GLM MULTIVARIATE, MANOVA, MANCOVA

Multivariate GLM is the version of the general linear model now often used to implement two long-established statistical procedures - MANOVA and MANCOVA. Multivariate GLM, MANOVA, and MANCOVA all deal with the situation where there is more than one dependent variable and one or more independents. MANCOVA also supports use of continuous control variables as covariates.

Multiple analysis of variance (MANOVA) is used to see the main and interaction effects of categorical variables on multiple dependent interval variables. MANOVA uses one or more categorical independents as predictors, like ANOVA, but unlike ANOVA, there is more than one dependent variable. Where ANOVA tests the differences in means of the interval dependent for various categories of the independent(s), MANOVA tests the differences in the centroid (vector) of means of the multiple interval dependents, for various categories of the independent(s). One may also perform planned comparison or post hoc comparisons to see which values of a factor contribute most to the explanation of the dependents.

There are multiple potential purposes for MANOVA.

To compare groups formed by categorical independent variables on group differences in a set of interval dependent variables.

To use lack of difference for a set of dependent variables as a criterion for reducing a set of independent variables to a smaller, more easily modeled number of variables.

To identify the independent variables which differentiate a set of dependent variables the most.

Multiple analysis of covariance (MANCOVA) is similar to MANOVA, but interval independents may be added as "covariates." These covariates serve as control variables for the independent factors, serving to reduce the error term in the model. Like other control procedures, MANCOVA can be seen as a form of "what if" analysis, asking what would happen if all cases scored equally on the covariates, so that the effect of the factors over and beyond the covariates can be isolated. The discussion of concepts in the separate Statistical Associates volume on GLM ANOVA also applies, including the discussion of assumptions.

See also separate “blue book” volumes from Statistical Associates on Univariate GLM, ANOVA, and ANCOVA ; Repeated measures GLM; and Discriminant Function Analysis, which yields results equivalent to one-way MANOVA.

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```MULTIVARIATE GLM, MANOVA, AND MANCOVA	1
Overview	6
Key Concepts	7
General Linear Model (GLM)	7
GLM vs. MANOVA procedures	8
SPSS Example	8
SPSS syntax	8
Variables	9
Models	10
Multiple and multivariate regression models	10
Contrasts	11
Plots	12
Post hoc tests	12
Save	13
Options	14
Statistical output in SPSS	15
Significance	15
SAS Example	17
SAS syntax	17
SAS Output	18
F-tests of individual effects for each dependent variable	19
Multivariate tests	19
Multivariate tests of model effects	20
Parameter estimates	22
Factors	24
Covariates	25
Interaction effects	25
Multiple comparison and post-hoc tests	26
Contrast Tests	28
Contrast Results table	28
Effect size measures	30
Partial eta-squared	30
R-Squared	31
Profile Analyis	31
Estimated marginal means	31
Pairwise comparison tables	32
Profile plots	33
Equality of means tests	36
Canonical Correlation	37
Overview	37
SPSS example	37
Canonical roots or linear discriminant functions, LDF	37
SPSS output	38
Eigenvalues	39
Canonical correlation	39
Diagnostics	40
Lack of fit test	40
Residual analysis with observed*predicted*standardized residual plots	41
Assumptions	42
Observations are independent of one another	42
Measurement level	42
Low measurement error of the covariates	42
Similar group sizes	42
Appropriate sums of squares	43
Random residuals	44
Homogeneity of variances	44
Box's M	44
Levene's Test	45
Homogeneity of regressions	45
Linearity	46
Sphericity	46
Bartlett's test of sphericity	47
Mauchly's test of sphericity	48
Corrections for violation of sphericity	48
Multivariate normal distribution	48
No outliers	49
Covariates are linearly related or in a known relationship to the dependents	49