Computational Methods for Multilevel Modelling
Computational Methods for Multilevel Modelling
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发表时间：
1998
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通讯作者：
D. Bates;J. Pinheiro
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作者：
D. Bates;J. Pinheiro
多级混合效应模型在观察到的响应的分组的若干嵌套级别中的每一个都具有随机效应。例如，我们可以使用这些来模拟随时间对学生进行的观察，这些学生被分到不同的班级，这些学生被分成不同的学校，这些学校被分成不同的学区。如果随机效果的每个分布是高斯的，并且如果分组的最低级别的干扰项也是高斯的，则直接定义固定效果的似然和定义随机效果分布的参数。我们证明，通过用相对精度因子来表示随机效应分布，并使用矩阵分解，这种可能性可以被描绘出来并且可以被紧凑地表示。同样的分解为REML估计提供了对分布的对数受限似然的快速评估。给定数据的随机效应的条件分布可以从分解的矩阵中得到。由此可以推导出EM迭代的紧凑且快速计算的表达式。仅从设计本身就可以得出相对精度因子的合理起始估计。这些开始估计，通过适度次数的EM迭代改进，为对数似然或对数约束似然的牛顿-拉夫森或准牛顿优化提供了极好的起始值。我们所描述的方法很容易推广到具有非球面分布的组内误差模型和非线性多水平模型。
A multilevel mixed-effects model has random effects at each of several nested levels of grouping of the observed responses. We may use these, for example, when modelling observations taken over time on students who are grouped into classes that are grouped into schools that are grouped into districts. If each of the distributions of the random effects is Gaussian and if the disturbance term at the lowest level of grouping is also Gaussian it is straightforward to define a likelihood for the fixed effects and the parameters defining the random effects distribution. We show that by expressing the random effects distribution in terms of relative precision factors and using matrix decompositions, this likelihood can be profiled and can be compactly expressed. The same decompositions provide rapid evaluation of the profiled log-restricted-likelihood for REML estimation. The conditional distribution of the random effects given the data can be derived from the decomposed matrices. From this a compact and rapidly evaluated expression for the EM iterations can be derived. Reasonable starting estimates for the relative precision factors can be derived from the design alone. These starting estimates, refined by a moderate number of EM iterations, provide excellent starting values for a Newton-Raphson or quasi-Newton optimization of the log-likelihood or the log-restricted-likelihood. The methods we describe extend easily to models with non-spherical distributions for the within-group errors and to nonlinear multilevel models.