## Analysis Of Covariance

The analysis of covariance (ANCOVA) has goals similar to those of analysis of variance; that is, it uses estimates of variability to test hypotheses about group means. However, ANCOVA differs from standard analysis of variance (ANOVA) because it uses not only information about the dependent variable, Y, but also information about an additional variable, X, called the covariate, which is correlated with the dependent variable. The ANCOVA procedure attempts to control statistically for differences in the covariate that would result in error variability and hence would reduce the efficiency of an ANOVA. The potentially greater efficiency of ANCOVAis obtained at the cost of additional complexity and stronger assumptions that must be made about the data. ANCOVA results are also frequently misunderstood.

Consider an example. Suppose we wish to test the effectiveness of four different software packages designed to develop problem-solving skills in fourth graders. Children are randomly assigned to work with each of the packages, and the dependent variable, Y, is the score on a problem-solving test given after the students have worked with the pack ages for 3 months. We also have available scores, X, on a pretest of problem-solving skills given before the children started working with the packages. Suppose we use ANOVAto test the null hypothesis that the software packages are equally effective. The means of the instructional groups will reflect not only the effects of the software packages, but also other sources of variability, including individual differences in problem-solving ability. The ANOVA can be thought of as a test of whether a model in which there is a treatment effect—that is, a component corresponding to the effect of working with one of the packages,

—accounts for the data better than a restricted model in which there are no treatment effects,

Yj = m + Bj where Yy is the test score of the ith participant in the jth treatment (here, software package) group, m is a common component, oj is the effect of the jth treatment, and eij is the error variability associated with the score. The larger the error variability, the more the treatment effects will be obscured. Because the children were randomly assigned to treatment groups, preexisting individual differences in problem-solving ability will not differ systematically across groups; however, they will contribute to the error variability and, thus, to the between-group variability. If all children had equal problem-solving ability (indicated by equal scores on the covariate) before working with the software packages, we would have a much better chance of assessing how effective the packages were. ANCOVA attempts to remove the component of the dependent variable predictable on the basis of the pretest by adding a regression component to each of the above models. It tests the model

Yj = m + Oj + p( Xj - x) + etj against the restricted model

Yj = m + P( xy - x) + ey where p is the regression coefficient or slope. An increase of power may be achieved because, if the treatment and error components are adjusted by removing the variability accounted for by the regression on X, the test statistic may be much larger. In effect, the ANOVA tries to assess whether there would be a treatment effect if all of the children had equal scores on the covariate. 