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

2. 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 (0.25, 0.5) are equal Response – urchin #, Factors – distance, quadrat One-way, Two-way… If our data was balanced – it is not!

3. The two-way ANOVA procedure does not support multiple comparisons or multiple factors 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 & Multiway– ANOVA

4. MANOVA Multivariate Analysis of Variance (MANOVA) – compare means of multiple responses Responses: #Urchins, #Species Factors: Distance, Quadrat 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.

5. 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!

6. MANOVA 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

7. Analysis of Covariance For 2-Way ANOVA 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 For MANOVA Covariance – relationship between two responses; when two responses are not independent e.g. - # of urchins and # species

8. 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)

9. 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

10. Co-whattheheckareyoutalkingabout? Pearson correlation (just like our RJ test) #Urchins and # Species = 0.642 P-Value = 0.000 (greater than 0 – linear relationship) Covariances: #Urchins, #Species #Urchins 11.991055 #Species 1.582084 0.506967 (positive # = relationship; negative = negative

11. 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

12. Assessing Covariance using Correlation Scatterplot – graph of one response (x-axis) plotted versus another (y-axis)

13. Correlation coefficient (Pearson) – measures the extent of a linear relationship between two continuous variables (responses) Null:Correlation = 0 Alternative:Correlation ≠ 0 Pearson correlation of #Urchins and #Species = 0.642 P-Value = 0.000 (Correlation is Significantly different than Zero) IF p < 0.05 THEN the linear correlation between the two variables is significantly different than 0 IF p > 0.05 THEN you cannot assume a linear relationship between the two variables Conclusion – there IS a linear relationship Assessing Covariance using Correlation

14. Null:Correlation = 0 Alternative:Correlation ≠ 0 Pearson correlation of #Urchins and Depth = 0.109 P-Value = 0.071 IF p < 0.05 THEN the linear correlation between the two variables is significantly different than 0 IF p > 0.05 THEN you cannot assume a linear relationship between the two variables Conclusion – there IS NO linear relationship Assessing Covariance using Correlation

15. MANOVA Multivariate Analysis of Variance - compare means of multiple responses at multiple factors MANOVA for Method s = 1 m = 0.0 n = 26.5 Test DF Criterion Statistic F Num Denom P Wilks' 0.63099 16.082 2 55 0.000 Lawley-Hotelling 0.58482 16.082 2 55 0.000 Pillai's 0.36901 16.082 2 55 0.000 Roy's 0.58482 16.082 2 55 0.000

16. 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 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 Wilks' test is the most widely used method – we will use Wilks

17. MANOVA Multivariate Analysis of Variance – compare means of multiple responses at multiple factors MANOVA for Method s = 1 m = 0.0 n = 26.5 Test DF Criterion Statistic F Num Denom P Wilks' 0.63099 16.082 2 55 0.000 Lawley-Hotelling 0.58482 16.082 2 55 0.000 Pillai's 0.36901 16.082 2 55 0.000 Roy's 0.58482 16.082 2 55 0.000