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Single-Factor Between-Subjects Analysis of Covariance

Single-factor between-subjects analysis of covariance (ANCOVA), also known as one-way ANCOVA, is an extension of single-factor between-subjects ANOVA. If some of the variations in the dependent variable scores are caused by the effect of another continuous variable (covariate), use of ANCOVA will remove this variation from the error or random variance, resulting in increased sensitivity of the test for treatment effects.

One-Way Between-Subjects ANCOVA Output

Multiple Comparisons/Post-Hoc Tests

Supported Worksheet Layout

One-Way Between-Subjects ANCOVA Output

The default output includes the one-way ANCOVA table. The additional output depends on selected options, and can include the means table, the test results for homogeneity of variance, the test results for homogeneity of regression, and table report for the chosen multiple comparison test.

The ANCOVA Table

This table includes:

(a) Output from analysis of variance for covariate
(b) Output from analysis of variance for dependent variable (Y)
(c) Output from analysis of covariance
(d) The correlation coefficients: total (r_T), between-groups (r_BG), and within-groups (r_WG)

In the analysis of covariance, the sum of the squared deviations from the mean (Sum of Squares), the degrees of freedom (df), and Mean Square computed for each of the variability components, are all adjusted values.

Testing for Homogeneity of Regression

The homogeneity of regression assumption is met if within each of the k groups there is a linear correlation between the dependent variable and the covariate, and that the k group regression lines have the same slope. The aim of the homogeneity of regression test is to define whether or not the homogeneity of regression is plausible.

The images below shows the test results for the homogeneity of regression and the corresponding regression plot for the ANCOVA report shown above.

Multiple Comparisons/Post-Hoc Tests

The multiple comparisons/post-hoc tests accompanying the analysis of covariance are conducted on The Adjusted Group/Treatment Means, and include:

Tukey's HSD test
Scheffé test

The example below is the Tukey's HSD test on adjusted means one-way between-subjects ANCOVA example illustrated above. For more information regarding multiple comparisons accompanying ANCOVA, click here.

Supported Worksheet Layout

(i) A categorical columns (preferentially textual) for storing levels of the design between-subjects single factor, (ii) two numeric columns, one for storing the covariate scores for all levels of the single factor(X), and the other one for the corresponding dependent variable (Y).
Each worksheet row corresponds to a given subject/object.
The covariate, dependent, and factor variables can be stored in any order or column positions.