Qual Quant (2015) 49:967–976
Speciﬁcation of random effects in multilevel models:
Leonardo Grilli · Carla Rampichini
Published online: 29 July 2014
© Springer Science+Business Media Dordrecht 2014
Abstract The analysis of highly structured data requires models with unobserved compo-
nents (random effects) able to adequately account for the patterns of variances and corre-
lations. The speciﬁcation of the unobserved components is a key and challenging task. In
this paper, we ﬁrst review the literature about the consequences of misspecifying the dis-
tribution of the random effects and the related diagnostic tools; we then outline the main
alternatives and generalizations, also considering some issues arising in Bayesian inference.
The relevance of suitably structuring the unobserved components is illustrated by means of
an application exploiting a model with heteroscedastic random effects.
Keywords Diagnostic tools · Finite mixture · Heteroscedasticity · Misspeciﬁcation ·
Mixed model · Non-parametric maximum likelihood · Prior distribution
Random effects models are a key tool for the analysis of multilevel data in a wide range
of ﬁelds. These models are also known as mixed (Demidenko 2013) or multilevel models
(Raudenbush and Bryk 2002; Goldstein 2011; Snijders and Bosker 2012). Multilevel data
can also be analysed with methods avoiding random effects, such as ﬁxed effects models,
marginal models via generalized estimating equations (GEE), and cluster-robust standard
errors derived from sandwich estimators or bootstrap (Scott et al. 2013). However, in this
review we explicitly focus on the random effects approach.
This research was supported by the FIRB 2012 project “Mixture and latent variable models for causal
inference and analysis of socio-economic data”, Grant No. RBFR12SHVV_003.
L. Grilli · C. Rampichini (
Dipartimento di Statistica, Informatica, Applicazioni ‘G. Parenti’ - University of Florence, Florence, Italy