<p>In this thesis, we apply the Ordinary Least Squares (OLS) and the Generalized Least Squares (GLS) methods for the estimation of Autoregressive Conditional Duration (ACD) models, as opposed to the typical approach of using the Quasi Maximum Likelihood Estimation (QMLE).</p> <p>The advantages of OLS and GLS as the underlying methods of estimation lie in their theoretical ease and computational convenience. The latter property is crucial for high frequency trading, where a transaction decision needs to be made within a minute. We show that both OLS and GLS estimates are asymptotically consistent and normally distributed. The normal approximation does not seem to be satisfactory in small samples. We also apply Residual Bootstrap to construct the confidence intervals based on the OLS and GLS estimates. The properties of the proposed methods are illustrated with intensive numerical simulations as well as by a case study on the IBM transaction data.</p> / Master of Science (MSc)

Spatial-temporal autologistic models are useful models for binary data that are measured repeatedly over time on a spatial lattice. They can account for effects of potential covariates and spatial-temporal statistical dependence among the data. However, the traditional parametrization of spatial-temporal autologistic model presents difficulties in interpreting model parameters across varying levels of statistical dependence, where its non-negative autocovariates could bias the realizations toward 1. In order to achieve interpretable parameters, a centered spatial-temporal autologistic regression model has been developed. Two efficient statistical inference approaches, expectation-maximization pseudo-likelihood approach (EMPL) and Monte Carlo expectation-maximization likelihood approach (MCEML), have been proposed. Also, Bayesian inference is considered and studied. Moreover, the performance and efficiency of these three inference approaches across various sizes of sampling lattices and numbers of sampling time points through both simulation study and a real data example have been studied. In addition, We consider the imputation of missing values is for spatial-temporal autologistic regression models. Most existing imputation methods are not admissible to impute spatial-temporal missing values, because they can disrupt the inherent structure of the data and lead to a serious bias during the inference or computing efficient issue. Two imputation methods, iteration-KNN imputation and maximum entropy imputation, are proposed, both of them are relatively simple and can yield reasonable results. In summary, the main contributions of this dissertation are the development of a spatial-temporal autologistic regression model with centered parameterization, and proposal of EMPL, MCEML, and Bayesian inference to obtain the estimations of model parameters. Also, iteration-KNN and maximum entropy imputation methods have been presented for spatial-temporal missing data, which generate reliable imputed values with the reasonable efficient imputation time.

This thesis looks at the problem of developmental respiration in Sarcophaga crassipalpis Macquart from the biological and instrumental points of view and adapts mathematical and statistical tools in order to analyze the data gathered. The biological motivation and current state of research is given as well as instrumental considerations and problems in the measurement of carbon dioxide production. A wide set of mathematical and statistical tools are used to analyze the time series produced in the laboratory. The objective is to assemble a methodology for the production and analysis of data that can be used in further developmental respiration research.

Longitudinal data arise when individuals are measured several times during an ob- servation period and thus the data for each individual are not independent. There are several ways of analyzing longitudinal data when different treatments are com- pared. Multilevel models are used to analyze data that are clustered in some way. In this work, multilevel models are used to analyze longitudinal data from a case study. Results from other more commonly used methods are compared to multilevel models. Also, comparison in output between two software, SAS and R, is done. Finally a method consisting of fitting individual models for each individual and then doing ANOVA type analysis on the estimated parameters of the individual models is proposed and its power for different sample sizes and effect sizes is studied by simulation.

Tornado intensity and impacts vary drastically across space, thus spatial and statistical analyses were used to identify patterns of tornado severity in the Southeastern United States and to assess the vulnerability and estimated recurrence of tornadic activity. Records from the Storm Prediction Center's tornado database (1950-2014) were used to estimate kernel density to identify areas of high and low tornado frequency at both the regional- and state-scales. Return periods (2-year, 5-year, 10-year, 25-year, 50-year, and 100-year) were calculated at both scales as well using a composite score that included EF-scale magnitude, injury counts, and fatality counts. Results showed that the highest density of tornadoes occur in Alabama, Mississippi, and Arkansas, while the highest return period intensities occur in Alabama and Mississippi. Scaledependent analysis revealed finer details of density and intensity for each state. Better communication of high hazard areas and integration into existing mitigation plans is suggested.

