MARE 250 Dr. Jason Turner

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

2. ANOVA Analysis of Variance (ANOVA) Method for comparing the means of more than two populations 1-Sample t-test – 1R, 1F, 1 Level 2-Sample t-test – 1R, 1F, 2 Levels 1-Way ANOVA – 1R, 1F, >2 Levels

3. ANOVA Research Question: Are there differences in the mean number of total urchins across locations (transects) at Onekahakaha?

4. ANOVA Research Question: Are there differences in the mean number of total urchins across locations at Onekahakaha? Null hypothesis: Ho: μ(urch shallow)= μ(urch middle) = μ (urch deep) Ha: All means not equal

5. ANOVA Why not run multiple T-test? μ1 μ2 μ3

6. ANOVA Why not run multiple T-test? μ1 μ2 μ3 1. Number of t-tests increases with # of groups becomes cognitively difficult 2. ↑ Number of analyses = ↑ probability of committing Type I error Probability of committing at least one type I error = experiment-wise error rate

7. Similarities T & ANOVA “I pity the fools that think T and ANOVA are similar!”– Mr. T A one-way analysis of variance (ANOVA) tests the hypothesis that the means of several populations are equal. The method is an extension of the two-sample t-test, specifically for the case where the population variances are assumed to be equal.

8. Assumptions for One-Way ANOVA Look Familiar? One-Way ANOVA Four assumptions for t-test hypothesis testing: 1. Random Samples 2. Independent Samples 3. Normal Populations (or large samples) 4. Variances (std. dev.) are equal

9. ANOVA A one-way analysis of variance (ANOVA) tests the hypothesis that the means of several populations are equal The null hypothesis for the test is that all population means (level means) are the same – H0: μ1 = μ2 = μ3 The alternative hypothesis is that one or more population means differ from the others – Ha: Not all means are equal

10. ANOVA H0: μ1 = μ2 = μ3 Ha: Not all means are equal One-way ANOVA: _ Urchins versus Location Source DF SS MS F P Location 2 60.54 30.27 10.00 0.000 Error 177 535.77 3.03 Total 179 596.31 We reject the null that all means are equal Accept alternative that all means not equal Is that all?

11. Multiple Comparisons Allow you to determine the relations among all the means Several methods: Tukey, Fisher’s LSD, Dunnett’s, Bonferroni, Scheffe, etc Most focus on Tukey

12. Multiple Comparisons • 3 ways to test: • 1) Confidence Intervals • - default on “older” Minitab versions • - less intuitive than other methods • 2) Grouping Information • - Just answers, no details • - easy to interpret • 3) Simultaneous Tests • - t-tests run after ANOVA • - provides details; interpret like t-test

13. Tukey's method Tukey's method compares the means for each pair of factor levels using a family error rate to control the rate of type I error Results are presented as a set of confidence intervals for the difference between pairs of means Use the intervals to determine whether the means are different: If an interval does not contain zero, there is a statistically significant difference between the corresponding means If the interval does contain zero, the difference between the means is not statistically significant

14. Tukey 95% Simultaneous Confidence Intervals

15. Tukey 95% Simultaneous Confidence Intervals Deep vs. Middle = Not significantly different Deep vs. Shallow = Significantly different Middle vs. Shallow = Significantly different

16. Tukey Grouping Information Deep vs. Middle = Not significantly different Deep vs. Shallow = Significantly different Middle vs. Shallow = Significantly different

17. Tukey Test (using GLM) Deep vs. Middle = Not significantly different Deep vs. Shallow = Significantly different Middle vs. Shallow = Significantly different

18. Non-Parametric Version of ANOVA Kruskal-Wallis If samples are independent, similarly distributed data Use nonparametric test regardless of normality or sample size Is based upon mean 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

19. Non-Parametric Version of ANOVA Kruskal-Wallis Does not have multiple comparisons test (Tukey’s) Will need to run separate “t-tests” (Mann-Whitney) to test for differences between individual “means”

20. Non-Parametric Version of ANOVA Kruskal-Wallis

21. When Do I Do the What Now? “Well, whenever I'm confused, I just check my underwear. It holds the answer to all the important questions.” – Grandpa Simpson If Data are normal –use ANOVA Otherwise – use Kruskal-Wallis