• Introduction
• Journal Articles
• Reference Works
• History
• Assumptions and Interpretation Issues
• Alternatives Accommodating Certain Assumption Violations and Design Flaws

## Introduction

The analysis of covariance (ANCOVA) is a method for testing the hypothesis of the equality of two or more population means, ideally in the context of a designed experiment. It is similar in purpose to the analysis of variance (ANOVA), but it differs in that an adjustment is made to both the dependent variable means and the error term to provide both descriptive and inferential advantages. The adjustments are made on the basis of information on one or more variables (called covariates) that are measured on each participant before treatments are applied. The advantages of incorporating the covariate information are typically (1) more meaningful outcome means and (2) a smaller error term than is associated with ANOVA. These adjustments result in more interpretable effects, narrower confidence intervals, and an increase in the statistical power of the analysis. Suppose an experiment is carried out to evaluate effects of two treatments. The randomly assigned treatment groups differ somewhat in average age, and age is correlated with the achievement measure used as the dependent variable. Differences between groups on achievement will be somewhat ambiguous to interpret because the groups differ in terms of both age and treatment condition. The analysis of covariance will provide “adjusted means” that estimate the value the outcome means would have been if the groups had been exactly the same with respect to age. At the same time, within-group variation in achievement scores predictable from the covariate (age) will be removed from the error variation to increase the precision of the test for differences between the adjusted means. The application of ANCOVA in some observational studies (rather than randomized experiments) is controversial and has led to a large literature that explores the concerns surrounding the adequacy of the analysis when used in this context. The label “analysis of covariance” is now viewed as anachronistic by some research methodologists and statisticians because this analysis can be both conceptualized and computed as a variant of the general linear model (GLM). But the term remains useful because it immediately conveys to most researchers the notion that a categorical variable (the treatment conditions) and two continuous variables (the covariate and the dependent variable) are involved in a single analysis. Researchers should be warned, however, that ANCOVA is not the same as the “analysis of covariance structures,” a term that was frequently used in the 1970s and 1980s to refer to what is currently known as a “structural equation model.” Additionally, some sources of information regarding ANCOVA subsume several analyses related to (but different from) ANCOVA under this general heading. Examples of these related analyses include the test of the significance of the covariate, the test for homogeneous regression slopes, and the Johnson-Neyman technique.

## General Overviews

The following three subsections list sources containing general overviews and introductions to analysis of covariance (ANCOVA). This list begins with the most elementary sources, progresses through those that are of intermediate length and sophistication, and ends with advanced treatments in the form of journal articles and comprehensive reference works. Short elementary presentations designed for readers interested in only the general ideas on ANCOVA are found in encyclopedia articles written for beginning researchers. Several intermediate and advanced level general statistics texts also provide solid introductions to ANCOVA. More extensive coverage is presented in full chapters on the topic found in several textbooks on experimental design. These textbooks provide the main exposure to ANCOVA for most researchers in the behavioral sciences. More technical presentations are available in articles published in methodology and statistics journals.