# MANOVA

Multivariate analysis of variance (MANOVA) is an extension of the univariate analysis of variance (ANOVA).  In an ANOVA, we examine for statistical differences on one continuous dependent variable by an independent grouping variable.  The MANOVA extends this analysis by taking into account multiple continuous dependent variables, and bundles them together into a weighted linear combination or composite variable.  The MANOVA will compare whether or not the newly created combination differs by the different groups, or levels, of the independent variable.  In this way, the MANOVA essentially tests whether or not the independent grouping variable simultaneously explains a statistically significant amount of variance in the dependent variable.

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Assumptions:

1. Independent Random Sampling: MANOVA assumes that the observations are independent of one another, there is not any pattern for the selection of the sample, and that the sample is completely random.
2. Level and Measurement of the Variables: MANOVA assumes that the independent variables are categorical and the dependent variables are continuous or scale variables.
3. Absence of multicollinearity: The dependent variables cannot be too correlated to each other.  Tabachnick & Fidell (2012) suggest that no correlation should be above r = .90..
4. Normality: Multivariate normality is present in the data.
5. Homogeneity of Variance: Variance between groups is equal.

Key concepts and terms:

• Levene’s Test of Equality of Variance: Used to examine whether or not the variance between independent variable groups are equal; also known as homogeneity of variance  Non-significant values of Levene’s test indicate equal variance between groups.
• Box’s M Test: Used to know the equality of covariance between the groups.  This is the equivalent of a multivariate homogeneity of variance.    Usually, significance for this test is determined at α = .001 because this test is considered highly sensitive.
• Partial eta square: Partial eta square (η2) shows how much variance is explained by the independent variable. It is used as the effect size for the MANOVA model.
• Post hoc test: If there is a significant difference between groups, then post hoc tests are performed to determine where the significant differences lie (i.e., which specific independent variable level significantly differs from another).
• Multivariate F-statistics: The F- statistic is derived by essentially dividing the means sum of the square (SS) for the source variable by the source variable mean error (ME or MSE).

SPSS: Can be performed using the analysis menu, selecting the “GLM” option, and then choosing the “Multivariate” option from the GLM option.

Resources

Tabachnick, B. G. & Fidell, L. S. (2012). Using multivariate statistics (6th ed.).  Boston, MA: Pearson.

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