1. Chapter 21prepared by Elizabeth Bauer, Ph.D. 1 Ranking Data –Sometimes your data is ordinal level –We can put people in order and assign them ranks Common to assign the lowest rank to the one with the most of the variable in question Comparing Ranks from 2 Separate Groups –We sum the ranks for each group: a test statistic is needed Chapter 21: Statistical Tests for Ordinal Data

2. Chapter 21prepared by Elizabeth Bauer, Ph.D. 2 Mann-Whitney U-test –Wilcoxon rank-sum test (modification of the Mann- Whitney) –The Sum of Ranks (S R ) (where N is the total number of people ranked) –The U Statistic ΣR A represents the sum of ranks for members of group A

3. Chapter 21prepared by Elizabeth Bauer, Ph.D. 3 Dealing with Tied Scores –These statistical procedures rest on the assumption that the variables are continuous (even though they are measured on an ordinal scale) Implies no ties in theory; not true in practice If many ties –Either variable is not continuous or –Ability to discern differences is too crude –Compromises the validity of tests in this chapter Sometimes we assign average ranks to tied scores

4. Chapter 21prepared by Elizabeth Bauer, Ph.D. 4 When to Use the Mann- Whitney Test –Usually used to answer the same type of questions answered by a two groups independent t test –When the variable of interest cannot be measured precisely but subjects can be put into meaningful order Many variables do not lend themselves to precise measurement (e.g. assertiveness) Controversy over using some variables as interval scale although they are ordinal (e.g. self-report measures on introversion)

5. Chapter 21prepared by Elizabeth Bauer, Ph.D. 5 –When the interval/ratio data do not meet the parametric assumptions Repeated Measures or Matched Samples –Difference scores that are not normal can be analyzed with the Sign test Only uses direction of effect (quantitative data are thrown away); less powerful –Wilcoxon matched-pairs signed- ranks test (Wilcoxon’s T test) More powerful alternative Difference scores are put in order, regardless of sign Ranks are summed separately for the negative and positive differences

6. Chapter 21prepared by Elizabeth Bauer, Ph.D. 6 The Normal Approximation to the Mann-Whitney Test –With fairly large samples, we can use a normal approximation –Use Table A.1 for critical values –Normal approximation becomes quite reasonable when n B is larger than 10 Effect size of the Mann-Whitney –Rank biserial correlation coefficient

7. Chapter 21prepared by Elizabeth Bauer, Ph.D. 7 –Relates the difference in average rank between the two samples to the total size of the two samples –Ranges from -1 to +1 –1.0 is attained when there is no overlap in the rankings of the two groups –Sign of r G will always match z Assumptions of the Mann- Whitney –Independent random sampling –Dependent variable is continuous

8. Chapter 21prepared by Elizabeth Bauer, Ph.D. 8 Ranking the Differences between Paired Scores: The Wilcoxon Signed-Ranks Test –Can be used with interval/ratio data that fail to meet assumtions –Also used when the DV is measured on an ordinal scale that lacks precise intervals Power of the Wilcoxon Test –Less power than matched t test –More power than Mann-Whitney (unless matching is poor: look at r S ) –More power than the Sign test

9. Chapter 21prepared by Elizabeth Bauer, Ph.D. 9 Normal Approximation to the Wilcoxon Test –Use for more than 20 pairs Effect Size of the Wilcoxon (Signed Ranks) Test –Matched-pairs rank biserial correlation coefficient

10. Chapter 21prepared by Elizabeth Bauer, Ph.D. 10 Assumptions of the Wilcoxon Signed-Ranks Test –Independent random sampling –The dependent variable is continuous When to Use the Wilcoxon –Differences between pairs of observations are ranked directly –Difference scores are derived from ordinal data but can be accurately ranked –Difference scores are derived from interval/ratio data but do not meet distributional assumptions

11. Chapter 21prepared by Elizabeth Bauer, Ph.D. 11 Correlation with Ordinal Data –The Spearman Correlation Coefficient (r S ) At least one variable is ordinal Calculate Pearson r using ranks of ordinal data Sometimes r S is useful even with interval/ratio data –Outliers –Curvilinear relationships –Determining Statistical Significance for Spearman H 0 : The two variables are independent in the population Need special tables (can’t use Pearson tables)

12. Chapter 21prepared by Elizabeth Bauer, Ph.D. 12 When to Use Spearman –Both variables have been measured on an ordinal scale –One variable has been measured on as ordinal scale and the other on an interval or ratio scale –Both variables have been measured on an interval/ratio scale Distributional assumptions were violated and sample size is small Non-linear relationship and you want to assess monotonicity

13. Chapter 21prepared by Elizabeth Bauer, Ph.D. 13 Testing for Differences in Ranks among Several Groups: The Kruskal-Wallis Test –Extension of the Mann-Whitney Combine all subjects from all subgroups into one large group and rank order them on the DV; ties are given average ranks Sum the ranks separately for each subgroup in study The test statistic: H (sometimes called Kruskal-Wallis H test) T i = sum of the ranks in one subgroup N i = number of subjects in one of the subgroups k = number of subgroups N = total # of subjects in study

14. Chapter 21prepared by Elizabeth Bauer, Ph.D. 14 H is minimized when the ranks are equally divided among the groups We can use the χ 2 table (A.14) for critical values; df = k – 1 –If none of our subgroups were larger than 5, the χ 2 distribution would not be a good approximation –Assumptions are the same as Mann-Whitney test and used in same circumstances –Ties are handled as in Mann- Whitney –Correction factor for H if more than about 25% of the observations result in ties

15. Chapter 21prepared by Elizabeth Bauer, Ph.D. 15 Follow-up Tests & Effect Size for the Kruskal-Wallis H Test –Mann-Whitney rank-sum test on each pair –Three groups: α =.05 –More than three: Bonferroni or some other nonparametric adjustment –K-W test is a good alternative to one-way independent ANOVA Comparable power –Effect size measure; provides proportion of variance accounted for:

16. Chapter 21prepared by Elizabeth Bauer, Ph.D. 16 Testing for Differences in Ranks among Matched Subjects: The Friedman Test –Order the subjects (or repeated observations) within each matched set When results are inconsistent from subject to subject, the sums of ranks are more similar; less likely to be significantly different c = number of (repeated) treatments N = number of matched sets of observations T i = sum of ranks for the ith treatment –Use χ 2 distribution with df = c – 1 for significance if total number of observations is at least about eight

17. Chapter 21prepared by Elizabeth Bauer, Ph.D. 17 Kendall’s Coefficient of Concordance –Could use the Friedman test to assess interrater reliability Interested in consistency of results Need significance and magnitude An average of the Spearman coefficients is useful. First transform your value for F r into Kendall’s coefficient (W) Then transform into average: