  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.
