A Formula for Type III Sums of Squares

TL;DR

This paper derives an explicit mathematical formula for Type III sums of squares, clarifying their properties and usage in two-factor ANOVA models, addressing a gap in the theoretical understanding of these widely used methods.

Contribution

It provides the first explicit mathematical formulation of Type III sums of squares, enhancing understanding and application in statistical modeling.

Findings

Explicit formula for Type III SSs derived

Illustration of properties in two-factor ANOVA

Clarification of Type III SSs in practice

Abstract

Type III methods were introduced by SAS to address difficulties in dummy-variable models for effects of multiple factors and covariates. They are widely used in practice; they are the default method in several statistical computing packages. Type III sums of squares (SSs) are defined by a set of instructions; an explicit mathematical formulation does not seem to exist. An explicit formulation is derived in this paper. It is used to illustrate Type III SSs and their properties in the two-factor ANOVA model.