Yates's MWSM SS in the General Linear Model

TL;DR

This paper extends Yates's sum of squares method from two-factor models to general linear models, establishing its properties and demonstrating its equivalence to the restricted model - full model difference in error sum of squares.

Contribution

It introduces a generalized approach to Yates's sum of squares for linear hypotheses in general linear models, clarifying its properties and optimality.

Findings

Yates's sum of squares is equivalent to the restricted minus full model error sum of squares.

The method provides a unique sum of squares that tests the specified hypothesis.

Extension of Yates's approach to broader linear model settings.

Abstract

In 1934, F. Yates described a sum of squares for testing factor main effects in saturated unbalanced models for effects of two factors. He claimed no particular properties of this sum of squares other than that it provided an "efficient estimate of the variance from the A means of the sub-class means... ." Although it became widely regarded as the gold standard in the two-factor model, its fundamental properties and relations to other sums of squares for the same model were not established until decades later. Its method has not been extended to more general settings. This paper shows how Yates's approach can be extended to construct numerator sums of squares for test statistics for linear hypotheses in general linear models. It is shown that Yates's sum of squares is equivalent to the restricted model - full model difference in error sum of squares, which in turn is shown to be the unique sum of squares that tests exactly the hypothesis in question.