Comparisons of parameter and hypothesis definitions in a general linear model

In the context of the general linear model E(Y)=Xβ possibly subject to restrictions Rβ=r two secondary parameters may be well defined by Θ_i=G_iE(Y)-Θ_oi=C_i β-Θ_oi,i=1,2, and corresponding nonconstant hypotheses, H_i:Θ_i=0. The following possible relations are defined: Θ₁: is dependent upon /equivalent to/identical to Θ₂:H₁is a subhypothesis of/is identical to H₂. Necessary and sufficient conditions, involving straightforward matrix computations, are presented for each relation. Comparisons of secondary parameters and hypotheses are illustrated with an incomplete, unbalanced 3 × 4 factorial design from Searle in which, using a constrained version of Searle's model, parameters and hypotheses in the incomplete, unbalanced design are shown to be indentical to parameters one would define if complete balanced data were available. Techniques for computing simplified definitions are illustrated.