Analysis of Variance and Dependent Variables
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, [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
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
References: 1. ^ Stevens, J. P. (2002). Applied multivariate statistics for the social sciences. Mahwah, NJ: Lawrence Erblaum. [edit] © Gregory Carey, 1998 MANOVA: I - 1