**ANOVA** = analysis of variance

**MANOVA** = multivariate analysis of variance

The ANOVA/MANOVA module includes a subset of the functionality within the General Linear Models module. The ANOVA/MANOVA module can perform univariate and multivariate analyses of the variance of factorial designs with or without one repeated measures variable. For more complicated linear models with categorical and continuous predictor variables, random effects, and multiple repeated measures factors you need the General Linear Models module.

In the ANOVA/MANOVA module, you can specify all designs in the most straightforward, functional terms of actual variables and levels (not in technical terms, e.g., by specifying matrices of dummy codes), and even less-experienced ANOVA users can analyze very complex designs. Like the General Linear Models module, ANOVA/MANOVA provides three alternative user interfaces for specifying designs: (1) A Design Wizard, that will take you step-by-step through the process of specifying a design, (2) a simple dialog-based user-interface that will allow you to specify designs by selecting variables, codes, levels, and any design options from well-organized dialogs, and (3) a Syntax Editor for specifying designs and design options using keywords and a common design syntax. Computational methods. The program will use, by default, the sigma restricted parameterization for factorial designs, and apply the effective hypothesis approach (see Hocking, 1980) when the design is unbalanced or incomplete. Type I, II, III, and IV hypotheses can also be computed, as can Type V and Type VI hypotheses that will perform tests consistent with the typical analyses of fractional factorial designs in industrial and quality-improvement applications.

*Historical** note:*

*a genius who almost single-handedly created the foundations for modern statistical science*" and "

*the single most important figure in 20th century statistics*".

### Results

The ANOVA/MANOVA module is not limited in any of its computational routines for reporting results, so the full suite of detailed analytic tools available in the General Linear Models module is also available. Results include summary ANOVA tables, univariate and multivariate results for repeated measures factors with more than 2 levels, the Greenhouse-Geisser and Huynh-Feldt adjustments, plots of interactions, detailed descriptive statistics, detailed residual statistics, planned and post-hoc comparisons, testing of custom hypotheses and custom error terms, detailed diagnostic statistics and plots (e.g., histogram of within-cell residuals, homogeneity of variance tests, plots of means versus standard deviations, etc.).

### Overparameterized and sigma-restricted models

In one line of literature, the analysis of multi-factor ANOVA designs is generally discussed as the Sigma-restricted model. The ANOVA parameters are constrained to sum to zero. In this manner, given k levels of a factor, the k-1 parameters (corresponding to the k-1 degrees of freedom) can readily be estimated (e.g., Lindeman, 1974, Snedecor and Cochran, 1989, p. 322). Another tradition discusses ANOVA in the context of the unconstrained and thus over-parameterized general linear model (e.g., Kirk, 1968). The results for mixed random and fixed effect models can be different applying the two approaches.

This module uses, by default, the means model approach. It will construct F-tests for mixed models that are consistent with the sigma restricted model. This is an ANOVA "tradition" most commonly discussed in statistics textbooks in the biological and social sciences.

**Note:** The Variance Components & Mixed Model ANOVA ANCOVA module uses the over-parameterized model.

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