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, [pic], or [pic]. The hypothesis that Σmodel = Σresidual implies that the product A∼I[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))|

| |1...p | |

distributed as lambda (Λ)

▪ the Pillai-M. S. Bartlett trace,

|ΛPillai |∑ |(1 / (1 + λp))|

|= | | |

| |1...p| |

▪ the Lawley-Hotelling trace,

|ΛLH = |∑ |(λp)|

| |1...p| |

▪ Roy's greatest root (also called Roy's largest root), ΛRoy = maxp(λp) 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.

[edit]References

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. [edit]

© Gregory Carey, 1998 MANOVA: I - 1

Multivariate Analysis of Variance (MANOVA): I. Theory Introduction

The purpose of a t test is to assess the likelihood that the means for two groups are sampled from the same sampling distribution of means. The purpose of an ANOVA is to test whether the means for two or more groups are taken from the same sampling distribution. The multivariate equivalent of the t test is Hotelling’s T 2

. Hotelling’s T2 tests whether the two vectors of means for the two groups are sampled from the same

sampling distribution. MANOVA is the multivariate analogue to Hotelling's T2

. The purpose of MANOVA is to test whether the vectors of means for the two or more groups are sampled from the same sampling distribution. Just as Hotelling's T2 will provide a measure of the likelihood of picking two random vectors of means out of the same hat, MANOVA gives a measure of the overall likelihood of picking two or more random vectors of means out of the same hat.

There are two major situations in which MANOVA is used. The first is when there are several correlated dependent variables, and the...