8.2 - The Covariate as a Regression Variable

ANCOVA by definition is a general linear model that includes both ANOVA (categorical) predictors and Regression (continuous) predictors. The simple linear regression model is:

\(Y_i=\beta_0+\beta_1 (X_i)+ \epsilon_i\)

Where \(\beta_0\) is the intercept and \(\beta_1\) is the slope of the line. The significance of a regression is tested by calculating a sums of squares due to the regression variable SS(Regr), calculating a mean squares for regression, MS(Regr), and using an F-test with F = MS(Regr) / MSE. In the case of a simple linear regression, this test is equivalent to the t-test for \(H_0 \colon \beta_1=0\).

However, In adding the regression variable to our one-way ANOVA model, we can envision a notational problem. In the balanced one-way ANOVA we have the grand mean (μ), but now we also have the intercept \(\beta_0\). To get around this, we can use

\(X^*=X_{ij}-\bar{X}_{..}\)

and get the following as an expression of our covariance model:

\(Y_{ij}=\mu+\tau_i+\gamma(X^* )+\epsilon_{ij}\)

The Type III (model fit) sums of squares for the treatment levels in this model are being corrected (or adjusted) for the regression relationship. This has the effect of evaluating the treatment levels ‘on the same playing field’, that is, comparing the means of the treatment levels at the mean value of the covariate. This process effectively removes variation that was originally seen in the treatment level means due to the covariate.