Multivariate analysis of variance: Difference between revisions

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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_\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}lambda_p/(1 + \lambda_{p}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|author=|date=|website=www.real-statistics.com|accessdate=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>