Statistical Modelling 11 (2011), 389408
A note on a hierarchical interpretation for negative variance components
Geert Molenberghs
I-BioStat, Universiteit Hasselt
Agoralaan 1
B3590 Diepenbeek
and
I-BioStat, Katholieke Universiteit Leuven
Belgium
eMail: geert.molenberghs@uhasselt.be
Geert Verbeke
I-BioStat, Universiteit Hasselt
and
I-BioStat, Katholieke Universiteit Leuven
Belgium
Abstract:
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.
Keywords:
linear mixed model; marginal model; random effect
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