MANOVA
Encyclopedia
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.
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,
and
inverse of the error variance matrix,
, or 
. The hypothesis that 
implies that the product 
. 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-dimension
al nature of the alternative hypothesis.
The most common statistics are summaries based on the roots (or eigenvalues)
of the 
matrix:
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.
Analysis of variance
In statistics, analysis of variance 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 variation...
(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.
Where sums of squares appear in univariate analysis of variance, in multivariate analysis of variance certain positive-definite matrices
Positive-definite matrix
In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....
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
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
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,
andinverse of the error variance matrix,
, or . The hypothesis that implies that the product . Invariance considerations imply the MANOVA statistic should be a measure of magnitudeMagnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
of the singular value decomposition
Singular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
of this matrix product, but there is no unique choice owing to the multi-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al nature of the alternative hypothesis.
The most common statistics are summaries based on the roots (or eigenvalues)
of the matrix:
- Samuel Stanley Wilks'
distributed as lambdaWilks' lambda distributionIn statistics, Wilks' lambda distribution , is a probability distribution used in multivariate hypothesis testing, especially with regard to the likelihood-ratio test and Multivariate analysis of variance...
(Λ) - the Pillai-M. S. BartlettM. S. BartlettMaurice Stevenson Bartlett FRS was an English statistician who made particular contributions to the analysis of data with spatial and temporal patterns...
trace,
- the Lawley-HotellingHarold HotellingHarold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...
trace,
- Roy's greatest root (also called Roy's largest root),
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
Null hypothesis
The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
is not straightforward and can only be approximated except in a few low-dimensional cases. The best-known approximation
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
for Wilks' lambda was derived by C. R. Rao
C. R. Rao
Calyampudi Radhakrishna Rao FRS known as C R Rao is an Indian statistician. He is currently professor emeritus at Penn State University and Research Professor at the University at Buffalo. Rao has been honored by numerous colloquia, honorary degrees, and festschrifts and was awarded the US...
.
In the case of two groups, all the statistics are equivalent and the test reduces to Hotelling's T-square.