1. Mann-Whitney U test Non-parametric test Tests the central tendency between two populations, does NOT test the means. Use sample sizes instead of degrees of freedom to look up critical values. Also known as the Wilcox rank test, or the Kendall’s S test INTRODUCTION

2. Mann-Whitney U test EXAMPLE 21 plants Treatment A n A = 11 Treatment A n A = 11 Treatment B n B = 10 Treatment B n B = 10 Ho: no difference between treatments A and B

3. Mann-Whitney U test Group AGroup B 4.64.74.95.25.35.4 5.15.25.55.66.26.3 5.86.16.56.87.78.0 6.57.28.1 mean5.66.5 EXAMPLE

4. Mann-Whitney U test EXAMPLE

5. Mann-Whitney U test EXAMPLE the two samples are randomly and independently drawn independence within samples and mutual independence between samples the measurement scale is at least ordinal Assumptions for test

6. Mann-Whitney U test EXAMPLE Raw measure 4.64.74.95.15.2 5.35.45.55.65.86.16.26.36.5 6.87.27.78.08.1 rank12345.5 789101112131415.5 1718192021 From sample AAAAABBBABAABBAABABBB Assemble measures into a single set Rank-order from lowest to highest

7. Mann-Whitney U test EXAMPLE Ranked measures Group AGroup B 15.5 27 38 410 5.513 814 1117 1219 15.520 15.521 18A & B tog. Sum of ranks96.5134.5231 Avg or ranks8.813.511 Rank values substitute for raw measures in table. Used to calculate critical value (test statistic)

8. Mann-Whitney U test EXAMPLE n i is the sample size in group i R i is the sum of ranks in group i UiUi = n 1 n 2 + ni(ni+1)ni(ni+1)- R i 2

9. Mann-Whitney U test EXAMPLE U = n 1 n 2 + n1(n1+1)n1(n1+1)- R 1 2 U'U' = n 1 n 2 + n2(n2+1)n2(n2+1)- R 2 2 U A = 110 + 66 – 96.5 = 79.5 U B = 110 + 55 – 134.5 = 30.5 See ZAR Table B.10

10. Mann-Whitney U test EXAMPLE Degrees of freedom n 1 = 10 and n 2 = 11 alpha = 0.05 Two-tailed - 84 One-tailed - 79