1. Today’s Lecture One more test for normality Testing variances
Shapiro-Wilk Test
Testing variances
Equality of Variance via the F-Distribution
Levene’s Test for Equality of Variances

2. Reference Material
Shapiro and Wilk, Biometrika (52:3 and 4) pgs
Burt and Barber, page 325
Levene, In Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling, I. Olkin et al. eds., Stanford University Press, pp

3. More Pretests
The tests presented in today’s lecture are pretests that can help to verify the assumptions of a parametric hypothesis test
The first is one of the strongest tests for normality
The second is one of the simplest tests for determining if a pooled or non-pooled variance t-test is required
The last allows for a comparison of variances in a multiple category layout like the analysis of variance

4. Shapiro-Wilk
The Shapiro-Wilk is one of the strangest tests that I have encountered thus far in my statistical explorations
But it is either the best or the second best test of normality in existence
It excels at normality testing small samples and is the definitive test for n<30

5. Curiouser and Curiouser
A brief rundown of the strangeness associated with the Shapiro-Wilk
You fail reject the null when your observed value is greater than your critical value (that’s right, the critical region on this test is in the small tail)
The test actually pairs observations from within the sample to determine normality
The number of pairs is determined by nearly the same equation that you would use to determine the median

6. So How Does It Work? The W-Statistic:
Recall that the variance of a sample is s2
So really all we are required to give is the sum of the squared deviations from the mean (plus this term b2)
b2is a bit more complex, but it is more odd than difficult

7. Getting to B-Squared
The b term is actually a weighted comparison of all the pairs within the sample
The way that it works is that you sort all of your data from least to greatest
Then you create k number of pairs from the sample with k=n/2 if n is even and k=n+1/2 if n is odd (note that k is the median of the sample)
Each pair has a companion that is from the other end of the sample
Example: Given the following set of numbers- 1,2,3,4,5,100 your pairs would be as follows:
100 and 1, 5 and 2, and 4 and 3

8. Once You Have Your Pairs
The pairs are important because you will be taking the difference between the large value and the small value (100-1=99, 5-2=3, 4-3=1)
Once you have all your differences, you then assign them weights (from a W-weight table
Once you have the weights, you multiply each one by its pair and then sum them all
This sum is b, which you then square

9. Strange, don’t you think?
Let’s go to Excel
But first let’s show you the equation for b
The median
Big and Little Pairs
ai weight (from math that you don’t want to have to learn) – basically the weights are the result of an expected normal distribution and its resulting covariance matrix

10. Results W=
This isn’t very small, so we are going to fail to reject
H0: Normal HA: Not Normal (note the wording here, we are not saying that this test shows that the data is normal, we are only saying that it fails to show that the data is not normal)
W(critical) for 0.05 and n=20 is 0.905
Note that this distribution is severely skewed so our result of has a p-value of around 0.40
This sample is suitable for parametric analysis

11. Shapiro-Wilk Tables Pair Coefficients (weights)
Critical levels for significance

12. Equality of Variance via Ratio
Assumptions:
s12 and s22 are independent estimates of σ2
The population from which the samples are drawn is normal (This means you had better check for normality first)
H0: σ12 = σ22 Ha: σi2 ≠ σj2
Statistic: s12/s22 (I typically place the larger variance in the numerator of the equation, but it doesn’t matter for two tailed tests)
Once you compute the statistic you find the F-distribution in the appendix of your book (page 613) and then use n1-1 and n2-1 for your degrees of freedom

13. Example
A couple of weeks ago we used two samples in a t-test. The first sample had an n=12 and a variance of 17.3, the second sample had an n=10 and a variance of 18.9
18.9/17.3=1.092
A look at our tables with 11 and 9 degrees of freedom at alpha=0.05 will tell us that a critical value of 3.96 (we have to use 10 for n1, because there is no 11 column)
Since 1.09<3.96, we fail to reject the null

14. Levene’s L-Statistic
Test for the equality of variance in multiple categories
H0: σ12 = σ22 = … = σk2
Ha: σi2 ≠ σj2 for at least one pair (i,j).
The statistic is run on the deviations from the mean but is very similar to the ANOVA in terms of computation
The test uses the F-distribution to determine significance

15. The Equation
All data in each category is “differenced” by its category mean
This is a categorical mean of differences
This is the global mean of differences
This is a sum of squares between, but on the xij differences
dfb
This is a sum of squares within, but on the xij differences
dfw

16. Off to Excel

17. Results After all of our computations, we find an L value of 2.41
Since our degrees of freedom are k-1=2 and N-k=12 an alpha of 0.05 would require a critical value for L of 3.88
Since 2.41<3.88 we fail to reject the null of equal variances between categories
This data set is suitable for parametric analysis via an ANOVA

18. Homework
Given two data sets, test for normality using the Shapiro Wilk and then test for equality of variance via ratio.
Once you have completed both tests, recommend the correct test for comparing the samples.
Your choices are the T-Test (pooled variance), T-Test (non-pooled variance) and the Wilcoxon Rank-Sum Test