Generalizability Theory With One-Facet Nonadditive Models

Abstract

In generalizability theory (G theory), one-facet models are specified to be additive, which is equivalent to the assumption that subject-by-facet interaction effects are absent. In this article, the authors first derive estimators of variance components (VCs) for nonadditive models and show that, in some cases, they are different from their counterparts in additive models. The authors then demonstrate and later confirm with a simulation study that when the subject-by-facet interaction exists, but the additive-model formulas are used, the VC of subjects is underestimated. Consequently, generalizability coefficients are also underestimated. Thus, depending on the nature of interaction effects, an appropriate model, either additive or nonadditive, should be used in applications of G theory. The nonadditive G theory developed in this article generalizes current G theory and uses data at hand to determine when additive or nonadditive models should be used to estimate VCs. Finally, the implications of the findings are discussed in light of an analysis of real data.