2. NON-PARAMETRIC SUBSTITUTES FOR t-TESTS ParametricNon-Parametric Independent tWilcoxon Rank-Sum or Mann-Whitney Dependent tWilcoxon T

3. WILCOXON RANK-SUM TEST Purpose: Compare the medians of two groups. Design: between subjects Assumptions: – independent observations – at least ordinal level data

4. Example Four students with ADD were assigned to receive counseling, and four others with ADD were assigned to receive Ritalin. The number of minutes out of their seats per day was recorded.

5. COUNSELINGRITALIN 3141 6025 7231 1954

6. STEP 1: Rank all scores from lowest to highest. COUNSELINGRITALIN 31 Rank =3.541 Rank = 5 60 Rank = 725 Rank = 2 72 Rank = 831 Rank = 3.5 19 Rank = 154 Rank = 6

7. STEP 2: Sum the ranks for each group. COUNSELINGRITALIN 31 Rank = 3.541 Rank = 5 60 Rank = 725 Rank = 2 72 Rank = 831 Rank = 3.5 19 Rank = 154 Rank = 6  R = 19.5  R = 16.5

8. STEP 3: W is the smaller  R. W = 16.5

9. STEP 4: Compare to critical value of W. Observed W must be EQUAL OR LESS THAN W-critical to be significant. For N1 = 4, N2 = 4,  =.05 two-tailed W-crit = 10 W = 16.5, not significant

10. APA Format Sentence A Wilcoxon Rank-Sum test showed that the Ritalin and Counseling groups were not significantly different, W (n 1 = 4, n 2 = 4) = 16.50, p >.05.

11. WILCOXON T Purpose: Test whether two distributions are different Design: within subjects or matched Assumptions: – at least ordinal level data – populations are identical except for means – minimum N of 6

12. Example: Eight patients are exposed to a placebo and an experimental treatment (at different times). They are measured on severity of symptoms. Was there a significant difference between conditions? (scores on next page) Wilcoxon T Calculation

13. PatientPE 1108 21410 31213 41515 564 6811 796 81412

14. STEP 1: Determine difference scores PatientPEDiff 1108+2 21410+4 31213-1 415150 564+2 6811-3 796+3 81412+2

15. STEP 2: Rank the difference scores from smallest to largest, ignoring differences of zero. PatientPEDiffRank 1108+23 21410+47 31213-11 415150 564+23 6811-35.5 796+35.5 81412+23

16. STEP 3: Compute the sum of the positive ranks and the sum of the negative ranks.  R positive = 3 + 7 + 3+ 5.5 + 3 = 21.5  R negative = 1 + 5.5 = 6.5 STEP 4: The Wilcoxon T is the smaller  R. T = 6.5

17. STEP 5: Compare to critical value. N = number of NONZERO differences N = 7 Tcrit = 2 Your T must be LESS THAN or equal to Tcrit to be significant!

18. APA Format Sentence A Wilcoxon T showed no significant difference between the placebo and experimental treatments,  = 6.5, p >.05.