Exact methods in the unbalanced, two-way analysis of variance - a geometric view

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Abstract
In the days when analyses of variance were typically performed on a desk calculator, approximate methods such as unweighted means were generally preferred to exact least squares methods when dealing with disproportionate cell frequencies. Difficulties in computation and interpretation were the chief reasons why exact methods were not favored. Today, the widespread availability of computers has largely removed the computational difficulties associated with the exact methods, but the difficulties of; interpretation remain. it is only through a better understanding of the complexities introduced by disproportionate cell frequencies that the diificulties of interpretation can be resolved. Th:s paper will attempt to achieve a better understanding of the various exact methods by studying the geometry of the disproportionate cell size case. A geometric definition of the general linear model is given and applied to the two-way analysis of variance. Geometric and parametric specifications of linear hypotheses are related. Previously proposed exact rnethods are interpreted geometrically.

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