  Type III methods, introduced by SAS in 1976, formulate estimable functions
that substitute, somehow, for classical ANOVA effects in multiple linear
regression models. They have been controversial since, provoking wide use and
satisfied users on the one hand and skepticism and scorn on the other. Their
essential mathematical properties have not been established, although they are
widely thought to be known: what those functions are, to what extent they
coincide with classical ANOVA effects, and how they are affected by cell sample
sizes, empty cells, and covariates. Those properties are established here.
