Multivariate analysis of variance

Multivariate analysis of variance

Multivariate analysis of variance (MANOVA) 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.[1]

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, Σmodel and inverse of the error variance matrix, \Sigma_{res}^{-1}, or A=\Sigma_{model} \times \Sigma_{res}^{-1}. The hypothesis that Σmodel = Σresidual implies that the product AI[2] . 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[3][4] statistics are summaries based on the roots (or eigenvalues) λp of the A matrix:

  • Samuel Stanley Wilks'
ΛWilks = (1 / (1 + λp))

distributed as lambda (Λ)

ΛPillai = (1 / (1 + λp))
ΛLH = p)
  • Roy's greatest root (also called Roy's largest root), ΛRoy = maxpp)

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.


  1. ^ Stevens, J. P. (2002). Applied multivariate statistics for the social sciences. Mahwah, NJ: Lawrence Erblaum.
  2. ^ Carey, Gregory. "Multivariate Analysis of Variance (MANOVA): I. Theory". Retrieved 2011-03-22. 
  3. ^ Garson, G. David. "Multivariate GLM, MANOVA, and MANCOVA". Retrieved 2011-03-22. 
  4. ^ UCLA: Academic Technology Services, Statistical Consulting Group.. "Stata Annotated Output -- MANOVA". Retrieved 2011-03-22. 

See also

External links

Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Analysis of variance — In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of… …   Wikipedia

  • multivariate analysis — Univariate analysis consists in describing and explaining the variation in a single variable. Bivariate analysis does the same for two variables taken together (covariation). Multivariate analysis (MVA) considers the simultaneous effects of many… …   Dictionary of sociology

  • analysis of variance — noun a statistical method for making simultaneous comparisons between two or more means; a statistical method that yields values that can be tested to determine whether a significant relation exists between variables • Syn: ↑ANOVA • Topics:… …   Useful english dictionary

  • multivariate analysis — noun a generic term for any statistical technique used to analyze data from more than one variable • Topics: ↑statistics • Hypernyms: ↑statistical method, ↑statistical procedure • Hyponyms: ↑multiple regression, ↑ …   Useful english dictionary

  • Multivariate statistics — is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis. Methods of bivariate statistics, for example simple linear… …   Wikipedia

  • Multivariate normal distribution — MVN redirects here. For the airport with that IATA code, see Mount Vernon Airport. Probability density function Many samples from a multivariate (bivariate) Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the… …   Wikipedia

  • Multivariate kernel density estimation — Kernel density estimation is a nonparametric technique for density estimation i.e., estimation of probability density functions, which is one of the fundamental questions in statistics. It can be viewed as a generalisation of histogram density… …   Wikipedia

  • Variance — In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value (mean). Whereas… …   Wikipedia

  • Multivariate Normalverteilung — Die gemeinsame Wahrscheinlichkeitsverteilung mehrerer Zufallsvariablen nennt man multivariate Verteilung oder auch mehrdimensionale Verteilung. Inhaltsverzeichnis 1 Formale Darstellung 2 Ausgewählte multivariate Verteilungen 3 Die multivariate… …   Deutsch Wikipedia

  • Multivariate stable distribution — multivariate stable Probability density function Heatmap showing a Multivariate (bivariate) stable distribution with α = 1.1 parameters: exponent shift/location vector …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”