We study an EM algorithm for estimating product-term regression models with missing data. The study of such problems in the likelihood tradition has thus far been restricted to an EM algorithm method using full numerical integration. However, under most missing data patterns, we show that this problem can be solved analytically, and numerical approximations are only needed under specific conditions. Thus we propose a hybrid EM algorithm, which uses analytic solutions when available and approximate solutions only when needed. The theoretical framework of our algorithm is described herein, along with two numerical experiments using both simulated and real data. We show that our algorithm confers higher accuracy to the estimation process, relative to the existing full numerical integration method. We conclude with a discussion of applications, extensions, and topics of further research.

As item response theory models are more frequently applied to psychological assessment, understanding the ramifications of failing to account for non-normality is of utmost importance, especially, considering the likelihood of encountering traits that are non-normally distributed in the population (e.g., anxiety, depression). Previous research has established concerns with regard to bias in item parameter and trait score estimates when non-normality is ignored, and, as such developed models to aid in minimizing bias. Once such model is the heteroscedastic GRM with a skewed latent trait (HSGRM). This research provides an in-depth examination of the viability and utility of the HSGRM. Under various degrees of skew and heteroscedasticity, including extreme on both, this research addresses the consequences of ignoring non-normality and how to address it. A simulation study was conducted to evaluate the ability of the HSGRM to provide improved item parameter estimates, and recover the shape of the distribution (i.e., skew). Results support the HSGRM as a major improvement over the traditional GRM when faced with non-normality in data due to skew in the trait and heteroscedastic errors.

An important aspect of empirical research is the construction of a model that represents the data. In psychological and educational measurement, models are typically evaluated regarding their ability to fit well to the observed data. Philosophers of science have long recognized that goodness-of-fit to the realized data is an insufficient metric of a model’s usefulness; models should also be appraised regarding their generalizability to unseen data. Frequentist statistics, Bayesian inference, and information theory seem to offer philosophically and methodologically dissimilar perspectives on model evaluation. However, this dissertation develops a simple theoretical framework that integrates these three perspectives. Within this framework, the information-theoretic principle of minimum description length is explored in the context of item response theory modeling. The findings reveal that complexity in item response theory is defined not by the number of freely estimated parameters in a model, but by its functional form. The frequentist, Bayesian, and information-theoretic approaches are then utilized in evaluating the usefulness of a unidimensional 3-parameter logistic model of item response data from the Program for International Student Assessment. Philosophical ramifications, future research directions, and implications for educational and psychological measurement are discussed.

Case diagnostics in categorical factor analysis include Mahalanobis distance-based statistics, which measure residual and leverage, and adaptations of existing influence diagnostics such as individual contribution to chi-square and generalized Cook’s distance which measure each case’s influence on statistical results. This dissertation uses two simulation studies to explore issues related to the use of case diagnostics in categorical factor analysis in order to assess the feasibility and utility of an iteratively reweighted least squares estimator for categorical factor analysis and structural equation modeling. In the first simulation, I used large data sets simulated according to a hypothesized model structure to examine the null distributions of Mahalanobis distance-based measures of residual and leverage in categorical factor analysis. Specifically, this study examined the validity of statistical cut-off values derived from continuous distributions in categorical factor analysis and assessed the differences between theoretical and empirical critical values in these models. In most conditions, the distributions of leverage and residual diagnostics in polytomous data, and of leverage diagnostics in dichotomous data, were similar enough to those in continuous data that existing critical values can safely be used to identify high-leverage cases. In contrast, residual diagnostics in dichotomous data had severely truncated distributions, a result which complicates the choice of critical value for identifying high-residual cases in residual analysis or down-weighting cases in robust estimation. In the second simulation, I examined the relationships between leverage, residual, and influence in categorical and continuous factor analysis and compared those relationships across continuous, polytomous, and dichotomous test conditions. Results were largely consistent between continuous and polytomous data but differed markedly in dichotomous data with high variability across dichotomous test conditions. Together, these findings reveal that, while categorical case diagnostics are well-behaved in polytomous tests under ideal conditions, these diagnostics can behave unpredictably in dichotomous data, and thus caution should be used in interpreting their values directly in dichotomous tests, whether as a means for screening for outliers or for down-weighting cases in robust estimation.

Exploratory factor analysis (EFA) almost always involves two unique steps: 1) initial extraction of m orthogonal, "unrotated" factors from a covariance matrix, followed by 2) rotation of said factors to an interpretable structure. The present research is focused entirely on the second step, rotation; specifically, a new, semi-analytic rotation algorithm called iterated target rotation (ITR) is developed and tested. The goal of ITR is to improve upon basic target rotation (Hurley & Cattell, 1962) by automatically searching for a viable target matrix. Whereas basic target rotation requires a user to specify a target matrix a priori, ITR uses an iterative search procedure to find a viable (often ideal) target matrix. Using Monte Carlo simulation of raw data (N = 250, 500, and 1000), the performance of ITR was tested in simple, complex, and highly complex population factor structures. Further, two characteristics of the ITR algorithm itself were varied, resulting in the following four simulation condition variables: sample size (three conditions), complexity of population factor structure (four conditions), method for iterating solutions (three conditions), and method for beginning the iterations (seven conditions). Performance was defined as the ability to accurately recover a true factor structure, as measured by the root mean-square error (RMSE) between the ITR solution and the population structure. Results suggest that, with few exceptions, ITR performs well even in highly complex structures. Limitations and future directions are discussed, and one potential improvement on the target-iteration algorithm is tested preliminarily.

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