### ANOVA and ANCOVA (GLM)

#### Between-Subjects ANCOVA: General Linear Model

• ANCOVA is an extension of ANOVA. If some of the variations in the dependent variable scores are caused by the effect of covariates, which are quantitative variables that are related to the dependent variable, using ANCOVA may remove this variation from the error or random variance (i.e., the treatment/group means are adjusted for the effects of the covariate(s)), resulting in increased sensitivity of the test for treatment effects.
• The null and alternative hypotheses tested using the between-subjects analysis of covariance are the same as those tested in the between-subjects analysis of variance. In addition, the use of ANCOVA requires that the homogeneity of regression is plausible (i.e., the regression coefficients are equal across treatments (homogeneity of regression).
##### The Analysis Output

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

• Multifactor cell means tables, including:
• A table representing means on the dependent variable
• A table representing means on each of the covariates
• Testing the homogeneity of regression, which includes:
• ANCOVA model that includes CxF terms*
• R2 comparison from a model that omits CxF terms* and a model that includes the terms
• ANCOVA test results omitting the product terms of covariate(s) and factors
• Measures of effect size/association
• Regression coefficients and model summary output
• Normality test on regression residuals of total model:
• D'Agostino-Pearson test
• Shapiro-Wilk test
• Homoscedasticity test on regression residuals of total model:
• Bartlett test
• Levene test
• Brown-Forsythe test
• Pairwise multiple comparisons (PMC) accompanying the between-subjects ANCOVA module:
• Bonferroni-Dunn
• Dunn-Sidak

* CxF terms: The product terms of covariate and the design factors/treatments.

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

The Design Cell Means

The Design Factors

###### Testing the Homogeneity of Regression
• The use of ANCOVA requires that the homogeneity of regression is plausible.
• The homogeneity of regression assumption in the ANCOVA is met if within each of the k groups/treatments there is a linear correlation between the dependent variable and the covariate, and that the k group regression lines have the same slope.
• Testing whether the design factors interact with covariate(s) is the easiest way to test the homogeneity of regression. For the homogeneity of regression to be regarded as plausible, no significant interaction should be indicated between the covariate(s) and the design factors.
• In Addition, Aabel will provide an omnibus test for evaluating the homogeneity of regression, which tests the statistical significance between the R2 values from two ANCOVA models (one model that includes interactions between covariate(s) and the design factor(s), and one that omits these interaction terms).

Testing Whether the Design Factors Interact With Covariate(s)

Omnibus Test for Evaluating the Homogeneity of Regression

###### ANCOVA Test Results
• With the homogeneity of regression being plausible, we can proceed with the ANCOVA analysis on the basis of the model that does not include the product terms of covariate and the design factors

Measures of Effect Size and Strength of Association

The Regression Analysis Report Corresponding to ANCOVA Test Results

Normality and Homoscedasticity Tests on Regression Residuals of Total Model

Normality Test on Regression Residuals

Homoscedasticity Test on Regression Residuals