1. Model Diagnostics and Tests

2. Tests Among t ≥ 2 Population Variances
Hartley’s Fmax Test
Must have equal sample sizes (n1 = … = nt)
Test based on assumption of normally distributed data
Uses special table for critical values
Levene’s Test
No assumptions regarding sample sizes/distributions
Uses F-distribution for the test
Bartlett’s Test
Can be used in general situations with grouped data
Uses Chi-square distribution for the test

3. Hartley’s Fmax Test H0: s12 = … = st2 (homogeneous variances)
Ha: Population Variances are not all equal
Data: smax2 is largest sample variance, smin2 is smallest
Test Statistic: Fmax = smax2/smin2
Rejection Region: Fmax  F* (Values from class website, indexed by a (.05, .01), t (number of populations) and df2 (n-1, where n is the individual sample sizes)

4. Levene’s Test H0: s12 = … = st2 (homogeneous variances)
Ha: Population Variances are not all equal
Data: For each group, obtain the following quantities:

5. Bartlett’s Test
General Test that can be used in many settings with groups
H0: s12 = … = st2 (homogeneous variances)
Ha: Population Variances are not all equal
MSE ≡ Pooled Variance

6. Welch’s Test – Unequal Variances

7. CRD with Non-Normal Data Kruskal-Wallis Test
Extension of Wilcoxon Rank-Sum Test to k > 2 Groups
Procedure:
Rank the observations across groups from smallest (1) to largest ( N = n1+...+nk ), adjusting for ties
Compute the rank sums for each group: T1,...,Tk . Note that T1+...+Tk = N(N +1)/2

8. Kruskal-Wallis Test H0: The k population distributions are identical
HA: Not all k distributions are identical
An adjustment to H is suggested when there are many ties in the data.

9. Nonparametric Comparisons
Based on results from Kruskal-Wallis Test
Makes use of the Rank-Sums for the t treatments
If K-W test is not significant, do not compare any pairs of treatments
Compute the t(t-1)/2 absolute mean differences
Makes use of the Studentized Range Table
For large-samples, conclude treatments are different if: