Using Statistics To Make Inferences 6. Wilcoxon Matched Pairs Signed Ranks Test. Wilcoxon Rank Sum Test/ Mann-Whitney Test

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1 Using Statistics To Make Inferences 6 Summary Non-parametric tests Wilcoxon Signed Ranks Test Wilcoxon Matched Pairs Signed Ranks Test Wilcoxon Rank Sum Test/ Mann-Whitney Test Goals Perform and interpret Wilcoxon Signed Ranks Test Perform and interpret Wilcoxon Matched Pairs Signed Ranks Test Perform and interpret Wilcoxon Rank Sum Test/ Mann-Whitney Test If appropriate employ a normal approximation Know when each test is appropriate Practical Perform a series of Mann-Whitney tests and compare the results to those obtained from t-tests Mike Cox 6.1 Version

2 Non-parametric Tests A single sample test Wilcoxon Signed Ranks Test Procedure 1. Take the difference between each observation and the median η.. Rank the absolute differences from 1 to n, allowing for ties. (1 smallest, n largest) 3. Sum the rank values for those observations above η, let this be W + 4. Sum the rank values for those observations below η, let this be W - Note: If two or more differences are equal (tied) they are assigned the average of the ranks If a difference is zero, it is omitted 5. Use the smaller of W + or W - to use as the test statistic, called W or W calc. Critical values of the test statistic are given in tables for various significance levels. For n greater than 8 a normal approximation may be employed where z 1 n( n 1) W 4 n( n 1)(n 1) 4 In this case the continuity correction is added since a lower tail is being considered. Mike Cox 6. Version

3 Example It has been established that an individuals median reaction time is 0.50 seconds. Twelve trials are conducted after the individual has consumed alcohol. The measured times are Test whether the data are consistent with the median value. The first step is to subtract the median from every value. raw data difference absolute difference Mike Cox 6.3 Version

4 Now rank on the absolute differences difference absolute difference rank true rank Mike Cox 6.4 Version

5 Now separate the contributions for positive (negative) differences difference absolute difference true rank total W calc W + =68.5 W - =9.5 min W, W 9. 5 W ( 0.05) 14 W ( 0.01) 7 crit crit Two Tail Probability n Therefore the result is significant at the 5% level, so the null hypothesis can be rejected. The median is apparently not consistent with 0.50 seconds. Mike Cox 6.5 Version

6 Normal approximation Employing the normal approximation, 1 n( n 1) 1 1(1 1) W 9.5 z using normal tables the p n( n 1)(n 1) 1(1 1)(1 1) 4 4 value is p remarkably close to the exact value reported by software. Z Mike Cox 6.6 Version

7 Wilcoxon Matched Pairs Signed Ranks Test Example Certain mental tasks are performed before and after exercise. The scores were recorded. Subject Exercise Relaxed Is there any evidence of a significant difference in the levels of performance under the two conditions? Still effectively a single sample since we seek a change. Mike Cox 6.7 Version

8 Subject Exercise Relaxed Absolute Difference Difference Now rank the absolute differences Subject Absolute Difference Difference Rank True Rank Mike Cox 6.8 Version

9 Now separate the contributions for positive (negative) differences Subject Difference Absolute Difference True Rank Total W calc W + =50 W - =5 min W, W 5 W ( 0.05) 8 W ( 0.01) 3 crit crit Two Tail Probability n Therefore the result is significant at the 5% level, so the null hypothesis can be rejected. The scores differ. Mike Cox 6.9 Version

10 A two sample test Wilcoxon Rank Sum Test/ Mann-Whitney Test 1. Combine the observations from the two samples (sizes n 1 and n ).. Rank the sorted data from 1 to (n 1 +n ). 3. Calculate R 1, as the sum of the ranks of the first sample and R for the second. 4. Form U U 1 n 1 n 1 n n n 1 n n1 1 R n 1 1 R (mid-point ½ n 1 n so only need calculate one) U calc min U U or U 1, the Mann-Whitney test statistic For n 1 and n greater than 8 a normal approximation may be employed where z 1 n1n U n1n n1 n 1 1 In this case the continuity correction is added since a lower tail is being considered. Mike Cox 6.10 Version

11 Wilcoxon Rank Sum Test A study of patients suffering from Parkinsons disease was conducted. An operation was performed on 8 of them, while it improved their general condition it might adversely affect their speech. In the data a higher value indicates a greater difficulty in speaking. Operated Others Mike Cox 6.11 Version

12 Speech Source Rank True Rank Others Operated 1. Others Others Others Others Others Others Others Operated Others Others Operated Others Others Others Operated Operated Operated Operated Operated Others Operated Others Total R 1 =16.5, n 1 = 8, n = 14 n1 n U 1 n1 n R R =16.5, n = 14, n 1 = 8 n n U n1 n R Mike Cox 6.1 Version

13 R 1 =16.5, n1 n U 1 n1 n R R =16.5, n n U n1 n R U 1 = 1.5 U = 90.5 (mid-point ½ n 1 n = 56 so only need calculate one) U calc U, U 1. 5 min 1 For n 1 =8, n =14, the critical value from the tables for p=0.05 is 6. The result is significant at the 5% level, the two samples appear to differ. n n Mike Cox 6.13 Version

14 Normal approximation Employing the normal approximation z U n n 1 1 n1n n n , using normal tables the p value is p remarkably close to the exact value reported by software. Z Mike Cox 6.14 Version

15 Parametric vs Non-Parametric Tests Parametric Tests They are robust with respect to violations of their assumptions. They are more powerful- more likely to detect an effect when one is present. They are more versatile there are tests for every experimental design. Non-Parametric Tests They make fewer assumptions. They are ideal for ordinal data, which is common in Psychology, whereas parametric tests require interval or ratio data. Read Howitt and Cramer pages , Read Russo pages Read Davis and Smith pages Mike Cox 6.15 Version

16 Critical Values For Wilcoxon's Signed-Rank Test The body of the table contains the critical values for Wilcoxon's signed-rank test. Always enter the table with W+, the sum of the ranks of the positive deviations. If a critical value is missing, the hypothesis can not be rejected for this combination of n and α. One Tail Probability OneTail Probability n Two Tail Probability TwoTail Probability n n Mike Cox 6.16 Version

17 Critical Values Of U In The Mann-Whitney Test Critical Values of U at α = 0.05 with direction predicted or at α = 0.05 with direction not predicted. n n Mike Cox 6.17 Version

18 Critical Values Of U In The Mann-Whitney Test Critical Values of U at α = 0.05 with direction predicted or at α = 0.10 with direction not predicted. n n Mike Cox 6.18 Version

19 Critical Values Of U In The Mann-Whitney Test Critical Values of U at α = with direction predicted or at α = 0.01 with direction not predicted. n n Mike Cox 6.19 Version