Quality & Quantity 34: 33–50, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Application of Hierarchical Linear Models to
Meta-Analysis: Study of the Monte Carlo
Simulation on the Functioning of Traditional and
Empirical-Bayes Effect Size
MARÍA CASTRO and JOSÉ-LUIS GAVIRIA
Faculty of Education, Complutense University of Madrid, Paseo de Juan XXIII s/n, 28040 Madrid,
Abstract. An empirical study and comparison is made of the functioning of the effect size proposed
from the traditional meta-analytical points of view and the hierarchical linear models. These esti-
mators are compared in bias, efﬁciency and robustness, via a Monte Carlo simulation. The results
show the best behaviour of Empirical Bayes estimator (EB). This estimator shows a slightly biased
behaviour compared with the traditional, although it is obviously more efﬁcient and robust than the
alternative traditionally used. Thus, it is empirically proved that the hierarchical linear models are an
alternative to the traditional methods of meta-analytical synthesis.
Meta-analysis is a systematic procedure for the integration of the results of re-
search. The synthesis of the results obtained is intended in a set of experimental
studies which cover the same research problem in order to check whether the
convergence of the hypothesis and the varied range of investigation designs lead
to the same or similar results for the problem that is the object of study (Glass,
1976; Glass et al., 1981). To sum up, an attempt is made to ﬁnd common solution
patterns in a set of studies that cover the same problem from different points of
view and using different procedures.
From the methodological point of view, meta-analysis implies a landmark in the
procedural evolution of the revisions of the research results. The introduction of a
statistic, the effect size, which allows the results from different primary research to
be put on the same scale, is one of the main contributions.
The effect size, traditionally conceived, implies the standardisation of the differ-
ences produced by different treatments. Therefore, a single effect size is obtained
for each investigation to be included in the meta-analytical study which will
be synthesised using different traditional meta-analytical integration models and