Journal
PSYCHOMETRIKA
Volume 78, Issue 4, Pages 685-709Publisher
SPRINGER
DOI: 10.1007/s11336-013-9328-2
Keywords
Bayes modal estimation; hierarchical linear model; mixed model; multilevel model; penalized likelihood; variance estimation; weakly informative prior
Categories
Funding
- Institute of Education Sciences [R305D100017]
- National Science Foundation [SES-1023189]
- Department of Energy [DE-SC0002099]
- National Security Agency [H98230-10-1-0184]
- Divn Of Social and Economic Sciences
- Direct For Social, Behav & Economic Scie [1023189, 1023176] Funding Source: National Science Foundation
Ask authors/readers for more resources
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,lambda) penalty with lambda -> 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.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available