Psychometrika

, Volume 78, Issue 4, pp 685–709 | Cite as

A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models

  • Yeojin Chung
  • Sophia Rabe-Hesketh
  • Vincent Dorie
  • Andrew Gelman
  • Jingchen Liu
Article

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.

Key words

Bayes modal estimation hierarchical linear model mixed model multilevel model penalized likelihood variance estimation weakly informative prior 

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Copyright information

© The Psychometric Society 2013

Authors and Affiliations

  • Yeojin Chung
    • 1
  • Sophia Rabe-Hesketh
    • 2
    • 3
  • Vincent Dorie
    • 4
  • Andrew Gelman
    • 4
  • Jingchen Liu
    • 4
  1. 1.School of Business AdministrationKookmin UniversitySeoulSouth Korea
  2. 2.Graduate School of EducationUniversity of California, BerkeleyBerkeleyUSA
  3. 3.Institute of EducationUniversity of LondonLondonUK
  4. 4.Department of StatisticsColumbia UniversityNew YorkUSA

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