A researcher assigns 33 subjects to 3 groups receiving dietary information via different modes: an online website, a nurse practitioner, or a video. The researcher measures 3 dependent variables related to the presentation: difficulty, usefulness, and importance. MANOVA is an appropriate analysis to determine if the modes of presentation have a significant effect on a combination of the dependent variables, while accounting for correlations between them. The researcher can use MANOVA to test whether the interactive website is superior to the other modes in conveying the information in a comprehensive yet cost-effective manner.

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

4. 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)

5. The MANOVA procedure identifies (inferentially) whether:

6. Different levels of the IVs have a significant effect on a linear combination of each of the DVs

7. There are interactions between the IVs and a linear combination of the DVs.

9. 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

11. Anova vs. Manova Consider the following 2 group and 3 group scenarios, regarding two DVs Y1 and Y2 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 Y1 and Y2 together (ellipses), we may see differences otherwise undetectable

12. Anova vs. Manova Now we can look for the greatest possible effect along some linear combination of Y1 and Y2 The linear combination of the DVs created makes the differences among group means on this new dimension look as large as possible

13. Anova vs. Manova 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

15. Rule of thumb: the n in each cell > the number of DVs

16. Larger samples make the procedure more robust to violation of assumptions

17. Normality

18. MANOVA sig. tests assume multivariate normality, however when cell size > ~20 to 30 the procedure is robust violating this assumption

19. Note that univariate normality is not a guarantee of multivariate normality, but it does help.

20. Check univariate normality via histograms, normal probability plots, skewness, kurtosis, etc

21. Linearity

22. Linear relationships among all pairs of DVs

23. Assess via scatterplots and bivariate correlations (check for each level of the IV(s

24. Homogeneity of regression

25. Homogeneity of variance-covariance matrix (Box's M)

26. Multicollinearityand Singularity

28. This assumption is only important if using stepdown analysis, i.e., there is reason for ordering the DVs.

29. Covariates must have a homogeneity of regression effect (must have equal effects on the DV across the groups)

30. Homogeneity of variance-covariance matrix (Box's M)

31. 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.

33. MANOVA works best when the DVs are only moderately correlated.

34. When correlations are low, consider running separate ANOVAs

35. When there is strong multicollinearity, there are redundant DVs (singularity) which decreases statistical efficiency.

36. Correlations above .7, and particularly above .8 or .9 are reason for concern.

37. Outliers

38. MANOVA is sensitive to the effect of outliers (they impact on the Type I error rate)

40. DECISION TREE

42. Tests for differences on only the first discriminant function

43. Most appropriate when DVs are strongly interrelated on a single dimension

44. Highly sensitive to violation of assumptions - most powerful when all assumptions are met

45. Wilks' lambda (λ)

46. Most commonly used statistic for overall significance

47. Considers differences over all the characteristic roots

49. Considers differences over all the characteristic roots

50. Pillai's criterion

51. Considers differences over all the characteristic roots

53. 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

54. Roy’s gcris 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

56. It has the power of convergence (no single operationally defined dependent variable is likely to capture perfectly the conceptual variable of interest)

58. It reduces error rate compared with performing a series of univariate tests

59. It provides interpretive advantages over a series of univariate ANOVAs

63. It can be very sensitive to outliers, for small N

64. It assumes a linear relationship (some sort of correlation) between the dependent variables