Model Diagnostics and Tests

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

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)

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

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

Welch’s Test – Unequal Variances

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

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.

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: