A Reformulation of Linear Models

SUMMARY Dissatisfaction is expressed with aspects of the current exposition of linear models, including the neglect of marginality, unnecessary differences between models for finite and infinite populations, failure to distinguish different kinds of random terms, impositon of unnecessary and inconsistent constraints on parameters, and lack of an adequate notation for negative components of variance. The reformulation, exemplified for crossed and nested classifications of balanced data, and for simple orthogonal designed experiments, is designed to integrate finite and infinite populations, random and fixed effects, excess and deficit of variance, to avoid unnecessary constraints on parameters, and to lead naturally to interesting hypotheses about the model terms.