# Multivariate analysis of variance

Multivariate analysis of variance (MANOVA) is a statistical test procedure for comparing multivariate (population) means of several groups. Unlike ANOVA, it uses the variance-covariance between variables in testing the statistical significance of the mean differences.

It is a generalized form of univariate analysis of variance (ANOVA). It is used when there are two or more dependent variables. It helps to answer : 1. do changes in the independent variable(s) have significant effects on the dependent variables; 2. what are the interactions among the dependent variables and 3. among the independent variables.1 Statistical reports however will provide individual p-values for each dependent variable, indicating whether differences and interactions are statistically significant.

Where sums of squares appear in univariate analysis of variance, in multivariate analysis of variance certain positive-definite matrices appear. The diagonal entries are the same kinds of sums of squares that appear in univariate ANOVA. The off-diagonal entries are corresponding sums of products. Under normality assumptions about error distributions, the counterpart of the sum of squares due to error has a Wishart distribution.

Analogous to ANOVA, MANOVA is based on the product of model variance matrix, $\Sigma_{model}$ and inverse of the error variance matrix, $\Sigma_{res}^{-1}$, or $A=\Sigma_{model} \times \Sigma_{res}^{-1}$. The hypothesis that $\Sigma_{model} = \Sigma_{residual}$ implies that the product $A \sim I$.2 Invariance considerations imply the MANOVA statistic should be a measure of magnitude of the singular value decomposition of this matrix product, but there is no unique choice owing to the multi-dimensional nature of the alternative hypothesis.

The most common34 statistics are summaries based on the roots (or eigenvalues) $\lambda_p$ of the $A$ matrix:

• Samuel Stanley Wilks' $\Lambda_{Wilks} = \prod _{i=1...p}(1/(1 + \lambda_{i}))$ distributed as lambda (Λ)
• the Pillai-M. S. Bartlett trace, $\Lambda_{Pillai} = \sum _{i=1...p}(\lambda_{i}/(1 + \lambda_{i}))$
• the Lawley-Hotelling trace, $\Lambda_{LH} = \sum _{i=1...p}(\lambda_{i})$
• Roy's greatest root (also called Roy's largest root), $\Lambda_{Roy} = max_i(\lambda_i)$

Discussion continues over the merits of each, though the greatest root leads only to a bound on significance which is not generally of practical interest. A further complication is that the distribution of these statistics under the null hypothesis is not straightforward and can only be approximated except in a few low-dimensional cases.citation needed The best-known approximation for Wilks' lambda was derived by C. R. Rao.

In the case of two groups, all the statistics are equivalent and the test reduces to Hotelling's T-square.

## Correlation of dependent variables

MANOVA is most effective when dependent variables are moderately correlated (.4 - .7). If dependent variables are too highly correlated it could be assumed that they may be measuring the same variable.

## References

1. ^ Stevens, J. P. (2002). Applied multivariate statistics for the social sciences. Mahwah, NJ: Lawrence Erblaum.
2. ^ Carey, Gregory. "Multivariate Analysis of Variance (MANOVA): I. Theory". Retrieved 2011-03-22.
3. ^ Garson, G. David. "Multivariate GLM, MANOVA, and MANCOVA". Retrieved 2011-03-22.
4. ^ UCLA: Academic Technology Services, Statistical Consulting Group. "Stata Annotated Output -- MANOVA". Retrieved 2011-03-22.

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