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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Below is the unformatted table of contents.

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 Spread-versus-level plots 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 Adequate sample size 43 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 Frequently Asked Questions 50 Why can't I just use multiple univariate ANOVA tests rather than MANOVA? 50 How do I write up the results of my MANOVA analysis? 51 How many dependents can I have in MANOVA? 52 Explain the syntax for MANOVA in SPSS 52 What is analysis of residuals for in MANOVA 53 Is there a limit on the number of covariates which can be included in an multiple analysis of variance? 53 What is step-down MANOVA? 54 What is the "protected F" or least significant difference (LSD) test in MANOVA? How does it relate to the use of discriminant analysis in MANCOVA? 55 What is the multivariate GLM syntax in SPSS? 55 What is the MANOVA syntax in SPSS? 56 Bibliography 59