### 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