1. Analysis of Variance 2-Way ANOVA MARE 250 Dr. Jason Turner

2. Two-way ANOVA - procedure to test the equality of population means when there are two factors 2-Sample T-Test (1R, 1F, 2 Levels) One-Way ANOVA (1R, 1F, >2 Levels) Two-Way ANOVA (1R, 2F, >1 Level) Two-Way – ANOVA

3. For Example… One-Way ANOVA – means of urchin #’s from each distance (shallow, middle, deep) are equal Response – urchin #, Factor – distance Two-Way ANOVA – means of urchin’s from each distance collected with each quadrat (¼m, ½m) are equal Response – urchin #, Factors – distance, quadrat Two-Way – ANOVA

4. SeaWall Deep Intermed. Shallow Factor 1 Location (S, M, D) Factor 2 Quad Size (¼m, ½m)

5. Two-Way – ANOVA SeaWall Deep Intermed. Shallow Factor 1 Location (S, M, D) Factor 2 Quad Size (¼m, ½m)

6. Two-Way – ANOVA SeaWall Deep Intermed. Shallow Factor 1 Location (S, M, D) Factor 2 Quad Size (¼m, ½m) INTERACTION Factor 1 X Factor 2 Location X Quad Size

7. If the effect of a fixed factor is significant, then the level means for that factor are significantly different from each other (just like a one-way ANOVA) If the effect of an interaction term is significant, then the effects of each factor are different at different levels of the other factor(s) Two-Way – ANOVA Results

8. Two-Way – ANOVA Results

9. Two-Way – ANOVA Results Urchins Location Quad Size

10. Two-Way ANOVA : Analysis of Variance Table Source DF SS MS F P Location 1 228.17 228.167 8.99 0.008 Quadsize 2 308.33 154.167 6.07 0.010 Interaction 2 76.33 38.167 1.50 0.249 Error 18 457.00 25.389 Total 23 1069.83 Two-Way – ANOVA Results

11. For the urchin analysis, the results indicate the following: The effect of Location (p = 0.008) is significant This indicates that urchin populations numbers were significantly different a different distances from shore The effect of Quad Size (p = 0.010) is significant This indicates quadrat type had a significant effect upon the number of urchins collected The interaction between Distance and Quadrat (p = 0.249) is not significant This means that the distance and quadrat size results were not influencing the other Thus, it is okay to interpret the individual effects of either factor alone

12. Two-Way ANOVA : Analysis of Variance Table Source DF SS MS F P Location 1 228.17 228.167 8.99 0.008 Quadsize 2 308.33 154.167 6.07 0.010 Interaction 2 76.33 38.167 1.50 0.009 Error 18 457.00 25.389 Total 23 1069.83 Two-Way – ANOVA Results

13. For the urchin analysis, the results indicate the following: The effect of Location (p = 0.008) is significant This indicates that urchin populations numbers were significantly different a different distances from shore The effect of Quad Size (p = 0.010) is significant This indicates quadrat type had a significant effect upon the number of urchins collected The interaction between Distance and Quadrat (p = 0.009) is not significant This means that the distance and quadrat size results WERE INFLUENCING the other Thus, the individual Factors must be analyzed alone

14. Use interactions plots to assess the two-factor interactions in a design Evaluate the lines to determine if there is an interaction: If the lines are parallel, there is no interaction If the lines cross, there IS Interaction The greater the lines depart from being parallel, the greater the degree of interaction Interactions

15. Interactions Plots

17. Why is there interaction? Because we get a different answer regarding #Urchins by Location (S,M,D) when using different Quadrats (¼m, ½m)

18. Interactions Plots Why is there interaction? Because we get a different answer regarding #Urchins by Quad Size (¼m, ½m) at different Locations (S,M,D)

19. The two-way ANOVA procedure does not support multiple comparisons To compare means using multiple comparisons, or if your data are unbalanced – use a General Linear Model General Linear Model - means of urchin #’s and species #’s from each distance (shallow, middle, deep) are equal Responses – urchin #, Factor – distance, quadrat Unbalanced…No Problem! Or multiple factors… General Linear Model - means of urchin #’s and species #’s from each distance (shallow, middle, deep) are equal Responses – urchin #, Factor – distance, quadrat, transect Two-Way – ANOVA

20. Two-Way ANOVA is a statistical test – there is a parametric (Two-Way ANOVA) and nonparametric version (Friedman’s) There are 3 ways to run a Two-Way ANOVA in minitab: 1) Two-Way ANOVA – for parametric (normal) balanced (equal n among levels) data 2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not 3) Friedman – nonparametric (not normal) data Two-Way – ANOVA

21. 1) Two-Way ANOVA – for parametric (normal) balanced (equal n among levels) data - See examples of Two-Way ANOVA above * Note – Two-Way ANOVA program cannot run Multiple Comparisons Tests (Tukey) Two-Way – ANOVA

22. 2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA

23. 2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA

24. 2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA

25. 2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA

26. 2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA

27. 2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA Location Quad Size

28. 2) General Linear Model (GLM) – for all parametric (normal) data – balanced or not Two-Way – ANOVA Location*Quad Size

29. 3) Friedman – nonparametric (not normal) data Two-Way – ANOVA

30. 3) Friedman – nonparametric (not normal) data Two-Way – ANOVA