Statistical Methods II

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

2. STAT 3130 – Non Parametric Testing 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.

3. STAT 3130 – Non Parametric Testing

4. STAT 3130 – Non Parametric Testing 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 H0: η> 40 H1: η < 40

5. STAT 3130 – Non Parametric Testing Step 1: Subtract η0 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.

6. STAT 3130 – Non Parametric Testing 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

7. STAT 3130 – Non Parametric Testing 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)

8. STAT 3130 – Non Parametric Testing 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.

9. STAT 3130 – Non Parametric Testing 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.

10. STAT 3130 – Non Parametric Testing 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