ANOVA and ANCOVA (GLM)

Between-Subjects ANOVA: General Linear Model

• Between-subjects evaluates (a) whether at least two of the levels of each factor represent populations with different mean values; and (b) whether there is a significant interaction between the n factors (i.e., it evaluates the variation among the differences between means for different levels of each factor over different levels of the other factor).
• The data can be balanced or unbalanced (unequal group sizes).
• The method allows using:
• Fixed-effects model: Any number of factors can be used with fixed-effects model (1, 2, 3, ..., n-way)
• Random-effects model: Not more than two factors can be used with random-effects model.
• Mixed-effects model: Can use two factors, one to be treated as random and one as fixed.
The Analysis Output

The analysis output including the default and optional ones can include:

• Multifactor cell means table
• Between-subjects ANOVA table
• Measures of association (variance-accounted for statistics)
• Effect size measures in standardized units of mean difference
• Regression coefficients and model summary output
• Normality test on regression residuals of total model:
• D'Agostino-Pearson test
• Shapiro-Wilk test
• Homoscedasticity tests on regression residuals of total model (Bartlett, Levene, and Brown-Forsythe)
• Pairwise multiple comparisons (PMC) accompanying the between-subjects ANOVA module:  Bonferroni-Dunn Dunn-Sidak Dunnett Scheffé Tukey-Kramer Fisher's LSD Tukey's HSD Newman-Keul Tukey B Dunnett's C Games-Howell

The tables on this page are examples from a three-way design whose factors and cell means are shown below:

The Design Cell Means The Design Factors Between-Subjects ANOVA Table: A Three-Way Example Measures of Effect Size and Strength of Association Homoscedasticity Tests on Regression Residuals  Normality Tests on Regression Residuals The Regression Analysis Report Corresponding to ANOVA Test Results Pairwise Multiple Comparisons (PMC) Output Results

Depending on the selected test(s) and analysis options, the PMC output can include:

• PMC pooled over all factors
• PMC at fixed levels of one other factor (this option requires running two-way or higher dimensions ANOVA)
• PMC at fixed levels of two other factors (this option requires running three-way or higher dimensions ANOVA) (see the example below).
PMC Pooled Over All Factors
• This option provides a PMC table for each factor with k>=3 levels. In the example shown here, the ANOVA test results indicate:
• The main effects of Factor A and Factor B (each havng k=2 levels) are significant, and the main effect of Factor C is insignificant. However, as shown below, the PMC at fixed levels of two other factors reveal differences that were otherwise masked.
PMC at Fixed Levels of One Other Factor
• In the current example, this option provides three PMC tables as follows:

PMC Within Levels of Factor A at Fixed Levels of B and C PMC Within Levels of Factor B at fixed levels of A and C PMC Within Levels of Factor C at fixed levels of A and B PMC at Fixed Levels of Two Other Factors
• In the current example, this option provides Two PMC tables as follows:

PMC Within Levels of Factor A at Combined Fixed Levels of B and C PMC Within Levels of Factor B at Combined Fixed Levels of A and C PMC Within Levels of Factor C at Combined Fixed Levels of A and B 