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A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models

Published online by Cambridge University Press:  01 January 2025

Yeojin Chung ,
Sophia Rabe-Hesketh ,
Vincent Dorie ,
Andrew Gelman  and
Jingchen Liu
Yeojin Chung*
Affiliation:
School of Business Administration, Kookmin University
Sophia Rabe-Hesketh
Affiliation:
Graduate School of Education, University of California, Berkeley Institute of Education, University of London
Vincent Dorie
Affiliation:
Department of Statistics, Columbia University
Andrew Gelman
Affiliation:
Department of Statistics, Columbia University
Jingchen Liu
Affiliation:
Department of Statistics, Columbia University
*
Requests for reprints should be sent to Yeojin Chung, School of Business Administration, Kookmin University, Seoul, South Korea. E-mail: [email protected]
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Abstract

Group-level variance estimates of zero often arise when fitting multilevel or hierarchical linear models, especially when the number of groups is small. For situations where zero variances are implausible a priori, we propose a maximum penalized likelihood approach to avoid such boundary estimates. This approach is equivalent to estimating variance parameters by their posterior mode, given a weakly informative prior distribution. By choosing the penalty from the log-gamma family with shape parameter greater than 1, we ensure that the estimated variance will be positive. We suggest a default log-gamma(2,λ) penalty with λ→0, which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero, thus remaining consistent with the data while being nondegenerate. We also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior.

Our default method provides better estimates of model parameters and standard errors than the maximum likelihood or the restricted maximum likelihood estimators. The log-gamma family can also be used to convey substantive prior information. In either case—pure penalization or prior information—our recommended procedure gives nondegenerate estimates and in the limit coincides with maximum likelihood as the number of groups increases.

Keywords

Bayes modal estimationhierarchical linear modelmixed modelmultilevel modelpenalized likelihoodvariance estimationweakly informative prior
Type
Original Paper
Information
Psychometrika , Volume 78 , Issue 4 , October 2013 , pp. 685 - 709
Copyright
Copyright © 2013 The Psychometric Society

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