Steele B M
Department of Mathematical Sciences, University of Montana, Missoula 59812-1032, USA.
Biometrics. 1996 Dec;52(4):1295-310.
Application of the EM algorithm for estimation in the generalized mixed model has been largely unsuccessful because the E-step cannot be determined in most instances. The E-step computes the conditional expectation of the complete data log-likelihood and when the random effect distribution is normal, this expectation remains an intractable integral. The problem can be approached by numerical or analytic approximations; however, the computational burden imposed by numerical integration methods and the absence of an accurate analytic approximation have limited the use of the EM algorithm. In this paper, Laplace's method is adapted for analytic approximation within the E-step. The proposed algorithm is computationally straightforward and retains much of the conceptual simplicity of the conventional EM algorithm, although the usual convergence properties are not guaranteed. The proposed algorithm accommodates multiple random factors and random effect distributions besides the normal, e.g., the log-gamma distribution. Parameter estimates obtained for several data sets and through simulation show that this modified EM algorithm compares favorably with other generalized mixed model methods.
在广义混合模型中应用期望最大化（EM）算法进行估计在很大程度上并不成功，因为在大多数情况下无法确定E步。E步计算完整数据对数似然的条件期望，当随机效应分布为正态时，该期望仍然是一个难以处理的积分。这个问题可以通过数值或解析近似来解决；然而，数值积分方法带来的计算负担以及缺乏精确的解析近似限制了EM算法的使用。在本文中，拉普拉斯方法适用于E步内的解析近似。所提出的算法计算简单，尽管不能保证具有常规EM算法的收敛特性，但仍保留了许多传统EM算法的概念简单性。所提出的算法除了正态分布外，还适用于多个随机因素和随机效应分布，例如对数伽马分布。通过对几个数据集的参数估计以及模拟结果表明，这种改进的EM算法与其他广义混合模型方法相比具有优势。
