1. MARE 250 Dr. Jason Turner Analysis of Variance (ANOVA) II

2. Assumptions for One-Way ANOVA Normality and Equal variance is more difficult to test with multiple populations Another way to assess: Residual – the difference between the observation and the mean of the sample containing it IF Normality and equal variances assumptions are met THEN normal probability plot should be roughly linear THEN residuals plot should be centered and symmetric about the x-axis

3. Assumptions for One-Way ANOVA A. Residuals centered and symmetric about the x-axis normally distributed, equal variances B. Residuals curved data not normal C. Residuals cone shaped variances not equal

4. Assumptions for One-Way ANOVA Four-in-one Plot: Probability plot, Residuals versus fitted Histogram, Residuals versus order Are residuals centered and symmetric? Are residuals distributed in a random pattern?

5. Non-Parametric Version of ANOVA If samples are independent, similarly distributed data Use nonparamentric test regardless of normality or sample size Is based upon median of ranks of the data – not the mean or variance (Like Mann-Whitney) If the variation in mean ranks is large – reject null Uses p-value like ANOVA Last Resort/Not Resort –low sample size, “bad” data Kruskal-Wallis

6. Kruskal-Wallis Test: _ Urchins versus Distance Kruskal-Wallis Test on _ Urchins Distance N Median Ave Rank Z Deep 50 3.000000000 208.2 6.90 Middle 75 0.000000000 153.2 1.94 Shallow 150 0.000000000 107.0 -7.08 Overall 275 138.0 H = 64.49 DF = 2 P = 0.000 H = 103.96 DF = 2 P = 0.000 (adjusted for ties) Non-Parametric Version of ANOVA

7. When Do I Do the What Now? If you are reasonably sure that the distributions are normal –use ANOVA Otherwise – use Kruskal-Wallis “Well, whenever I'm confused, I just check my underwear. It holds the answer to all the important questions.” – Grandpa Simpson 1. Test all samples for normality Use Kruskal-Wallis test Data Not normal 2. Test samples for equal variance (Bartlett’s test) Data normal Use Kruskal-Wallis test Use single factor ANOVA Variances equal Variances not equal