Theses and Dissertations from UMD
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Item A Mean-Parameterized Conway–Maxwell–Poisson Multilevel Item Response Theory Model for Multivariate Count Response Data(2024) Strazzeri, Marian Mullin; Yang, Ji Seung; Measurement, Statistics and Evaluation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Multivariate count data arise frequently in the process of measuring a latent construct in human development, psychology, medicine, education, and the social sciences. Some examples include the number of different types of mistakes a student makes when reading a passage of text, or the number of nausea, vomiting, diarrhea, and/or dysphagia episodes a patient experiences in a given day. These response data are often sampled from multiple sources and/or in multiple stages, yielding a multilevel data structure with lower level sampling units (e.g., individuals, such as students or patients) nested within higher level sampling units or clusters (e.g., schools, clinical trial sites, studies). Motivated by real data, a new Item Response Theory (IRT) model is developed for the integrative analysis of multivariate count data. The proposed mean-parameterized Conway--Maxwell--Poisson Multilevel IRT (CMPmu-MLIRT) model differs from currently available models in its ability to yield sound inferences when applied to multilevel, multivariate count data, where exposure (the length of time, space, or number of trials over which events are recorded) may vary across individuals, and items may provide different amounts of information about an individual’s level of the latent construct being measured (e.g., level of expressive language development, math ability, disease severity). Estimation feasibility is demonstrated through a Monte Carlo simulation study evaluating parameter recovery across various salient conditions. Mean parameter estimates are shown to be well aligned with true parameter values when a sufficient number of items (e.g., 10) are used, while recovery of dispersion parameters may be challenging when as few as 5 items are used. In a second Monte Carlo simulation study, to demonstrate the need for the proposed CMPmu-MLIRT model over currently available alternatives, the impact of CMPmu-MLIRT model misspecification is evaluated with respect to model parameter estimates and corresponding standard errors. Treating an exposure that varies across individuals as though it were fixed is shown to notably overestimate item intercept and slope estimates, and, when substantial variability in the latent construct exists among clusters, underestimate said variance. Misspecifying the number of levels (i.e., fitting a single-level model to multilevel data) is shown to overestimate item slopes---especially when substantial variability in the latent construct exists among clusters---as well as compound the overestimation of item slopes when a varying exposure is also misspecified as being fixed. Misspecifying the conditional item response distributions as Poisson for underdispersed items and negative binomial for overdispersed items is shown to bias estimates of between-cluster variability in the latent construct. Lastly, the applicability of the proposed CMPmu-MLIRT model to empirical data was demonstrated in the integrative data analysis of oral language samples.Item Estimation of a Function of a Large Covariance Matrix Using Classical and Bayesian Methods(2018) Law, Judith N.; Lahiri, Partha; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this dissertation, we consider the problem of estimating a high dimensional co- variance matrix in the presence of small sample size. The proposed Bayesian solution is general and can be applied to dierent functions of the covariance matrix in a wide range of scientic applications, though we narrowly focus on a specic application of allocation of assets in a portfolio where the function is vector-valued with components which sum to unity. While often there exists a high dimension of time series data, in practice only a shorter length is tenable, to avoid violating the critical assumption of equal covariance matrix of investment returns over the period. Using Monte Carlo simulations and real data analysis, we show that for small sample size, allocation estimates based on the sample covariance matrix can perform poorly in terms of the traditional measures used to evaluate an allocation for portfolio analysis. When the sample size is less than the dimension of the covariance matrix, we encounter diculty computing the allocation estimates because of singularity of the sample covariance matrix. We evaluate a few classical estimators. Among them, the allocation estimator based on the well-known POET estimator is developed using a factor model. While our simulation and data analysis illustrate the good behavior of POET for large sample size (consistent with the asymptotic theory), our study indicates that it does not perform well in small samples when compared to our pro- posed Bayesian estimator. A constrained Bayes estimator of the allocation vector is proposed that is the best in terms of the posterior risk under a given prior among all estimators that satisfy the constraint. In this sense, it is better than all classi- cal plug-in estimators, including POET and the proposed Bayesian estimator. We compare the proposed Bayesian method with the constrained Bayes using the tradi- tional evaluation measures used in portfolio analysis and nd that they show similar behavior. In addition to point estimation, the proposed Bayesian approach yields a straightforward measure of uncertainty of the estimate and allows construction of credible intervals for a wide range of parameters.