:orphan:
:title: Multilevel Analysis with Few Clusters: Improving Likelihood-based Methods

.. bibsource:: ../publications.bib
.. pub-details:: elff.et.al:multilevel.improving

.. abstract::
  
   Quantitative comparative social scientists have long worried about the
   performance of multilevel models when the number of upper-level units is
   small. Adding to these concerns, an influential Monte Carlo study by
   Stegmüller (2013) suggests that standard maximum likelihood methods yield
   biased point estimates and severely anti-conservative inference with few
   upper-level units.  In this paper, we seek to rectify this negative
   assessment. First, we show that maximum likelihood estimators of
   coefficients are unbiased in linear multilevel models. The apparent bias
   in coefficient estimates found by Stegmueller can be attributed to Monte
   Carlo Error and a flaw in the design of his simulation study. Second, we
   show how inferential problems can be overcome by using *restricted*
   maximum likelihood estimators for variance parameters and a
   *t*-distribution with appropriate degrees of freedom for statistical
   inference. Thus, accurate multilevel analysis is possible within the
   framework that most practitioners are familiar with, even if there are
   only a few upper-level units.
   
.. figure:: multilevel-ci-coverage.png
   :figwidth: 700px

   Performance of Likelihood-based Confidence Intervals of Upper-Level Covariate
   Effect in Multilevel Linear and Probit Models

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   - `Replication material </files/multilevel/ReplicationFiles.zip>`__
   - `Example application </files/multilevel/AppliedExample.zip>`__
   - R package for improved inference: `iimm </software/iimm>`__

.. dropdown:: Open Access at Cambridge University Press
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   - `Full article <https://doi.org/10.1017/S0007123419000097>`__
   - `Online appendix <https://static.cambridge.org/content/id/urn:cambridge.org:id:article:S0007123419000097/resource/name/S0007123419000097sup001.pdf>`__
   - `Replication dataverse <https://doi.org/10.7910/DVN/CMMQRK>`__