Mathematical Sciences: Estimation in Generalized Linear Mixed Models

DMS 9625476 McCulloch广义线性混合模型从三个方面概括了标准线性模型：适应非正态分布的响应，说明响应的平均值和预测值之间可能存在的非线性联系，以及允许数据中的某些形式的相关性。不幸的是，像最大似然估计这样的标准技术对于GLMM来说在计算上是困难的，因此已经提出了许多替代方法。这项研究将为广义最小二乘模型的推理开发两种方法：1)基于计算密集型仿真的最大似然估计方法；2)基于联合最大化思想的方法。为了理论上的评估和改进，联合最大化方法将被看作是一组广义的估计方程。将对新方法进行评估，并与现有方法进行比较。广义线性混合模型是用于数据分析的一组重要的、应用广泛的统计模型。例如，为了环境评估的目的，它们可以用来模拟在固定采样地点通过时间重复的动物计数。它们适用于以各种格式收集的数据，并且能够对显示关联的数据进行建模。上例中的关联之所以会出现，是因为单个位置的重复数据值将是相似的。没有纳入这种关联可能会导致对数据的分析得出错误的结论。遗憾的是，广义线性混合模型的使用受到了限制，原因是计算困难，以及缺乏经过良好测试的具有已知性能特征的参数估计方法。这项研究将开发两种方法来分析这些数据，并评估它们的性能，无论是从绝对值上还是相对于现有方法而言。

DMS 9625476 McCulloch The generalized linear mixed model (GLMM) generalizes the standard linear model in three ways: accommodation of non-normally distributed responses, specification of a possibly nonlinear link between the mean of a response and the predictors, and allowance for some forms of correlation in the data. Unfortunately, standard techniques like maximum likelihood estimation are computationally difficult for GLMMs and hence a number of alternate approaches have been proposed. This research will develop two approaches to inference for GLMMs: 1) computationally-intensive simulation-based methods for maximum likelihood estimation, and 2) methods based on "joint-maximization" ideas. The joint maximization methods will be viewed as a set of generalized estimating equations for the purpose of theoretical evaluation and improvement. The new approaches will be evaluated and compared to extant methods. Generalized linear mixed models are an important and broadly applicable set of statistical models for the analysis of data. For example, they can be used to model repeated counts of animals through time at fixed sampling locations for the purpose of environmental assessment. They are applicable to data which are gathered in a wide variety of formats and are capable of modelling data exhibiting associations. Association in the above example would arise because repeated data values at a single location would be similar. Failure to incorporate such associations can lead to incorrect conclusions from the analysis of data. Unfortunately, the use of generalized linear mixed models has been limited due to computational difficulties and the lack of availability of well-tested parameter estimation methods which have known performance characteristics. This research will develop two approaches to the analysis of such data and evaluate their performance, both in absolute terms and in relation to extant methods.