second results box will contain only the H statistic, as calculated by the formula in the text, along with its df (one less than the number of groups being compared), and its approximate p value, labeled Asymp. Sig. (see the output boxes below). The H statistic is actually labeled Chi- Square in the output box, because that is the distribution from which the (asymptotic) p value is

4 obtained. In the output that follows you will see that the null hypothesis can be easily rejected at the.05 level, allowing us to conclude that, in the larger population, the undergrad majors represented do not all have the same average number of previous math classes (but see the text for a more nuanced version of what ordinal tests can allow you to conclude). Ranks ug_major N Mean Rank Prevmath Psych Pre-Med Bio Soc Total 90 Test Statistics a,b prevmath Chi-Square Df 3 Asymp. Sig..004 a. Kruskal Wallis Test b. Grouping Variable: ug_major Unlike the rank-sum test, the Exact test is not automatically included in the Test Statistics output box even when your total N is quite small. However, the Exact test can always be requested in the same way as for the rank-sum test but note for a large N and several groups, the Exact test may run out of time or computer memory. Fortunately, the chi-square distribution with df = k 1, serves as a good approximation for the H statistic, as long as the group sizes are not very small. The relation between the Kruskal-Wallis H and rank-sum tests is perfectly analogous to the relation between the one-way ANOVA and the independent-samples t test. If you run the Kruskal-Wallis test on just two groups, the chi-square value you will obtain is just the square of the Z score produced by the rank-sum test for the same two groups. And, of course, the p value will be the same for both tests. A significant H test can be followed by pairwise tests using either the H test with the range set for two groups at a time, or the rank-sum test however, your p values should be compared to a Bonferroni-adjusted alpha when more than three groups are involved. The Wilcoxon test. Anytime that your data has been entered in such a way that a paired-samples t test can be run, you can perform the Wilcoxon Signed Ranks test (as SPSS refers to it), instead. Just select 2 Related Samples from the Analyze/ Nonparametric Tests menu, and make sure that the first choice under Test Type ( Wilcoxon ) has been checked. Choose two

5 variables from the list on the left, and move the pair to the Test Pair(s) List, as you would for a paired-samples t test. Repeat this process for each pair of variables that you would like to test. The Wilcoxon test creates difference scores for each pair of variables to be tested, and then ranks these differences by magnitude, temporarily ignoring the signs. SPSS does not show you these difference scores, or their actual ranks, but the first output box for the Wilcoxon procedure gives the N, sum, and mean rank separately for the positive and negative differences. The number of ties (i.e., differences that are equal to zero not identical difference scores) is also noted. The second output box contains only the z score (labeled Z ) obtained from the Wilcoxon normal approximation formula (see text), along with its two-tailed p value. These two boxes are shown below for a comparison of heart rates measured before (hr_pre) and after (hr_post) the quiz that Sara gave. You will see that the null hypothesis of no difference in heart rate from before to after the quiz can be rejected at the.05 level. Ranks N Mean Rank Sum of Ranks hr_pre - hr_post Negative Ranks 39 a Positive Ranks 55 b Ties 6 c Total 100 a. hr_pre < hr_post b. hr_pre > hr_post c. hr_pre = hr_post

6 Test Statistics b hr_pre - hr_post Z a Asymp. Sig. (2-tailed).034 a. Based on negative ranks. b. Wilcoxon Signed Ranks Test As for the other tests described in this chapter, an Exact test is available for the Wilcoxon test. The results box produced by the Exact test is shown below. Test Statistics b hr_pre - hr_post Z a Asymp. Sig. (2-tailed).034 Exact Sig. (2-tailed).034 Exact Sig. (1-tailed).017 Point Probability.000 a. Based on negative ranks. b. Wilcoxon Signed Ranks Test As is often the case with large samples, the Exact Sig. is close to the asymptotic significance value (in this case, they are the same to the third decimal place). Also, note that the p value would be very similar if you were to perform a paired t test in place of the Wilcoxon test on pre vs. post heart rates. You can also perform a Sign test on these data, but that tests throws away all of the quantitative data and keeps track only of whether heart rate increases or decreases from before to after the quiz. Not surprisingly, given that in this case the Wilcoxon test barely reached significance, the Sign test for these same data yields a p value greater than.05. The Spearman rank-order correlation coefficient. The Spearman correlation cannot be obtained from the Analyze/ Nonparametric Tests menu, though it would have fit naturally as an option in the test for two related samples. Instead, as you may recall, Spearman is one of the choices (the default is Pearson ) when you open the dialog box for Bivariate Correlations (select Bivariate from the Analyze/Correlate menu). Unless the data for the two variables to be correlated have already been entered in terms of ranks, the Spearman correlation coefficient (labeled Spearman s rho by SPSS) will generally differ somewhat from the Pearson r, as will its Sig. value. However, either correlation coefficient can be the larger, depending on how the data are distributed for each variable, and how the data are distributed bivariately (see the text for further explanation).

7 Computer Exercises 1. Separately for each of the five majors, use the rank-sum test to determine whether male students differ significantly from female students with respect to: a. statistics quiz scores. b. pre-quiz anxiety scores. c. math phobia ratings. 2. Using college major as the grouping variable, perform the Kruskal-Wallis test to determine whether there are significant differences for the following DV s (compare your results with the corresponding ANOVA results you obtained for computer exercise #2 in Chapter 15): a. the math background quiz. b. the statistics quiz. 3. a. Use the Kruskal-Wallis procedure to test whether the different quiz conditions (last question easy, moderate, difficult, or impossible) had a significant effect on post-quiz anxiety scores. Regardless of the significance of your test, perform all of the possible pairwise comparisons as ordinal tests. Which of these pairwise tests would be significant after a Bonferroni adjustment of your alpha for each comparison? b. Repeat part a for the post-quiz heart rates. 4. a. Separately for male and female students, perform the Wilcoxon test to determine whether there is a significant increase in heart rate from baseline to the pre-quiz measurement. Compare your results to those you obtained in computer exercise #5 in Chapter 11. b. Repeat part a for the anxiety scores. 5. Separately for male and female students, compute the Spearman correlation coefficient between the baseline and the pre-quiz heart rates. How do these results relate to those you obtained in part a of the previous exercise? Compute the corresponding Spearman correlations for the anxiety scores, as well.

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