The response of a karst aquifer to storm events is often faster and more severe than that of a non-karst aquifer. This distinction is often problematic for planners and municipalities, because karst flooding does not typically occur along perennial water courses; thus, traditional flood management strategies are usually ineffective. The City of Bowling Green (CoBG), Kentucky is a representative example of an area plagued by karst flooding. The CoBG, is an urban karst area (UKA), that uses Class V Injection Wells to lessen the severity of flooding. The overall effectiveness, siting, and flooding impact of Injection Wells in UKA’s is lacking; their influence on groundwater is evident from decades of recurring problems in the form of flooding and groundwater contamination. This research examined Class V Injection Wells in the CoBG to determine how Injection Well siting, design, and performance influence urban karst hydrology. The study used high-resolution monitoring, as well as hydrologic modeling, to evaluate Injection Well and spring responses during storm and baseflow conditions. In evaluating the properties of the karst aquifer and the influences from the surrounding environment, a relationship was established between precipitation events, the drainage capacity of the Injection Wells, and the underlying karst system. Ultimately, the results from this research could be used to make sound data-driven policy recommendations and to inform stormwater management in UKAs.

The United States is in the midst of an opioid epidemic. In addition to other system level interventions, almost all states have responded to the crisis by implementing prescription drug monitoring programs (PDMPs). PDMPs are state-level interventions that track the dispensing of Controlled Substances. Data generated at the time of medication dispensing is uploaded to a central data server that may be used to assist in identifying drug diversion, medication misuse, or potentially aberrant prescribing practices. Prior studies assessing the impact of PDMPs on trends in opioid-related morbidity have often failed to take into account the wide heterogeneity of program features and how the effectiveness of these features may be mitigated by insurance status. Previous research has also failed to differentiate the effects of these programs on prescription vs. illicit opioid-related morbidity. The studies in this dissertation attempt to address these gaps using epidemiological techniques to examine the associations between specific PDMP features and trends in prescription and illicit opioid-related poisonings in populations of different insurance beneficiaries. Results of these studies demonstrate that implementation of specific PDMP features is significantly associated with differential trends in prescription and illicit-opioid related poisonings and that the effectiveness of these features vary depending on the insurance status of the population studied. These results suggest that PDMPs offer a valuable tool in addressing the United States’ opioid epidemic, and may be used as empirical evidence to support PDMP best practices in the future.

Increased opioid analgesic (OA) prescribing has been associated with increased risk of prescription opioid diversion, misuse, and abuse. States established prescription drug monitoring programs (PDMPs) to collect and analyze electronic records for dispensed controlled substances to reduce prescription drug abuse and diversion. PDMP data can be used by prescribers for tracking patient’s history of controlled substance prescribing to inform clinical decisions. The studies in this dissertation are focused on the less utilized potential of the PDMP data to enhance public health surveillance to monitor OA prescribing and co-prescribing and association with opioid overdose mortality and morbidity. Longitudinal analysis of OA prescribing and evaluation of the effect of recent policies and opioid prescribing guidelines require consensus measures for OA utilization and computational tools for uniform operationalization by researchers and agencies. Statistical macros and computational tools for OA utilization measures were developed and tested with Kentucky PDMP data. A set of covariate measures using mortality and morbidity surveillance data were also developed as proxy measures for prevalence of painful conditions justifying OA utilization, and availability of heroin and medication treatment for opioid use disorder. A series of epidemiological studies used the developed OA measures as outcomes, and adjusted for time-varying socio-demographic and health care utilization covariates in population-averaged statistical models to assess longitudinal trend and pattern changes in OA utilization in Kentucky in recent years. The first study, “Trends and Patterns of OA Prescribing: Regional and Rural-Urban Variations in Kentucky from 2012 to 2015,” shows significant downward trends in rates of residents with OA prescriptions. Despite the significant decline over time, and after accounting for prevalence of injuries and cancer, the rate of dispensed OA prescriptions among residents in Kentucky Appalachian counties remained significantly higher than the rest of the state. The second study, “Population-Level Measures for High-Risk OA Prescribing: Longitudinal Trends and Relationships with Pain-Associated Conditions,” shows significant reduction in high-risk OA prescribing (e.g., high daily dosage, long-term use, concurrent prescriptions for OA and benzodiazepines) from 2012 to 2016, significantly positive associations between high-risk OA prescribing and cancer mortality rates with no substantial change in the association magnitude over time, and declining strengths of positive associations between high-risk OA prescribing and acute traumatic injuries or chronic non-cancer pain over the study period. The third study, “A Reciprocal Association between Longitudinal Trends of Buprenorphine/Naloxone Prescribing and High-Dose OA Prescribing,” indicates a significant reciprocal relationship between high-dose OA prescribing and buprenorphine/ naloxone prescribing, and a clinically meaningful effect of buprenorphine/naloxone prescribing on reducing OA utilization. The results from the studies advanced the understanding of the epidemiology of opioid use and misuse in Kentucky, and identified actionable risk and protective factors that can inform policy, education, and drug overdose prevention interventions. The developed operational definition inventory and computational tools could stimulate further research in Kentucky and comparative studies in other states.

