Mathematical Sciences: Estimation in Generalized Linear Mixed Models

广义线性混合模型（GLMM）在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者之间可能存在的非线性联系，以及考虑数据中某些形式的相关性。不幸的是，像最大似然估计这样的标准技术在计算上对glmm来说是困难的，因此已经提出了许多替代方法。本研究将开发两种方法来推断glmm: 1)基于计算密集型模拟的最大似然估计方法，以及2)基于“联合最大化”思想的方法。将联合最大化方法视为一组广义估计方程，以进行理论评价和改进。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一种重要的、应用广泛的数据分析统计模型。例如，它们可以用来模拟动物在固定采样地点随时间的重复计数，以进行环境评估。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。上述示例中的关联会出现，因为在单个位置上重复的数据值可能是相似的。如果不考虑这些关联，就会从数据分析中得出不正确的结论。不幸的是，由于计算困难和缺乏具有已知性能特征的经过良好测试的参数估计方法的可用性，广义线性混合模型的使用受到限制。这项研究将发展两种方法来分析这些数据并评价它们的表现，无论是绝对的还是与现有方法的关系。

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.