Nonparametric Statistics - Dependent Samples How do we test differences from matched pairs of measurement data? If the differences are normally distributed, use the t-test. If the differences are nonnormal, use the Wilcoxon Signed Rank Sum test. Note, Wilcoxon Rank Sum test (Mann-Whitney on Minitab) is different than the Wilcoxon Signed Rank test.

Wilcoxon Signed Rank Test - Procedures 1.Compute the paired differences. 2.Eliminate all differences that are equal to 0. 3.Take the absolute value for each difference. 4.Rank the absolute value of the differences. (Adjust ties) 5.Sum the ranks of the positive differences (from step 1). 6.Call this sum of the ranks W, our test statistic. 7.We use Minitab to calculate p-values, etc. –Stat > Nonparametrics > 1-Sample Wilcoxon –Variables: [ diff ] –  Test Median: [ 0 ] – Alternative: [ less than ] > OK You may have to calculate the differences: calc > calculator > Store result: diff > Expression: c1-c2 > OK

Wilcoxon Signed Rank Test - Example Sample 1: Sample 2: Differences |Diff| Ranks |Diff| Sum of Ranks of the Positive Differences: T = = 19.5 H 0 : Populations have the same location. H 1 : Populations have different locations. p-value = (from Minitab)

Wilcoxon Signed Rank Test - Example Minitab Output: Wilcoxon Signed Rank Test: diff Test of median = versus median not = N for Wilcoxon Estimated N Test Statistic P Median diff Note: N is the number of matched pairs. N for Test is the number of non-zero differences. W is the Wilcoxon Statistic. P is the p-value for the test.

Wilcoxon Signed Rank Test – Ice Cream European vs Inexpensive label ice cream Diff = European rating minus Inexpensive rating Wilcoxon Signed Rank Test: diff Test of median = versus median > N for Wilcoxon Estimated N Test Statistic P Median diff With a p-value of 0.000, we can conclude there is overwhelming evidence that the label affects perception.