1. Wilcoxon Matched-Pairs Signed Rank Test A nonparametric alternative to the t test for related samples Before and After studies Studies in which measures are taken on the same person or object under different conditions Studies or twins or other relatives

2. Wilcoxon Matched-Pairs Signed Rank Test Differences of the scores of the two matched samples Differences are ranked, ignoring the sign Ranks are given the sign of the difference Positive ranks are summed Negative ranks are summed T is the smaller sum of ranks

3. Wilcoxon Matched-Pairs Signed Rank Test: Sample Size Consideration n is the number of matched pairs If n > 15, T is approximately normally distributed, and a Z test is used. If n  15, a special “small sample” procedure is followed. –The paired data are randomly selected. –The underlying distributions are symmetrical.

4. Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example Family PairPittsburghOakland 11,950 1,760 21,840 1,870 32,015 1,810 41,580 1,660 51,790 1,340 61,925 1,765 H 0 : M d = 0 H a : M d  0 n = 6  =0.05 If T observed  1, reject H 0.

5. Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example Family PairPittsburghOaklanddRank 11,950 1,760 190 21,840 1,870 -30 32,015 1,810 205 41,580 1,660 -80 51,790 1,340 450 61,925 1,765 160 +4 +5 -2 +6 +3 T = minimum( T +, T - ) T + = 4 + 5 + 6 + 3= 18 T - = 1 + 2 = 3 T = 3 T = 3 > T crit = 1, do not reject H 0.

6. Wilcoxon Matched-Pairs Signed Rank Test: Large Sample Formulas

7. Airline Cost Data for 17 Cities, 1997 and 1999 City19791999dRankCity19791999dRank 120.322.8-2.5-81020.320.9-0.6 219.512.76.8171119.222.6-3.4-11.5 318.614.14.5131219.516.92.69 420.916.14.8151318.720.6-1.9-6.5 519.925.2-5.3-161417.718.5-0.8-2 618.620.2-1.6-41521.623.4-1.8-5 719.614.94.7141622.421.31.13 823.221.31.96.51720.817.43.411.5 921.818.73.110 H 0 : M d = 0 H a : M d  0

8. Airline Cost: T Calculation

9. Airline Cost: Conclusion