Analysis of covariance ($E(Y_i)=\alpha +\beta x_i+{\tau }_{G_i}$)

Analysis of covariance combines a categorical treatment and a quantitative covariate in a normal linear model. A common-slope model is $Y_i=\alpha +\beta x_i+{\sum }_{g=2}^G{\tau }_g{1}_{\{G_i=g\}}+{\epsilon }_i$, with independent $N(0,{\sigma }^2)$ errors. Treatment differences compare fitted means at the same covariate value. Adding treatment-by-covariate interactions permits separate slopes, and a partial F-test compares this extension with the parallel-lines model. The treatment test after adding the covariate differs from an unadjusted comparison, because treatment groups can have different covariate distributions.