A Non-degenerate Estimator for Variance Parameters in Multilevel Models via Penalized Likelihood Estimation

In this paper we consider the problem of obtaining zero estimates for group-level variance parameters in multilevel or hierarchical linear models, a problem that occurs frequently when the number of groups or the value of the variance 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 default values for the shape and rate parameters so that the maximum penalized likelihood estimator is approximately one standard error from zero when the maximum likelihood estimate is zero, and thus remains consistent with the likelihood while being non-degenerate. We also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median with a non-informative prior. Our method with the default penalty 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 non-degenerate estimates and in the limit coincides with maximum likelihood as the number of groups increases.