Multivariate analysis of variance: Difference between revisions

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Where [[Partition of sums of squares|sums of squares]] 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]].

MANOVA is based on the product of model variance matrix, <math>\Sigma_\text{model}</math> and inverse of the error variance matrix, <math>\Sigma_\text{res}^{-1}</math>, or <math>A=\Sigma_\text{model} \times \Sigma_\text{res}^{-1}</math>. The hypothesis that <math>\Sigma_\text{model} = \Sigma_\text{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|accessdateaccess-date=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|accessdateaccess-date=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|accessdateaccess-date=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_\text{Wilks} = \prod_{1,\ldots,p}(1/(1 + \lambda_{p})) = \det(I + A)^{-1} = \det(\Sigma_\text{res})/\det(\Sigma_\text{res} + \Sigma_\text{model})</math> distributed as [[Wilks' lambda distribution|lambda]] (Λ)

* the [[K. C. Sreedharan Pillai]]–[[M. S. Bartlett]] [[trace of a matrix|trace]], <math>\Lambda_\text{Pillai} = \sum_{1,\ldots,p}(\lambda_p/(1 + \lambda_p)) = \operatorname{tr}(A(I + A)^{-1})</math><ref>{{cite web|url=http://www.real-statistics.com/multivariate-statistics/multivariate-analysis-of-variance-manova/manova-basic-concepts/|title=MANOVA Basic Concepts – Real Statistics Using Excel|website=www.real-statistics.com|accessdateaccess-date=5 April 2018}}</ref>

* the Lawley–[[Harold Hotelling|Hotelling]] trace, <math>\Lambda_\text{LH} = \sum_{1,\ldots,p}(\lambda_{p}) = \operatorname{tr}(A)</math>

* [[Roy's greatest root]] (also called ''Roy's largest root''), <math>\Lambda_\text{Roy} = \max_p(\lambda_p) = \|A\|_\infty </math>