A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models
A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models
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DOI：
10.1007/s11336-013-9328-2
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发表时间：
2013-10-01
期刊：
影响因子：
3
通讯作者：
Liu, Jingchen
中科院分区：
文献类型：
--
作者：
Chung, Yeojin;Rabe-Hesketh, Sophia;Liu, Jingchen
在拟合多水平或分层线性模型时，经常会出现组水平方差估计为零的情况，特别是当组数较小时。对于零方差先验不可信的情况，我们提出了一种最大惩罚似然方法来避免这种边界估计。这种方法等价于在弱信息先验分布下，通过后验模式估计方差参数。通过从形状参数大于1的log-gamma族中选择惩罚，我们确保估计的方差为正。我们建议一个默认的log-gamma（2，lambda）惩罚，其中lambda -> 0，这确保了当最大似然估计为零时，最大惩罚似然估计近似为零的一个标准误差，从而在非退化的同时与数据保持一致。我们还证明了在无信息先验下，最大惩罚似然估计是后验中值的一个很好的近似，我们的默认方法比最大似然估计和限制最大似然估计提供了更好的模型参数和标准误差的估计.对数伽马族也可用于传达实质性的先验信息。在这两种情况下，纯惩罚或先验信息，我们推荐的程序给出非退化估计，并在限制符合最大似然组的数量增加。
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.