In longitudinal studies, outcomes are measured repeatedly over time, but in reality clinical studies are full of missing data points of monotone and non-monotone nature. Often this missingness is related to the unobserved data so that it is non-ignorable. In such context, pattern-mixture model (PMM) is one popular tool to analyze the joint distribution of outcome and missingness patterns. Then the unobserved outcomes are imputed using the distribution of observed outcomes, conditioned on missing patterns. However, the existing methods suffer from model identification issues if data is sparse in specific missing patterns, which is very likely to happen with a small sample size or a large number of repetitions. We extend the existing methods using latent class analysis (LCA) and a shared-parameter PMM. The LCA groups patterns of missingness with similar features and the shared-parameter PMM allows a subset of parameters to be different among latent classes when fitting a model, thus restoring model identifiability. A novel imputation method is also developed using the distribution of observed data conditioned on latent classes. We develop this model for continuous response data and extend it to handle ordinal rating scale data. Our model performs better than existing methods for data with small sample size. The method is applied to two datasets from a phase II clinical trial that studies the quality of life for patients with prostate cancer receiving radiation therapy, and another to study the relationship between the perceived neighborhood condition in adolescence and the drinking habit in adulthood.

This thesis explores the problems of fitting antedependence (AD) models and partial antecorrelation (PAC) models to continuous non-Gaussian longitudinal data. AD models impose certain conditional independence relations among the measurements within each subject, while PAC models characterize the partial correlation relations. The models are parsimonious and useful for data exhibiting time-dependent correlations. Since the relation of conditional independence among variables is rather restrictive, we first consider an autoregressively characterized PAC model with independent asymmetric Laplace (ALD) innovations and prove that this model is an AD model. The ALD distribution previously has been applied to quantile regression and has shown promise for modeling asymmetrically distributed ecological data. In addition, the double exponential distribution, a special case of the ALD, has played an important role in fitting symmetric finance and hydrology data. We give the distribution of a linear combination of independent standard ALD variables in order to derive marginal distributions for the model. For the model estimation problem, we propose an iterative algorithm for the maximum likelihood estimation. The estimation accuracy is illustrated by some numerical examples as well as some longitudinal data sets. The second component of this dissertation focuses on AD multivariate skew normal models. The multivariate skew normal distribution not only shares some nice properties with multivariate normal distributions but also allows for any value of skewness. We derive necessary and sufficient conditions on the shape and covariance parameters for multivariate skew normal variables to be AD(p) for some p. Likelihood-based estimation for balanced and monotone missing data as well as likelihood ratio hypothesis tests for the order of antedependence and for zero skewness under the models are presented. Since the class of skew normal random variables is closed under the addition of independent standard normal random variables, we then consider an autoregressively characterized PAC model with a combination of independent skew normal and normal innovations. Explicit expressions for the marginals, which all have skew normal distributions, and maximum likelihood estimates of model parameters, are given. Numerical results show that these three proposed models may provide reasonable fits to some continuous non-Gaussian longitudinal data sets. Furthermore, we compare the fits of these models to the Treatment A cattle growth data using penalized likelihood criteria, and demonstrate that the AD(2) multivariate skew normal model fits the data best among those proposed models.