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**Statistical Methods II**

Session 9 Non Parametric Testing – The Wilcoxon Rank Sum Test (also known as the Mann Whitney Test)

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Wilcoxon Rank Sum Test Recall that Non-Parametric tests (in all forms) should be your “Plan B”. In the previous two sessions, we covered the Sign Test and the Wilcoxon Signed Rank Test – both of which can be used when testing the center location of a single population (or a pair). In the current session, we will be covering the Wilcoxon Rank Sum Test – used with two independent samples.

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**Wilcoxon Rank Sum Test Test Parametric Non Parametric**

One Quantitative Response Variable One Sample ttest Sign Test One Quantitative Response Variable – Two Values from Paired Samples Paired Sample ttest Wilcoxon Signed Rank Test One Quantitative Response Variable – One Qualitative Independent Variable with two groups Two Independent Sample ttest Wilcoxon Rank Sum or Mann Whitney Test One Quantitative Response Variable – One Qualitative Independent Variable with three or more groups ANOVA Kruskall Wallis

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Wilcoxon Rank Sum Test Although this test does not have parametric assumptions – specifically the number of observations can be small – it does require two things: The two groups being tested are independent of each other. The two groups should have approximately similar distributions (this test evaluates the “shift” of the distributions).

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Wilcoxon Rank Sum Test The hypothesis statements function the same way as the two sample ttest – but we are focused on the medians rather than on the means: H0: η1 – η2 = 0 H1: η1 – η2 ≠ 0 These could also be expressed as one tailed tests.

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Wilcoxon Rank Sum Test Step 1: List the data values from both samples in a single list arranged from smallest to largest. Step 2: In the next column, assign the numbers 1 to N (where N = n1+n2). These are the ranks of the observations. As before, if there are ties, assign the average of the ranks the values would receive to each of the tied values. Step 3: Let W denote the sum of the ranks for the obs from Population 1. Note that if there is no difference between the two medians (the null is true), the value of W will be around half the sum of the ranks – {(n1(1+N))/2}

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Wilcoxon Rank Sum Test The following data measures the reaction times of two samples of people – one set drank alcohol, one set drank a placebo. Alcohol Placebo 1.56 .90 .37 1.76 1.63 1.44 .83 1.11 .95 3.07 .78 .98 .86 1.27 .61 2.56 .38 1.32 1.97

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Wilcoxon Rank Sum Test From this dataset, the hypothesis statements will be: H0: The median reaction times for the placebo group is the same or slower than the median reaction time for the alcohol group. H1: The median reaction times for the placebo group is faster than the median reaction time for the alcohol group.

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**Wilcoxon Rank Sum Test Data Rank Alcohol or Placebo Group .37 1**

.38 2 .61 3 .78 4 .83 5 .86 6 .90 7 .95 8 .98 9 Alcohol 1.11 10 1.27 11 1.32 12 1.44 13 1.45 14 1.46 15 1.63 16 1.76 17 1.97 18 2.56 19 3.07 20

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Wilcoxon Rank Sum Test If we sum the ranks of the Placebo group, we get W = = 70. Since the middle point of the ranks is (10*21)/2 – and the placebo ranks is much lower, we have initial evidence to conclude that the placebo group had quicker reaction times than did the alcohol group. A z-score approximation can be found on page S2-11 of your book.

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Wilcoxon Rank Sum Test Lets do this same test using SAS…

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