Deconstructing Type III

TL;DR

This paper provides an explicit mathematical formulation of Type III sums of squares in linear models, proving their fundamental properties and clarifying their relationship with estimable effects in ANOVA.

Contribution

It derives an explicit formulation for Type III sums of squares and proves their key properties, addressing gaps in theoretical understanding.

Findings

Type III effects include all estimable ANOVA effects.

Type III SS tests exactly when the entire effect is estimable.

The results apply to general linear models with factors and covariates.

Abstract

SAS introduced Type III methods to address difficulties in dummy-variable models for effects of multiple factors and covariates. Type III methods are widely used in practice; they are the default method in many statistical computing packages. Type III sums of squares (SSs) are defined by an algorithm, and an explicit mathematical formulation does not seem to exist. For that reason, their properties have not been rigorously proven. Some that are widely believed to be true are not always true. An explicit formulation is derived in this paper. It is used as a basis to prove fundamental properties of Type III estimable functions and SSs. It is shown that, in any given setting, Type III effects include all estimable ANOVA effects, and that if all of an ANOVA effect is estimable then the Type III SS tests it exactly. The setting for these results is general, comprising linear models for the mean vector of a response that include arbitrary sets of effects of factors and covariates.