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No Need to Turn Bayesian in Multilevel Analysis with Few Clusters: How Frequentist Methods Provide Unbiased Estimates and Accurate Inference

Comparative political science has long worried about the performance of multilevel models when the number of upper-level units is small. Exacerbating these concerns, an inuential Monte Carlo study by Stegmueller (2013) suggests that frequentist methods yield biased estimates and severely anti-conservative inference with small upper-level samples. Stegmueller recommends Bayesian techniques, which he claims to be superior in terms of both bias and inferential accuracy. In this paper, we reassess and refute these results. First, we formally prove that frequentist maximum likelihood estimators of coe cients are unbiased. The apparent bias found by Stegmueller is simply a manifestation of Monte Carlo Error. 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 without turning to Bayesian methods, even if the number of upper-level units is small. Acknowledgements: Earlier versions of this paper were presented at the annual meetings of the European Political Science Association, of the Midwest Political Science Association, the German Statistical Society (DStatG), and the meeting of the Political Methodology Specialist Group of the Political Studies Association. We are grateful to Michael Peress, Sir David Hendry and the members of the audience of the panels at the various conferences and seminars for their constructive comments and valuable feedbacks. The authors are listed in alphatical order; they all contributed equally to the paper.