When there are multiple dependent variables in a design, the design is said to be multivariate. Multivariate measures of association are by nature more complex than their univariate counterparts (such as the correlation coefficient, for example). This is because multivariate measures of association must take into account not only the relationships of the predictor variables with responses on the dependent variables, but also the relationships among the multiple dependent variables. By doing so, however, these measures of association provide information about the strength of the relationships between predictor and dependent variables independent of the dependent variable interrelationships. A basic discussion of multivariate designs is also presented in the Multivariate Designs topic in the Introductory Overview of the ANOVA/MANOVA module.

The most commonly used multivariate measures of association all can be expressed as functions of the eigenvalues of the product matrix E-1H where E is the error SSCP matrix (i.e., the matrix of sums of squares and cross-products for the dependent variables that are not accounted for by the predictors in the between design), and H is a hypothesis SSCP matrix (i.e., the matrix of sums of squares and cross-products for the dependent variables that are accounted for by all the predictors in the between design, or the sums of squares and cross-products for the dependent variables that are accounted for by a particular effect). If

li = the ordered

then the 4 commonly used multivariate measures of association are

Wilks' Lambda = Õ[1/(1 + li)

Hotelling - Lawley trace = S li

Roy's largest root = li

These 4 measures have different upper and lower
bounds, with Wilks' Lambda perhaps
being the most easily interpretable of the 4 measures. Wilks' Lambda
can range from 0 to 1, with 1 indicating no relationship of predictors
to responses and 0 indicating a perfect relationship of predictors to
responses. 1 - Wilks' Lambda
can be interpreted as the multivariate counterpart of a univariate

The 4 measures of association are also used
to construct multivariate tests of significance. These multivariate tests
are covered in detail in a number of sources (e.g., Finn, 1974;

Between-subject designs

Analysis of Covariance (ANCOVA)

Within-subject (repeated measures) designs

One-Way Within-Subject Designs

Multi-way Within-Subject Designs

Doubly Multivariate Within-Subject Designs

Multivariate designs

See also GLM - Index.