= Analysis of covariance
{title2=$E(Y_i)=\alpha+\beta x_i+\tau_{G_i}$}
{wiki}
Analysis of covariance combines a categorical treatment and a quantitative covariate in a <normal linear model>. A common-slope model is $Y_i=\alpha+\beta x_i+\sum_{g=2}^G\tau_g\mathbf1_{\{G_i=g\}}+\varepsilon_i$, 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.
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