1. Statistical Methods II Session 8 Non Parametric Testing – The Wilcoxon Signed Rank Test

2. In the previous session, we introduced the concept of Non- Parametric tests (your Plan B tests). These tests come in handy when you have small datasets that are not normal. We reviewed the Sign Test as a simple test of the center (median) of a variable. Now we will introduce another non parametric test used to test the median of a dataset – The Wilcoxon Signed Rank Test. STAT 3130 – Non Parametric Testing

3. 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 STAT 3130 – Non Parametric Testing

4. Lets consider the same dataset, and hypotheses from the previous session and use the Wilcoxon Signed Rank Test: 22, 24, 25, 25, 26, 29, 32, 34, 38, 40, 40, 42, 44 H 0 : η > 40 H 1 : η < 40 STAT 3130 – Non Parametric Testing

5. Step 1: Subtract η 0 from each value. Step 2: Take the absolute value of all the deviations calculated in Step 1. Step 3: Delete all of the 0 values and let n be the number of values which remain. Step 4: Rank the absolute deviations from the smallest to the largest. Assign the average of the ranks in cases of ties. Step 5: Let T+ be the total of ranks given to deviations that were originally positive. STAT 3130 – Non Parametric Testing

6. Step 1: -18, -16, -15, -15, -14, -11, -8, -6, -2, 0, 0, 2, 4 Step 2: 18, 16, 15, 15, 14, 11, 8, 6, 2, 0, 0, 2, 4 Step 3: 18, 16, 15, 15, 14, 11, 8, 6, 2, 2, 4(n=11) Step 4: 11 (neg), 10 (neg), 8.5 (neg), 8.5 (neg), 7 (neg), 6 (neg), 5 (neg) 4 (neg), 1.5 (neg), 1.5 (pos), 3 (pos) Step 5: T+ = 1.5 + 3 = 4.5 STAT 3130 – Non Parametric Testing

7. What is 4.5? This is our test statistic. We can use SAS to determine the p-value associated with this statistic to determine if we will reject the null or not. A note of explanation – If the η is the true median of the data, then the sum of the ranks for positive deviations will be about the same as that of the negative deviations. In this case, the T- would be 61.5 – which is not even close to 4.5. (all of the rankings in aggregate would have been 66, which is 11+10+9+…1) STAT 3130 – Non Parametric Testing

8. The Wilcoxon Signed Rank Test can be used in a Paired Situation as well. This makes sense, since paired analysis evaluates a single variable – the difference between the two samples. STAT 3130 – Non Parametric Testing

9. A few notes about when to use the Sign Test and when to use the Signed Rank Test: The Sign Test just considers how many values there are above or below the hypothesized median. The Wilcoxon Signed Rank Test is slightly more sophisticated. The Wilcoxon Signed Rank Test considers how far the values actually lie from the hypothesized median – and could be affected by outliers. In a paired situation, you should use the Wilcoxon Signed Rank Test. STAT 3130 – Non Parametric Testing

10. Lets go through this analysis using SAS… Note regarding the SAS output – we will generate a test statistic of -28.5. How does this square with our value of 4.5? If we subtract the “shift parameter” – Nt*(Nt+1)/4, where Nt is the number of obs NOT EQUAL to the hypothesized median, from the sum of the positive ranks, we get the S stat as reported in SAS. So the math here is: 4.5 – ((11*12)/4) = 4.5 – 33 = -28.5 STAT 3130 – Non Parametric Testing