A nondegenerate penalized likelihood estimator for variance parameters in multilevel models

Area of Science:

• Statistics
• Multilevel Modeling
• Hierarchical Linear Models

Background:

• Zero variance estimates are common in multilevel and hierarchical linear models, particularly with small group numbers.
• Such boundary estimates are often implausible when variances are theoretically expected to be non-zero.

Purpose of the Study:

• To introduce a maximum penalized likelihood approach to prevent implausible zero variance estimates.
• To provide a statistically sound method for obtaining positive variance estimates in multilevel models.

Main Methods:

• Utilizing a maximum penalized likelihood approach with a log-gamma penalty.
• Implementing a default log-gamma(2,λ) penalty with λ → 0 for a weakly informative prior.
• Equating the penalized likelihood to the posterior mode with a weakly informative prior.

Main Results:

• The proposed method ensures positive variance estimates, avoiding boundary solutions.
• The default penalty approximates the posterior median under a noninformative prior.
• Penalized likelihood estimates offer improved model parameter and standard error estimates over maximum likelihood and restricted maximum likelihood.

Conclusions:

• The maximum penalized likelihood approach provides a robust solution for estimating variances in multilevel models.
• This method yields nondegenerate estimates that align with data while remaining statistically sound.
• The approach is flexible, allowing for the incorporation of substantive prior information if desired.