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Multivariate analysis of variance (MANOVA) is an extension of common analysis of variance (ANOVA). In ANOVA, differences among various group means on a single-response variable are studied. In MANOVA, the number of response variables is increased to two or more. The hypothesis concerns a comparison of vectors of group means. The multivariate extension of the F-test is not completely direct. Instead, several test statistics are available, such as Wilks' Lambda and Lawley's trace. Wilks' lambda can range from 0 to 1, with 1 indicating no relationship of predictors to responses and 0 indicating a perfect relationship of predictors to responses. 1 - Wilks' lambda can be interpreted as the multivariate counterpart of a univariate R-squared, that is, it indicates the proportion of generalized variance in the dependent variables that is accounted for by the predictors. When only two groups are being compared, the results are identical to Hotelling's T2 procedure. Once a multivariate test has found a term significant, use the univariate ANOVA to determine which of the variables and factors are "causing" the significance.
Assumptions
1. The response (dependent) variables are continuous.
2. The residuals follow the multivariate-normal probability distribution with means equal to zero.
3. The variance-covariance matrices of each group of residuals are equal.
4. The individuals are independent.
Limitations
Unequal Sample Size
You should have more observations per factor category than you have dependent variables so that you can test the equality of covariance matrices using Box's M test.
Multivariate Normality and Outliers
Non-normality caused by the presence of outliers (extreme cases) can cause severe problems that even the robustness of the test will not overcome. You should screen your data for outliers and run it through various univariate and multivariate normality tests and plots to determine if the normality assumption is reasonable.
Homogeneity of Covariance Matrices
MANOVA makes the assumption that the within-group covariance matrices are equal. If the design is balanced so that there is an equal number of observations in each cell, the robustness of the MANOVA tests is guaranteed. If the design is unbalanced, you should test the equality of covariance matrices using Box's M test. If this test is significant at less than 0.001, there may be severe distortion in the alpha levels of the tests. You should only use Pillai's trace criterion in this situation.
Linearity
MANOVA assumes linear (straight-line) relationships among the dependent variables within a particular cell. You should study scatter plots of each pair of dependent variables using a different color for each level of a factor. Look carefully for curvilinear patterns and for outliers. The occurrence of curvilinear relationships will reduce the power of the MANOVA tests.
Multicollinearity
Multicollinearity occurs when one dependent variable is almost a weighted average of the others.
Example: Dependent measure consists of four indicators of success in society, and each indicator represents a completely independent way in which a person could "make it" in life (e.g., successful professional, successful entrepreneur, successful homemaker, etc.).