Mathematical Sciences: Estimation in Generalized Linear Mixed Models

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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.