Changes in the general linear model: A unified approach

AbstractAnalysis of the general linear model with possibly rank-deficient design and dispersion matrices has sometimes generated some confusion and controversy, prompting some researchers to discuss it as quite distinct from the case of full-rank matrices. We show that linear zero functions, i.e., linear functions in observations which have zero expectations for all parameter values, provide an intuitive way of developing all the important results in connection with the general linear model, thus bridging this imaginary gap. We show that the effect of addition or deletion of a set of observations in this model can be clearly understood in statistical terms if viewed through such linear zero functions. The effect of adding or dropping a group of parameters is also explained well in this manner. Several sets of update equations were derived by a host of previous researchers in various special cases of the above set-up. The results derived here bring out the common underlying principles of these formulae and indeed help simplify most of them. These results also provide further insights into recursive residuals, design of experiments, deletion diagnostics and selection of subset models