Estimable functions and testable hypotheses in orthogonal and partially orthogonal designs with three factors

Abstract

In experimental situations, not rarely, the scientist faces the impossibility of planning balanced experiments. A serious problem arises immediately concerning the interpretation of the hypothesis tested with the statistical systems, mainly when a high unbalancing degree is present. Taking that into account, the objective of this work is focused on the study of the estimable functions and of the testable hypotheses in orthogonal and partially orthogonal designs with three factors. For the main effects, in which the generated subspace is individually orthogonal to the inherent subspaces to the other factors, are estimable and, therefore, the corresponding hypothesis is testable in models with no interactions. When interactions are present, the estimable functions present besides the parameters of the factor itself, parameters of the interactions in which the factor is present. In these cases, whether the model contains or not interactions, the hypotheses on weighted averages (type I) are equivalent to the hypothesis on proportional averages (type II) and, since the complete term is imperative for both full and partial orthogonalities, the equivalence between the hypotheses on non-proportional averages (types III and IV) also occurs. The equity between the hypotheses of the types I and II occurs in all interactions, in the orthogonal designs and, in the partially orthogonal designs, it also occurs in the interactions formed by the factors that were not orthogonal between themselves. In these interactions the estimable functions present parameters of the interaction itself, including parameters of the interactions of degree three.