1. MARE 250 Dr. Jason Turner Multiway, Multivariate, Covariate, ANOVA

2. For Example… One-Way ANOVA – means of urchin #’s from each location (shallow, middle, deep) are equal Response – urchin #, Factor – location Two-Way ANOVA – means of urchin’s from each location collected with each quadrat (0.25, 0.5) are equal Response – urchin #, Factors – location, quadrat One-way, Two-way… If our data was balanced – it is not!

3. 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 location (shallow, middle, deep) are equal Responses – urchin #, Factor – location, quadrat Unbalanced…No Problem! Two-Way – ANOVA

4. Multi-way ANOVA: Response:1 Factors: >2 Multi-Way ANOVA – means of urchin’s from each location (shallow, middle, deep) collected with each quadrat (¼m, ½m), across different years (2009, 2010, 2011, 2012) are equal Response – urchin #, Factors – location, quadrat, year Multi-Way – ANOVA

5. Multi-way ANOVA: Run using a General Linear Model (GLM) – very similar to the way a 2-way test is run Multi-Way – ANOVA

6. Multivariate ANOVA (MANOVA): Response:>1 Factors:1-? MANOVA– means of urchin’s and species from each location (shallow, middle, deep) are equal Responses – urchin #, species # Factor – location Multivariate – ANOVA (MANOVA)

7. MANOVA Multivariate Analysis of Variance - compare means of multiple responses at multiple factors Run using General MANOVA program – like GLM

8. MANOVA Multivariate Analysis of Variance - compare means of multiple responses at multiple factors – results to 4 different types of MANOVAs given as output

9. MANOVA By default, MINITAB displays a table of the four multivariate tests for each term in the model: Wilks' test - the most commonly used test because it was the first derived and has a well-known F approximation Lawley-Hotelling - also known as Hotelling's generalized T statistic or Hotelling’s Trace

10. MANOVA Pillai's - will give similar results to the Wilks' and Lawley- Hotelling's tests Roy's - use only when the mean vectors are collinear; does not have a satisfactory F approximation For this class we will use Wilks’

11. MANOVA Multivariate Analysis of Variance – compare means of multiple responses at multiple factors Responses: #Urchins, #Species Factors: Distance Q - Why not just run multiple one-way ANOVAs????? A - When you use multiple one-way ANOVAs to analyze data, you increase the probability of a Type I error. MANOVA controls the family error rate, thereby minimizing the probability of making one or more type I errors for the entire set of comparisons.

12. The probability of making a TYPE I Error (rejection of a true null hypothesis) is called the significance level (α) of a hypothesis test TYPE II Error Probability (β) – nonrejection of a false null hypothesis Error!

13. MANOVA In Conclusion… We run ANOVA instead of multiple t-tests to investigate 1 response versus multiple factors We run MANOVA instead of multiple one-way ANOVAs to investigate multiple responses versus multiple factors

14. MANOVA H 0 : (μUrchS = μUrchM = μUrchD)(μSpecS = μSpecM = μSpecD) Ha: All means not equal

15. Analysis of Covariance Interaction – relationship between two factors; when the effect of one factor is not independent of the effect of another e.g. – # of urchins at each distance is effected by quadrat size Covariance – relationship between two responses; when two responses are not independent e.g. - # of urchins and # species

16. Analysis of Covariance We can assess Covariance in 2 ways: 1. Run a covariance test 2. Run a correlation Both help us to determine whether (or not) there is a linear relationship between two variables (our responses)

17. Relationship between covariance and correlation Although both the correlation coefficient and the covariance are measure of linear association, they differ in the following ways: Correlations coefficients are standardized, thus a perfect linear relationship will result in a coefficient of 1. Covariance values are not standardized, thus the value for a perfect linear relationship will depend on the data. Assessing Covariance using Correlation

18. Relationship between covariance and correlation The correlation coefficient is a function of the covariance. The correlation coefficient is equal to the covariance divided by the product of the standard deviations of the variables Thus, a positive covariance will always result in a positive correlation and similarly, a negative covariance will always result in a negative correlation Assessing Covariance using Correlation

19. Co-whattheheckareyoutalkingabout? Pearson correlation (just like our RJ test) (greater than 0 – linear relationship; H 0 : r=0)

20. Co-whattheheckareyoutalkingabout? Pearson correlation (just like our RJ test) (greater than 0 – linear relationship; H 0 : r=0)

21. Co-whattheheckareyoutalkingabout? Covariances: #Urchins, #Species (positive # = relationship; negative = negative

22. Co-whattheheckareyoutalkingabout? Covariances: #Urchins, #Species (positive # = relationship; negative = negative

23. Co-whichoneshouldIuse? It is important to note that covariance does not imply causality (relationship between cause & effect) Can determine that using Correlation SO…run a Correlation between responses to determine if there is Covariance If Covariance than run MANOVA with other Response as a Covariate

24. Co-whichoneshouldIuse? If you run a MANOVA, and fail to accept the null hypothesis (H 0 : means are equal) Then need to run ANOVA w/ Tukeys on each individual Response Variable

25. MANOVA H 0 : (μUrchS = μUrchM = μUrchD) Ha: All means not equal H 0: (μSpecS = μSpecM = μSpecD) Ha: All means not equal