The Linear Model

Abstract

Suppose that we are to analyze n measurements or observations y _i to see how they depend upon q other sets of measurements or observations F_l … F _q If F _j is considered quantitative, we will refer to it as a variate. If F _j is considered qualitative, we will refer to it as a factor, and use the notation n _j to denote the number of levels of F _j . (In other words, n _j is the number of classes into which F _j divides the n measurements y.) In the balance of this report, we will deal almost exclusively with analysis of variance models, that is, models in which all the F _j are factors. Models in which some of the F _j are variates will be referred to as analysis of covariance models. We will use the phrase factorial design to describe any experiment in which all (or nearly all) of the combinations of the factors F_l … F _q are of interest. Depending on the nature of the factors or the design, a nested model might well be appropriate in such a design.