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Estimation of Complex Generalized Linear Mixed Models for Measurement and Growth


Maximum likelihood (ML) estimation of generalized linear mixed models (GLMMs) is technically challenging because of the intractable likelihoods that involve high dimensional integrations over random effects. The problem is magnified when the random effects have a crossed design and thus the data cannot be reduced to small independent clusters. A variety of methods have been developed for approximating the intractable likelihood functions, but there seems no method yet that is both computationally efficient and accurate in a wide range of situations. In this dissertation, I consider new estimation methods and applications of complex GLMMs for measurement and growth. The dissertation consists of three papers,

1) Variational maximization-maximization (MM) algorithm,

2) Monte Carlo local likelihood (MCLL) estimation,

and 3) Autoregressive item response theory (IRT) growth model for longitudinal item analysis.

In the first and second papers, I develop two ML methods for estimating GLMMs with crossed random effects. The variational MM algorithm is a modified expectation-maximization (EM) algorithm where a variational density is introduced in the expectation (E) step to approximate the true posterior density of the random effects given the data. The E-step is replaced by another maximization step that minimizes the Kullback-Leibler (KL) divergence between the posterior and the variational density, or equivalently, maximizes the lower bound of the log-likelihood with respect to the variational distribution. The MCLL algorithm uses the posterior samples of model parameters obtained from Markov chain Monte Carlo (MCMC) for likelihood inference. The posterior density is estimated by local likelihood density estimation and the likelihood function is approximated up to a constant by the local likelihood density estimate of the posterior divided by the prior. The performance of these new algorithms is evaluated using simulation and empirical studies and compared with other ML and Bayesian estimators. In the third paper, a new autoregressive IRT growth model is proposed to take into account serial correlations among responses to the same items over time. The consequences of ignoring serial dependence and the initial conditions problem are investigated using simulations. The new model is applied to longitudinal data of Korean students' self-esteem.

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