ANOVA

1. ANOVA Univariate, Multivariate and Repeated Measures

2. Analysis of Variance Assumptions: • DV is normally distributed within groups • DV is continuous (interval or ratio) • Observations are independent • IV consists of 2 or more independent categoricalgroups • There are no significant outliers • There is homogeneity of variances (homoscedasticity)

3. Recall Previous Example • A researcher has collected a species of lizard from three different island populations. Each island represents a different eco-zone. He collects 10 lizards from each island and measures their running speed Test whether lizards from the different islands differ in their running speeds

4. One-Way ANOVA • Use the following for a simple one-way ANOVA • From the menus choose: Analyze Compare Means One-Way ANOVA •  Select one or more dependent variables •  Select a single independent grouping variable (factor) • Variables must be numeric

5. One-Way ANOVA • Select your variables using the arrows • Click Options • Tick Homogeneity of Variances

6. One-Way ANOVA • Click on Post Hoc for Multiple Comparison Tests • Select Tukey

7. ANOVA Results Make sure you can interpret these results

8. GLM – Univariate • General Linear Model • Extension of the one-way ANOVA • Instead of only one grouping variable, you can select multiple grouping variables • Only one dependent variable • After an overall F test has shown significance, you can use post hoc tests to evaluate differences among specific means

9. Example • Suppose our lizard researcher had also recorded the sex (male or female) of the lizards • We now have two grouping variables Conduct a two-factor ANOVA on the data: Is there a difference in running speed according to island and sex?

10. GLM Univariate • From the menus choose: Analyze General Linear Model Univariate • Select one DV and multiple IVs (fixed factors)

11. General Linear Model • Fixed Factors are the main factors (predictor variables) of interest • Random Factors are variables that may explain excess variability in the dependent variable • Covariates are scale variables that may be correlated with the dependent variable • AVOVA becomes an ANCOVA

12. GLM Univariate • Specify the Model

13. GLM Univariate • Model: • A full factorial model contains all factor main effects, all covariate main effects, and all factor-by-factor interactions. It does not contain covariate interactions • Select Custom to specify only a subset of interactions or to specify factor-by-covariate interactions. You must indicate all of the terms to be included in the model • Sum of Squares: Type III sum-of-squares method is most commonly used • Include intercept in model: The intercept is usually included in the model. If you can assume that the data pass through the origin, you can exclude the intercept

14. GLM Univariate • Add a Post Hoc test for Island

15. GLM Univariate • Add a means plot under Plots

16. GLM Univariate • Test for Homogeneity of Variances under Options

17. Results What do the results tell us?

18. Multivariate Tests • Two or more dependent (response) variables • Examines the relationships/interactions of these variables • Takes into account the fact that: • Variables may not be independent of each other • Performing multiple comparisons increases the risk of making a Type I error • Performing a series of univariatetests would not be appropriate

19. Multivariate Analysis of Variance (MANOVA) • Extension of the ANOVA • Two or more response variables • Combines multiple response variables into a single new variable to maximise the differences between the treatment group means • Obtain a multivariate F value – Wilks’ lambda (value between 0 and 1) is most commonly used • If the overall test is significant, we can then go on to examine which of the individual variables contributed to the significant effect

20. MANOVA – Example • Suppose our lizard researcher also measured body length and limb length for each lizard • There are now three dependent variables Do lizards from different islands and of different sexes differ in their morphology and abilities?

21. GLM – Multivariate • The GLM Multivariate procedure in SPSS provides regression analysis and analysis of variance for multiple dependent variables by one or more factor variables or covariates • The factor variables divide the population into groups

22. GLM – Multivariate • From the menus choose: Analyze General Linear Model Multivariate • Select 2 or more DVs and one or more IVs (fixed factors)

23. GLM – Multivariate • Select the required additional tests and options as with the GLM univariate by clicking on the blue boxes on the right

24. Results Sig. Sig. Not sig.

26. Results

27. Repeated Measures ANOVA • Groups are not independent • E.g. The same individuals are measured at different time intervals or under different treatment conditions

28. Repeated Measures ANOVA

29. Repeated Measures ANOVA • Repeated measures must be entered as separate variables • From the menus choose: Analyze General Linear Model Repeated Measures

30. Example Four male and four female turtles had their plasma protein measured while they were well fed and after ten and twenty days of fasting. Conduct a repeated measures ANOVA on the data. What do you conclude?

31. Repeated Measures ANOVA • Give your IV a name under Within Subject Factor Name • Specify the number of levels (number of repeated measures) • Click Add • Give your DV a name under Measure Name • Click Add

32. Repeated Measures ANOVA • Click on Define • Use the arrow to move variables across to the right-hand box (Within-Subjects Variables) • Enter your grouping variables (if any) as Between-Subjects Factor(s)

33. Repeated Measures ANOVA • Click on Plots • Select your factor (IV) and move it across to the Horizontal Axis box using the arrow • Click Add and then Continue

34. Repeated Measures ANOVA • Click on Options

35. Results There is an overall significant result of the main effect (FastingLevel): Sig. Not sig.

36. Results • Test for sphericity: • The variances of the differences between all combinations of related groups must be equal • Equivalent of homogeneity of variances

37. Results Assumption of sphericity is not violated

38. Results There is no significant difference between males and females

39. Results • Post-Hoc Tests: Significant

40. Results • Visual summary of results: