1. EPI 809 / Spring 2008 Wilcoxon Signed Rank Test

2. EPI 809 / Spring 2008 Signed Rank Test Example You work in the finance department. Is the new financial package faster (.05 level)? You collect the following data entry times: UserCurrentNew Donna9.989.88 Santosha9.889.86 Sam9.909.83 Tamika9.999.80 Brian9.949.87 Jorge9.849.84 © 1984-1994 T/Maker Co.

3. EPI 809 / Spring 2008 Wilcoxon Signed Rank Test 1.Tests Probability Distributions of 2 Related Populations 2.Corresponds to t-test for Dependent (Paired) Means 3.Assumptions Random samples; Both populations are continuous; paired samples. Random samples; Both populations are continuous; paired samples. Can Use Normal Approximation if #(non-zero differences)  16 if #(non-zero differences)  16

4. EPI 809 / Spring 2008 Signed Rank Test Procedure 1. Obtain Difference Scores, D i = X 1i - X 2i 2. Take Absolute Value |D i | and rank them (Do not count D i = 0) 3. Assign Ranks, R i, with Smallest = 1 4. Calculate range and mean rank for |D i | 5. Sum ‘+’ Ranks (T + ) & ‘-’ Ranks (T - ) 6. Statistic (T + ). mean = n(n+1)/4, var = n(n+1)(2n+1)/24 if no ties. var = n(n+1)(2n+1)/24 if no ties.

5. EPI 809 / Spring 2008 Signed Rank Test Procedure Table method for (n < 16). Use Table 11 that provides the critical values at significant level. Use Table 11 that provides the critical values at significant level. 2. If n ≥ 16, use normal approximation. Let T 0 = n(n+1)/4, If T + =T 0, then T = 0 Let T 0 = n(n+1)/4, If T + =T 0, then T = 0 Otherwise, if no ties, Otherwise, if no ties, If ties, t i =R i If ties, t i =R i of ties.

6. EPI 809 / Spring 2008 Signed Rank Test Procedure (continued) If T > Z 1- α/2 then reject H 0. Otherwise accept H 0. P – value: p = 2 [1 - Φ(T)]. Condition: number of nonzero differences > 15.

7. EPI 809 / Spring 2008 Signed Rank Test Example Skin care treatments A vs B 1. Each subject received two treatments A and B randomly applied to left and right arms. 2. Measure redness after sunlight exposure of 10 min. 3. Data: subject 1: X 1A, X 1B subject 2: X 2A, X 2B.….…

8. EPI 809 / Spring 2008 Signed Rank Test Computation Illustration |di|di (-)fi (-)di(+)fi(+) k (# subj) Range (Rank) Mean Rank 8-818140 7-737337-3938 6-626235-3635.5 5-525233-3433.5 4-414132 3 -3532725-3128 2 -24261015-2419.5 1 4110141-147.5

9. EPI 809 / Spring 2008 Testing with positive statistic T +  T + = 10*7.5 + 6*19.5 + 2*28 = 248  E(T + ) = 40*(40+1)/4 = 410  Var = 40*(40+1)*(80+1)/24 – [(14 3 -14) + (10 3 -10) + (7 3 -7) +…+ (1 3 -1)]/48 = 5449.75  T = {|248-10|-.5}/√5449.75 = 2.19  P = 2*[1- Φ(2.19)] = 0.029.

10. EPI 809 / Spring 2008 Signed Rank Test Example You work in the finance department. Is the new financial package faster (.05 level)? You collect the following data entry times: UserCurrentNew Donna9.989.88 Santosha9.889.86 Sam9.909.83 Tamika9.999.80 Brian9.949.87 Jorge9.849.84 © 1984-1994 T/Maker Co.

11. EPI 809 / Spring 2008 Signed Rank Test Computation Table

12. EPI 809 / Spring 2008 Signed Rank Test Solution  H 0 : Identical Distrib.  H a : Current Shifted Right   =.05 (one-sided)  n’ = 5 (not 6; 1 elim.)  Critical Value(s): Test Statistic: Decision:Conclusion: Reject Do Not Reject TuTuTuTu TLTLTLTL

13. EPI 809 / Spring 2008 Wilcoxon Signed Rank Table (Portion)

14. EPI 809 / Spring 2008 Signed Rank Test Solution  H 0 : Identical Distrib.  H a : Current Shifted Right   =.05 (one-sided)  n’ = 5 (not 6; 1 elim.)  Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject 015

15. EPI 809 / Spring 2008 Signed Rank Test Solution  H 0 : Identical Distrib.  H a : Current Shifted Right   =.05 (one-sided)  n’ = 5 (not 6; 1 elim.)  Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject 015 Since one-sided Test & Current Shifted Right, Use T +

16. EPI 809 / Spring 2008 Signed Rank Test Solution  H 0 : Identical Distrib.  H a : Current Shifted Right   =.05 (one-sided)  n’ = 5 (not 6; 1 elim.)  Critical Value(s): Test Statistic: Conclusion: Border line at  =.05 Do Not Reject 015 Since One-sided Test & Current Shifted Right, Use T + T + = 15

17. EPI 809 / Spring 2008 Kruskal-Wallis test  A k-sample non-parametric test on the means (k > 2).  Pool observations together N = ∑n i and assign ranks to individuals.  Compute the rank sum R i for each sample.  Test statistic Chi-squares with df = (k-1) one tail prob. Compare with χ 2 k-1, 1-α  Should only be used if smallest n i ≥ 5.

18. EPI 809 / Spring 2008 Kruskal-Wallis test statistic  If no ties.  If ties. Denote t i, g as before.

19. EPI 809 / Spring 2008 Conclusion  1. Distinguished Parametric & Nonparametric Test Procedures  2. Explained a Variety of Nonparametric Test Procedures  3. Solved Hypothesis Testing Problems Using Nonparametric Tests

20. EPI 809 / Spring 2008 Summary NonparametricParametric Sign Rank testOne sample t-test Wilcoxon Rank – Sum test (Mann-Whitney U test) Two sample t-test Wilcoxon Signed-Rank testTwo paired sample t-test Kruskal-Wallis testMultiple sample test.