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

Multifactor Cell Means

The Design Factors

Factorial Parallel Coordinates

Between-Subjects ANOVA Table: A Three-Way Example

Between-Subjects ANOVA (General Linear Mdel): Test Results

Measures of Effect Size and Strength of Association

Between-Subjects ANOVA: Effect Size

Homoscedasticity Tests on Regression Residuals

Bartlett Test

Levene Test

Normality Tests on Regression Residuals

D'Agostino-Pearson and Shapiro-Wilk Normality Tests

The Regression Analysis Report Corresponding to ANOVA Test Results

Between-Subjects ANOVA (General Linear Model): Regression Coefficients and Model Summary

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

Pairwise Multiple Comparisons (PMC) at Fixed Levels of One Other Factor

PMC Within Levels of Factor B at fixed levels of A and C

Pairwise Multiple Comparisons (PMC) at Fixed Levels of One Other Factor

PMC Within Levels of Factor C at fixed levels of A and B

Pairwise Multiple Comparisons (PMC) at Fixed Levels of One Other Factor

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

Pairwise Multiple Comparisons (PMC) at Fixed Levels of Two Other Factors

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

Pairwise Multiple Comparisons (PMC) at Fixed Levels of Two Other Factors

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

Pairwise Multiple Comparisons (PMC) at Fixed Levels of Two Other Factors