Manova

1. MANOVA

2. Multivariate analysis When there is more than one dependent variable, it is inappropriate to do a series of univariate tests. Multivariate analysis of variance (MANOVA) is an extension of analysis of variance, used with two or more dependent variables

3. MANOVA Developed as a theoretical construct by Samual S. Wilks in 1932 An extension of univariate ANOVA procedures to situations in which there are two or more related dependent variables (ANOVA analyses only a single DV at a time ) The MANOVA procedure identifies (inferentially) whether: Different levels of the IVs have a significant effect on a linear combination of each of the DVs There are interactions between the IVs and a linear combination of the DVs. There are significant univariate effects for each of the DVs separately.

4. MANOVA USAGE MANOVA is appropriate when we have several DVs which all measure different aspects of some cohesive theme, e.g., several different types of academic achievement (e.g., Maths, English, Science). MANOVA works well in situations where there are moderate correlations between DVs. For very high or very low correlation in DVs, it is not suitable: if DVs are too correlated, there isn’t enough variance left over after the first DV is fit, and if DVs are uncorrelated, the multivariate test will lack power "Because of the increase in complexity and ambiguity of results with MANOVA, one of the best overall recommendations is: Avoid it if you can." ( Tabachnick & Fidell , 1983, p.230)

5. Anova vs. Manova Why not multiple Anovas ? Anovas run separately cannot take into account the pattern of covariation among the dependent measures It may be possible that multiple Anovas may show no differences while the Manova brings them out MANOVA is sensitive not only to mean differences but also to the direction and size of correlations among the dependents

6. Consider the following 2 group and 3 group scenarios, regarding two DVs Y 1 and Y 2 If we just look at the marginal distributions of the groups on each separate DV, the overlap suggests a statistically significant difference would be hard to come by for either DV However, considering the joint distributions of scores on Y 1 and Y 2 together (ellipses), we may see differences otherwise undetectable Anova vs. Manova

7. Now we can look for the greatest possible effect along some linear combination of Y 1 and Y 2 The linear combination of the DVs created makes the differences among group means on this new dimension look as large as possible Anova vs. Manova

8. So, by measuring multiple DVs you increase your chances for finding a group difference In this sense, in many cases such a test has more power than the univariate procedure, but this is not necessarily true as some seem to believe Also conducting multiple ANOVAs increases the chance for type 1 error and MANOVA can in some cases help control for the inflation Anova vs. Manova

9. MANOVA ASSUMPTIONS Sample size Rule of thumb: the n in each cell > the number of DVs Larger samples make the procedure more robust to violation of assumptions Normality MANOVA sig. tests assume multivariate normality, however when cell size > ~20 to 30 the procedure is robust violating this assumption Note that univariate normality is not a guarantee of multivariate normality, but it does help. Check univariate normality via histograms, normal probability plots, skewness, kurtosis, etc Linearity Linear relationships among all pairs of DVs Assess via scatterplots and bivariate correlations (check for each level of the IV(s Homogeneity of regression Homogeneity of variance-covariance matrix (Box's M) Multicollinearity and Singularity Outliers

10. MANOVA ASSUMPTIONS Homogeneity of regression This assumption is only important if using stepdown analysis, i.e., there is reason for ordering the DVs. Covariates must have a homogeneity of regression effect (must have equal effects on the DV across the groups) Homogeneity of variance-covariance matrix (Box's M) The F test from Box’s M statistics should be interpreted cautiously because it is a highly sensitive test of the violation of the multivariate normality assumption, particularly with large sample sizes. MANOVA is fairly robust to this assumption where there are equal sample sizes for each cell .

11. MANOVA ASSUMPTIONS Multicollinearity and Singularity MANOVA works best when the DVs are only moderately correlated. When correlations are low, consider running separate ANOVAs When there is strong multicollinearity , there are redundant DVs (singularity) which decreases statistical efficiency. Correlations above .7, and particularly above .8 or .9 are reason for concern. Outliers MANOVA is sensitive to the effect of outliers (they impact on the Type I error rate) MANOVA can tolerate a few outliers, particularly if their scores are not too extreme and there is a reasonable N . If there are too many outliers, or very extreme scores, consider deleting these cases or transforming the variables involved (see Tabachnick & Fidell )

12. DECISION TREE

13. DECISION TREE

14. Multivariate test statistics Roy's greatest characteristic root Tests for differences on only the first discriminant function Most appropriate when DVs are strongly interrelated on a single dimension Highly sensitive to violation of assumptions - most powerful when all assumptions are met Wilks ' lambda (λ) Most commonly used statistic for overall significance Considers differences over all the characteristic roots The smaller the value of Wilks ' lambda, the larger the between-groups dispersion

15. Multivariate test statistics Hotelling's trace Considers differences over all the characteristic roots Pillai's criterion Considers differences over all the characteristic roots More robust than Wilks '; should be used when sample size decreases, unequal cell sizes or homogeneity of covariances is violated

16. Test statistics - Preferences Pillai’s criterion or wilk’s lambda is the preferred measure when the basic design considerations( adequate sample size, no violations of assumptions, approx. equal cell sizes) are met Pillai’s criterion is considered more robust and should be used if sample size decreases, unequal cell sizes appear or homogeneity of covariances is violated Roy’s gcr is a more powerful test statistic if the researcher is confident that all assumptions are strictly met and the dependent measures are representative of a single dimension of effects In a vast majority of situations, all of statistical measures provide similar conclusions

17. MANOVA - ADVANTAGES It tests the effects of several independent variables and several outcome (dependent) variables within a single analysis It has the power of convergence (no single operationally defined dependent variable is likely to capture perfectly the conceptual variable of interest) independent variables of interest are likely to affect a number of different conceptual variables - for example: an organisation's non-smoking policy will affect satisfaction, production, absenteeism, health insurance claims, etc It can provide a more powerful test of significance than available when using univariate tests It reduces error rate compared with performing a series of univariate tests It provides interpretive advantages over a series of univariate ANOVAs Since only ‘one’ dependent variable is tested, the researcher is protected against inflating the type 1 error due to multiple comparisons.

18. MANOVA - DISADVANTAGES Discriminant functions are not always easy to interpret - they are designed to separate groups, not to make conceptual sense. In MANOVA, each effect evaluated for significance uses different discriminant functions (Factor A may be found to influence a combination of dependent variables totally different from the combination most affected by Factor B or the interaction between Factors A and B). Like discriminant analysis, the assumptions on which it is based are numerous and difficult to assess and meet.

19. HOW TO AVOID MANOVA Combine or eliminate dependent variables so that only one dependent variable need be analyzed Use factor analysis to find orthogonal factors that make up the dependent variables, then use univariate ANOVAs on each factor (because the factors are orthogonal each univariate analysis should be unrelated)

20. MANOVA - LIMITATIONS The number of people in the smallest cell should be larger than the total number of dependent variables It can be very sensitive to outliers, for small N It assumes a linear relationship (some sort of correlation) between the dependent variables MANOVA won't give you the interaction effects between the main effect and the repeated factor

21. MANOVA Question A researcher randomly assigns 33 subjects to one of three groups. The first group receives technical dietary information interactively from an on-line website. Group 2 receives the same information in from a nurse practitioner, while group 3 receives the information from a video tape made by the same nurse practitioner. The researcher looks at three different ratings of the presentation, difficulty, useful and importance, to determine if there is a difference in the modes of presentation. In particular, the researcher is interested in whether the interactive website is superior because that is the most cost-effective way of delivering the information.