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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 TestParametricNon Parametric One Quantitative Response Variable One Sample ttestSign Test One Quantitative Response Variable – Two Values from Paired Samples Paired Sample ttestWilcoxon 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 ANOVAKruskall 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: 1.The two groups being tested are independent of each other. 2.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: H 0 : η 1 – η 2 = 0 H 1 : η 1 – η 2 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 = n 1 +n 2 ). 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 – {(n 1 (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. AlcoholPlacebo

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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 DataRankAlcohol or Placebo Group.371Placebo.382Placebo.613Placebo.784Placebo.835Placebo.866Placebo.907Placebo.958Placebo.989Alcohol Alcohol Alcohol Alcohol Alcohol Alcohol Alcohol Placebo Alcohol Placebo Alcohol Alcohol

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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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