Analysis of Covariance

Form of the ANCOVA Model

The model for ANCOVA, in the case where there is just a single covariate, is:

$$y_{ij}=\mu +{\tau }_j+\beta z_{ij}+{\epsilon }_{ij},$$

[Page 82]where yij is the outcome score for participant i in group j; µ is the overall mean score on the outcome variable in the study; τj is the effect of the experimental factor in group j; zij is the covariate score for participant i in group j; β is the regression coefficient for z (estimated from the sample data); and εij is the residual for participant i in group j. Note that removing βzij would leave the basic ANOVA model. Normally, the term $\left(z_{ij}-\overline{z}\right)$, where $\overline{z}$ is the overall mean covariate score within the study, is used rather than zij so that the constant term in the model is set at the overall mean for the outcome variable, giving this model:

$$y_{ij}=\mu +{\tau }_j+\beta \left(z_{ij}-\overline{z}\right)+{\epsilon }_{ij}.$$

It follows that when ANCOVA is performed, each participant’s score is adjusted in relation to the covariate—it is the “hypothetical” score the individual would have if all participants had the mean value of the covariate, $\overline{z}$. Accordingly, the mean for each group in the experiment is also an adjusted mean. The adjusted mean for group j is:

$${\overline{y}}_j{}^{\prime }={\overline{y}}_j-\beta \left(z_j-\overline{z}\right),$$

where ${\overline{y}}_j{}^{\prime }$ and ${\overline{y}}_j$ are the adjusted and unadjusted means, respectively, for group j; ${\overline{z}}_j$ is the mean of the covariate for group j; and $\overline{z}$ is the overall covariate mean.

Functions of ANCOVA