A note on a hierarchical interpretation for negative variance components

A lot has been said about the relationship between hierarchical models, such as linear mixed-effects models, and the marginal models they imply. Generally, there is a many-to-one map of hierarchical models onto a given marginal model. Additionally, in some cases, no obvious hierarchical model leads to a given marginal model. For example, it is commonly known that the random-intercepts model produces, marginally, a compound-symmetry model with non-negative intraclass correlation, whereas, on the other hand, a compound-symmetry model with negative intraclass correlation is not induced by a conventional random-intercepts model. We show here that it is still possible, and even intuitively appealing, to formulate hierarchical models inducing structure such as negative compound-symmetry correlation. Thus, the aim of this note is to further clarify the relationship between hierarchical and marginal models, enhancing appeal and establishing symmetry of the concepts. Consequences for interpretation and sensitivity analysis are discussed. The ideas are illustrated in three sets of data.