en>HcG007 at 17:26, 8 April 2011

2011-04-08T17:26:00Z

New page

'''Multivariate analysis of covariance (MANCOVA)''' is an extension of [[analysis of covariance]] ([[ANCOVA]]) methods to cover cases where there is more than one dependent variable and where the control of concomitant continuous independent variables - [[covariate]]s - is required. The most prominent benefit of the MANCOVA design over the simple [[MANOVA]] is the 'factoring out' of [[statistical noise|noise]] or error that has been introduced by the covariant.<ref name="Statsoft" /> A commonly used multivariate version of the [[ANOVA]] [[F-distribution|F-statistic]] is [[Wilks' lambda distribution|Wilks' Lambda]] (Λ), which represents the ratio between the error variance (or covariance) and the effect variance (or covariance).<ref name="Statsoft">[http://www.statsoft.com/textbook/anova-manova/#multivariate] Statsoft Textbook, ANOVA/MANOVA.</ref>

==Goals of MANCOVA==
Similarly to all tests in the [[ANOVA]] family, the primary aim of the MANCOVA is to test for significant differences between group means.<ref name="Statsoft">[http://www.statsoft.com/textbook/anova-manova/#multivariate]. Statsoft Textbook, ANOVA/MANOVA.</ref> The process of characterising a covariate in a data source allows the reduction of the magnitude of the error term, represented in the MANCOVA design as ''MS<sub>error</sub>''. Subsequently, the overall [[Wilks' lambda distribution|Wilks' Lambda]] will become larger and more likely to be characterised as significant.<ref name="Statsoft"/> This grants the researcher more [[statistical power]] to detect differences within the data. The multivariate aspect of the MANCOVA allows the characterisation of differences in group means in regards to a linear combination of multiple dependent variables, while simultaneously controlling for covariates.

''Example situation where MANCOVA is appropriate:''
Suppose a scientist is interested in testing two new drugs for their effects on depression and anxiety scores. Also suppose that the scientist has information pertaining to the overall responsivity to drugs for each patient; accounting for this [[covariate]] will grant the test higher sensitivity in determining the effects of each drug on both dependent variables.

==Assumptions==
Certain assumptions must be met for the MANCOVA to be used appropriately:
# '''Normality''': For each group, each dependent variable must represent a [[normal distribution]] of scores. Furthermore, any linear combination of dependent variables must be normally distributed. Transformation or removal of outliers can help ensure this assumption is met.<ref name="Frenchpdf">[http://userwww.sfsu.edu/~efc/classes/biol710/manova/MANOVAnewest.pdf] French, A. et al., 2010. Multivariate analysis of variance (MANOVA).</ref> Violation of this assumption may lead to an increase in [[Type I error]] rates.<ref name="Davis">[http://schatz.sju.edu//multivar/guide/Mancova.pdf] Davis, K., 2003. Multiple analysis of variance (MANOVA) or multiple analysis of covariance (MANCOVA). Louisiana State University.</ref>
# '''Independence of observations''': Each observation must be independent of all other observations; this assumption can be met by employing [[random sample|random sampling]] techniques. Violation of this assumption may lead to an increase in [[Type I error]] rates.<ref name="Davis" />
# '''Homogeneity of variances''': Each dependent variable must demonstrate similar levels of variance across each independent variable. Violation of this assumption can be conceptualised as a correlation existing between the variances and the means of dependent variables. This violation is often called '[[homoscedasticity]]'<ref name="UoTS">[http://www.utsc.utoronto.ca/~bors/HomoVariance.ppt] Bors, D. A. University of Toronto at Scarborough.</ref> and can be tested for using [[Levene's test]].<ref name="USC">[http://www-bcf.usc.edu/~mmclaugh/550x/PPTslides/WeekElevenSlides/MANOVA.ppt] McLaughlin, M., 2009. University of Southern Carolina.</ref>
# '''Homogeneity of covariances''': The intercorrelation matrix between dependent variables must be equal across all levels of the independent variable. Violation of this assumption may lead to an increase in [[Type I error]] rates as well as decreased [[statistical power]].<ref name="Davis"/>

==Logic of MANOVA==
{{Main|MANOVA}}
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 _{1...p}(1/(1 + \lambda_{p}))</math> distributed as [[Wilks' lambda distribution|lambda]] (Λ)
* the Pillai-[[M. S. Bartlett]] [[trace of a matrix|trace]], <math>\Lambda_{Pillai} = \sum _{1...p}(1/(1 + \lambda_{p}))</math>
* the Lawley-[[Harold Hotelling|Hotelling]] trace, <math>\Lambda_{LH} = \sum _{1...p}(\lambda_{p})</math>
* [[Roy's greatest root]] (also called ''Roy's largest root''), <math>\Lambda_{Roy} = max_p(\lambda_p)</math>

==Covariates==
{{Main|Covariate}}

In statistics, a '''covariate''' represents a source of variation that has not been controlled in the experiment and is believed to affect the dependent variable.<ref name="Kirk">{{cite book|last=Kirk|first=Roger E.|title=Experimental design|year=1982|publisher=Brooks/Cole Pub. Co.|location=Monterey, Calif.|isbn=0-8185-0286-X|edition=2nd ed.}}</ref> The aim of such techniques as [[ANCOVA]] is to remove the effects of such uncontrolled variation, in order to increase statistical power and to ensure an accurate measurement of the true relationship between independent and dependent variables.<ref name="Kirk" />

An example is provided by the analysis of trend in sea-level by Woodworth (1987). Here the [[dependent variable]] (and variable of most interest) was the annual mean sea level at a given location for which a series of yearly values were available. The primary independent variable was "time". Use was made of a "covariate" consisting of yearly values of annual mean atmospheric pressure at sea level. The results showed that inclusion of the covariate allowed improved estimates of the trend against time to be obtained, compared to analyses which omitted the covariate.

==See also==
*[[Discriminant function analysis]]
*[[ANCOVA]]
*[[MANOVA]]

==References==
<references />

{{DEFAULTSORT:Mancova}}
[[Category:Analysis of variance]]

First variation of area formula - Revision history

en>HcG007 at 17:26, 8 April 2011