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'''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.<ref>Stevens, J. P. (2002). ''Applied multivariate statistics for the social sciences.'' Mahwah, NJ: Lawrence Erblaum.</ref>  Statistical reports however will provide individual p-values for each dependent variable, indicating whether differences and interactions are statistically significant.


Where [[sums of squares]]{{dn|date=December 2013}} appear in univariate analysis of variance, in multivariate analysis of variance certain [[positive-definite matrix|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 [[errors and residuals in statistics|error]] distributions, the counterpart of the sum of squares due to error has a [[Wishart distribution]].
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Analogous to [[ANOVA]], MANOVA is based on the product of model variance matrix, <math>\Sigma_{model}</math> and
inverse of the error variance matrix, <math>\Sigma_{res}^{-1}</math>, or <math>A=\Sigma_{model} \times \Sigma_{res}^{-1}</math>. The hypothesis that <math>\Sigma_{model} = \Sigma_{residual}</math> implies that the product <math>A \sim I</math>.<ref>{{cite web|last=Carey|first=Gregory|title=Multivariate Analysis of Variance (MANOVA): I. Theory|url=http://ibgwww.colorado.edu/~carey/p7291dir/handouts/manova1.pdf|accessdate=2011-03-22}}</ref> Invariance considerations imply the MANOVA statistic should be a measure of [[magnitude (mathematics)|magnitude]] of the [[singular value decomposition]] of this matrix product, but there is no unique choice owing to the multi-[[dimension]]al nature of the alternative hypothesis.  
 
The most common<ref>{{cite web|last=Garson|first=G. David|title=Multivariate GLM, MANOVA, and MANCOVA|url=http://faculty.chass.ncsu.edu/garson/PA765/manova.htm|accessdate=2011-03-22}}</ref><ref>{{cite web|last=UCLA: Academic Technology Services, Statistical Consulting Group.|title=Stata Annotated Output -- MANOVA|url=http://www.ats.ucla.edu/stat/stata/output/Stata_MANOVA.htm|accessdate=2011-03-22}}</ref>  statistics are summaries based on the roots (or [[eigenvalues]]) <math>\lambda_p</math> of the <math>A</math> matrix:
* [[Samuel Stanley Wilks]]' <math>\Lambda_{Wilks} = \prod _{i=1...p}(1/(1 + \lambda_{i}))</math> distributed as [[Wilks' lambda distribution|lambda]] (Λ)
* the Pillai-[[M. S. Bartlett]] [[trace of a matrix|trace]], <math>\Lambda_{Pillai} = \sum _{i=1...p}(\lambda_{i}/(1 + \lambda_{i}))</math>
* the Lawley-[[Harold Hotelling|Hotelling]] trace, <math>\Lambda_{LH} = \sum _{i=1...p}(\lambda_{i})</math>
* [[Roy's greatest root]] (also called ''Roy's largest root''), <math>\Lambda_{Roy} = max_i(\lambda_i)</math>
 
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|date=March 2013}} 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.
 
==See also==
*[[Discriminant function analysis]]
*[[Repeated measures design]]
 
==References==
{{reflist}}
 
==External links==
{{wikiversity}}
*[http://online.sfsu.edu/~efc/classes/biol710/manova/manovanewest.htm  Multivariate Analysis of Variance (MANOVA) by Aaron French, Marcelo Macedo, John Poulsen, Tyler Waterson and Angela Yu, San Francisco State University]
 
{{Statistics}}
{{Experimental design}}
 
[[Category:Analysis of variance]]
[[Category:Design of experiments]]

Latest revision as of 21:28, 12 January 2015


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