Wilcoxon Signed Rank Test EPI 809 Spring 2008

Wilcoxon Signed Rank Test 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: User Current New Donna 9. 98 9. 88 Santosha 9. 88 9. 86 Sam 9. 90 9. 83 Tamika 9. 99 9. 80 Brian 9. 94 9. 87 Jorge 9. 84 EPI 809 / Spring 2008 © 1984 -1994 T/Maker Co.

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. Can Use Normal Approximation if #(non-zero differences) 16 EPI 809 / Spring 2008

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

Signed Rank Test Procedure Table method for (n < 16). • 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 • Otherwise, if no ties, • If ties, ti =Ri of ties. 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. EPI 809 / Spring 2008

Signed Rank Test Example Skin care treatments A vs B 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 1 A, X 1 B subject 2: X 2 A, X 2 B EPI 809 / Spring 2008. … 1.

Signed Rank Test Computation Illustration k (# subj) Range (Rank) Mean Rank 8 1 40 40 3 7 3 37 -39 38 -6 2 35 -36 35. 5 5 -5 2 33 -34 33. 5 4 -4 1 32 32 -3 5 3 2 7 25 -31 28 -2 4 2 6 10 15 -24 19. 5 -1 4 1 10 14 1 -14 7. 5 |di| di (-) fi (-) di(+) 8 -8 1 7 -7 6 3 2 1 fi(+) EPI 809 / Spring 2008

Testing with positive statistic T+ Ø T+ = 10*7. 5 + 6*19. 5 + 2*28 = 248 Ø E(T+ ) = 40*(40+1)/4 = 410 Ø Var = 40*(40+1)*(80+1)/24 – [(143 -14) + (103 -10) + (73 -7) +…+ (13 -1)]/48 = 5449. 75 Ø T = {|248 -10|-. 5}/√ 5449. 75 = 2. 19 Ø P = 2*[1 - Φ(2. 19)] = 0. 029. 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: User Current New Donna 9. 98 9. 88 Santosha 9. 88 9. 86 Sam 9. 90 9. 83 Tamika 9. 99 9. 80 Brian 9. 94 9. 87 Jorge 9. 84 EPI 809 / Spring 2008 © 1984 -1994 T/Maker Co.

Signed Rank Test Computation Table EPI 809 / Spring 2008

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

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

Signed Rank Test Solution H 0: Identical Distrib. Ø Ha: Current Shifted Right Ø =. 05 (one-sided) Ø n’ = 5 (not 6; 1 elim. ) Ø Critical Value(s): Ø Do Not Reject 0 Test Statistic: Decision: Conclusion: 15 EPI 809 / Spring 2008

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

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

Kruskal-Wallis test Ø A k-sample non-parametric test on the means (k > 2). Ø Pool observations together N = ∑ni and assign ranks to individuals. Ø Compute the rank sum Ri for each sample. Ø Test statistic Chi-squares with df = (k-1) one tail prob. Compare with χ2 k-1, 1 -α Ø Should only be used if smallest ni ≥ 5. EPI 809 / Spring 2008

Kruskal-Wallis test statistic Ø If no ties. Ø If ties. Denote ti, g as before. EPI 809 / Spring 2008

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

Summary Nonparametric Parametric Sign Rank test One sample t-test Wilcoxon Rank – Sum test (Mann-Whitney U test) Two sample t-test Wilcoxon Signed-Rank test Two paired sample t-test Kruskal-Wallis test Multiple sample test. EPI 809 / Spring 2008