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2010 | OriginalPaper | Chapter
5. Adding additional covariates and the Analysis of Covariance
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Abstract
Suppose that having fitted the regression model we wish to introduce qadditional explanatory variables into our model. The augmented regression model, M
A
, say becomes We rewrite this as say, where Here Xis n×pand assumed to be of rank p, Zis n×qof rank q, and the columns of Zare linearly independent of the columns of X. This final assumption means that there is a sense in which the qadditional explanatory variables are adding genuinely new information to that already contained in the pre-existing Xmatrix.
$$\vec{y} = X\beta + \epsilon, $$
(M0)
$$\vec{y} = X\beta + Z\gamma + \epsilon. $$
(MA)
$$\begin{array}{rcl} \vec{y} = X\beta + Z\gamma + \epsilon & =& (X,Z){\left (\beta, \gamma \right )}^{T} + \epsilon, \\ & =& W\delta + \epsilon, \\ \end{array}$$
$$\begin{array}{rcl} W := (X,Z),\quad \delta := \left (\begin{array}{c} \beta \\ \gamma \end{array} \right )\!.& & \\ \end{array}$$