Multivariate analysis of variance or multiple analysis of variance (MANOVA) is a statistical test procedure for comparing multivariate (population) means of several groups. As a multivariate procedure, it is used when there are two or more dependent variables, although statistical reports provide individual p-values for each dependent variable in order to test for statistical significance.

It helps to answer:

إنها تساعد في الإجابة --~~~~ Rehabb

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Do changes in the independent variable(s) have significant effects on the dependent variables?

What are the interactions among the dependent variables?

And among the independent variables?

Relationship with ANOVA[edit]

MANOVA is a generalized form of univariate analysis of variance (ANOVA), although, unlike univariate ANOVA, it uses the variance-covariance between variables in testing the statistical significance of the mean differences.

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, and inverse of the error variance matrix, , or .

The hypothesis that implies that the product .

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 common statistics are summaries based on the roots (or eigenvalues) of the matrix:

Samuel Stanley Wilks' distributed as lambda (Λ) the Pillai-M.

S. Bartlett trace, the Lawley-Hotelling trace, Roy's greatest root (also called Roy's largest root),

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.

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[edit]

MANOVA is most effective when dependent variables are moderately correlated (0.4–0.7).

If dependent variables are too highly correlated it could be assumed that they may be measuring the same variable.