Analysis of Covariance (ANCOVA)

Behavioral sciences rely heavily on experiments and quasi experiments for evaluating the effects of, for example, new therapies, instructional methods, or stimulus properties. An experiment includes at least two different treatments (conditions), and human participants are randomly assigned one treatment. If assignment is not based on randomization, the design is called a quasi experiment. The dependent variable or outcome of an experiment [Page 21]or a quasi experiment, denoted by Y here, is usually quantitative, such as the total score on a clinical questionnaire or the mean response time on a perceptual task. Treatments are evaluated by comparing them with respect to the mean of the outcome Y using either analysis of variance (ANOVA) or analysis of covariance (ANCOVA). Multiple linear regression may also be used, and categorical outcomes require other methods, such as logistic regression. This entry explains the purposes of, and assumptions behind, ANCOVA for the classical two-group between-subjects design. ANCOVA for within-subject and split-plot designs is discussed briefly at the end.

Researchers often want to control or adjust statistically for some independent variable that is not experimentally controlled, such as gender, age, or a pretest value of Y A categorical variable such as gender can be included in ANOVA as an additional factor, turning a one-way ANOVA into a two-way ANOVA. A quantitative variable such as age or a pretest recording can be included as a covariate, turning ANOVA into ANCOVA. ANCOVA is the bridge from ANOVA to multiple regression. There are two reasons for including a covariate in the analysis if it is predictive of the outcome Y In randomized experiments, it reduces unexplained (within-group) outcome variance, thereby increasing the power of the treatment effect test and reducing the width of its confidence interval. In quasi experiments, it adjusts for a group difference with respect to that covariate, thereby adjusting the between-group difference on Y for confounding.

Model

The ANCOVA model for comparing two groups at posttest Y, using a covariate X, is as follows:

where Yij is the outcome for person i in group j (e.g., j = 1 for control, j = 2 for treated), and Xij is the covariate value for person i in group j, μ is the grand mean of Y αj is the effect of treatment j, β is the slope of the regression line for predicting Y from X within groups,

is the overall sample mean of covariate X, and eij is a normally distributed residual or error term with a mean of zero and a variance σ2e, which is the same in both groups. By definition, α1 + α2 = 0, and so α2 – α1 = 2α2 is the expected posttest group difference adjusted for the covariate X. This is even better seen by rewriting Equation 1 as

showing that ANCOVA is ANOVA of Y adjusted for X. Due to the centering of X, that is, the subtraction of

, the adjustment is on the average zero in the total sample. So the centering affects individual outcome values and group means, but not the total or grand mean μ of

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