Title: Deconstructing Type III,
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.