1. Residual Analysis for ANOVA Models KNNL – Chapter 18

2. Residuals

3. Model Departures Detected With Residuals and Plots Errors have non-constant variance Errors are not independent Existence of Outlying Observations Omission of Important Predictors Non-normal Errors Common Plots  Residuals versus Treatment  Residuals versus Treatment Mean  Aligned Dot Plot (aka Strip Chart)  Residuals versus Time  Residuals versus Omitted Variables  Box Plots, Histograms, Normal Probability Plots

4. Tests for Constant Variance H 0 :  1 2 =...=  t 2

5. Remedial Measures Normally distributed, Unequal variances – Use Weighted Least Squares with weights: w ij = 1/s i 2 Non-normal data (with possibly unequal variances) – Variance Stabilizing Transformations and Box-Cox Transformation – Variance proportional to mean: Y’=sqrt(Y) – Standard Deviation proportional to mean: Y’=log(Y) – Standard Deviation proportional to mean 2 : Y’=1/Y – Response is a (binomial) proportion: Y’=2arcsin(sqrt(Y)) Non-parametric tests – F-test based on ranks and Kruskal-Wallis Test

6. Effects of Model Departures Non-normal Data – Generally not problematic in terms of the F-test, if data are not too far from normal, and reasonably large sample sizes Unequal Error Variances – As long as sample sizes are approximately equal, generally not a problem in terms of F-test. Non-independence of error terms – Can cause problems with tests. Should use Repeated Measures ANOVA if same subject receives each treatment

7. Nonparametric Tests