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This research project investigates the effectiveness of mentors on rates of self-reported criminal offending for released offenders. I use data from the Serious and Violent Offender Reentry Initiative (SVORI) study (2004-2007), which sought to evaluate factors relating to high-risk offenders outcomes post release in an effort to reduce the societal problem of mass incarceration. Previous research has examined the use of mentors to improve the delinquent and criminogenic behavior of youth, but little is known about the effectiveness of mentors used to aid imprisoned adult males. I utilize negative binomial analysis to compare the number of self-reported criminal activities among released offenders with mentors versus those without mentors, and assess if the values varied between different reported levels of need for mentoring. Results indicate that mentoring did not reduce the rate of post-release offending at a statistically significant level. Reasons for the lack of significant results and policy implications are discussed.

Having the ability to predict enrollment is an important task for any school's recruiting team. The purpose of this study was to identify significant factors that can be used to predict the spatial distribution of enrollments. As a case study, we used East Tennessee State University (ETSU) pharmacy school, a regional pharmacy school located in the Appalachian Mountains. Through the application of a negative binomial regression model, we found that the most important indicators of enrollment volume for the ETSU pharmacy school were Euclidean distance, probability (based on competing pharmacy schools' prestige, driving distance between schools and home and tuition costs), and the natural barrier of the Appalachian Mountains. Using these factors, together with other control variables, we successfully predicted the spatial distribution of enrollments for ETSU pharmacy school. Interestingly, gender also surfaced as a variable for predicting the pharmacy school's enrollment. We found female students are more sensitive to the geographic proximity of home to school.

The objective of this thesis is to examine the relationship between CAIX (a biomarker for insufficient oxygen in tumor microenvironment) and CD8+ T cells (the immune cells for killing cancer cells) for ovarian cancer. We approach the problem from two perspectives. The first approach is to set up count models such as Poisson, negative binomial, and zero-inflated Poisson models to examine the cell counts between CAIX and CD8+ T cells in the tumor microenvironment. The second approach is to apply the cross-K function, which is a second-order property of the point pattern process. We find that the tissue microarray (TMA), which is a technique to assemble hundreds of tissue samples on one TMA block, has a fixed effect on the CD8+ T cell counts. There are two TMA blocks A2 and B1. The relationship between CAIX and CD8+ T cells highly depends on TMAs. On TMA B1 stroma, a negative relationship between CAIX and CD8+ T cell counts is observed in the negative binomial models. When taking the spatial domain into account and comparing the estimated cross-K function of CAIX and CD8+ T cells to the simulated envelopes generated by a homogeneous Poisson process, we find that CAIX and CD8+ T cells are regulated and repel each other on TMA B1. Tissue category also plays an influential role in analyzing the relationship. The estimated cross-K function of CAIX and CD8 + T cells is more dispersed on tumors than on stroma. / Graduate

It is estimated that one third of the world's population is infected with tuberculosis. Though once thought a "dead" disease, tuberculosis is very much alive. The rise of drug resistant strains of tuberculosis, and TB-HIV coinfection have made tuberculosis an even greater worldwide threat. While HIV, poverty, and public health infrastructure are historically assumed to affect the burden of tuberculosis, recent research has been done to implicate smoking in this list. This analysis involves combining data from multiple sources in order determine if smoking is a statistically significant factor in predicting the number of incident tuberculosis cases in a country. Quasi-Poisson generalized linear models and negative binomial regression will be used to analyze the effect of smoking, as well as the other factors, on tuberculosis incidence. This work will enhance tuberculosis control efforts by helping to identify new hypotheses that can be tested in future studies. One of the main hypotheses is whether or not smoking increases the number of tuberculosis cases above and beyond the effects of other factors that are known to influence tuberculosis incidence. These known factors include TB-HIV coinfection, poverty and public health infrastructure represented by treatment outcomes.

This thesis is an extension of the development of orthogonal statistics which can be used to investigate sampling properties of moment estimators. This work is particularized for estimators of parameters of the negative binomial distribution. / Master of Science

Ph. D.

This dissertation deals primarily with the development of the technique of orthogonal statistics and the use of this technique to investigate sampling properties of moment estimators of parameters of the negative binomial distribution. The general technique of orthogonal statistics which is based on the existence of an infinite set {q_r(x)} of orthogonal polynomials associated with a particular distribution, enables one to obtain expansions of sampling moments of statistics which are functions of say, the first k sample moments m₁, m₂,…, m_k. The thesis describes the technique in general, and gives tables which facilitate the expansion through terms in n⁻⁵ of sampling moments of statistics which are functions of any four sample moments. The need for the development of this technique resulted from an interest in the problem of investigating sampling properties of certain moment estimators for the case of the negative binomial distribution. Thus further work was done on the technique for this particular case. Tables are given in the thesis which simplify the procedure for moment statistics which result from a sample taken from this particular distribution. Sampling properties of moment estimators for the negative binomial distribution were investigated. The distribution forms considered in depth were due to Anscombe [Biometrika, 37 (1950}, pp. 358-362] with parameters λ and α, Evans [Biometrika, 40 (1953), pp. 186-211] with parameters m and a, and Fisher [Annals of Eugenics, 11 (1941), pp. 182-187] with parameters p and k. The purpose of this study was to obtain an insight into the behavior of expansions through high powers of 1/n (e.g., terms in n⁻⁴) of the bias, variance, and higher moments for these estimators. It was felt that the usual asymptotic properties described by the first term approximations might be misleading for practical cases (i.e., ordinary sample sizes). The results verified what was suspected. For the moment estimators of Ansaombe's form, when α > λ the sample sizes needed to make high order terms negligible for the expansion of the bias and variance were extremely large. (For one particular case, in order to use the usual asymptotic variance safely one would need an n of 2 million.) This then reveals the hazardous practice of using the first term approximation and resulting in a very serious under-assessment of the true variance of the estimate of α. Since for Fisher's form k̂ = α̂, the same applies. For Evans' form, the situation was in marked contrast. Higher order terms were "damped off" with much smaller sample sizes, and in most cases one is justified in using first term approximations. Studies for Evans' estimators were confined to the range λ > 1 and α > 1. The results for the estimators of Anscombe's form were compared with similar results for the maximum likelihood estimator of α, in order to ascertain the effect on efficiency of the chaotic nature of the n⁻³ term in the expansion of the covariance determinant of α̂. The maximum likelihood results were taken from Bowman [Thesis submitted for Ph.D. degree, Virginia Polytechnic Institute, Moments to Higher Orders for Maximum Likelihood Estimators with an Application to the Negative Binomial Distribution]. This study revealed that there is a striking similarity in the n⁻³ term in the covariance determinant for the two estimators. This made the "true" efficiency almost identical to the asymptotic efficiency in cases when sufficiently large sample sizes are used to "sink" terms beyond n⁻³. This statement cannot be generalized, however, to include any sample size, since for α > λ only relatively large sample sizes "damp off' further terms in the covariance determinants for both estimators. Hence one cannot be sure of the behavior of these determinants beyond n⁻³ unless these large sample sizes are used. Tables and charts are given which display the nature of the expansions given in the text. In particular, charts are given of minimum sample size needed in order that the expansions given can safely be used as approximations. / Ph. D.

This dissertation deals with the properties of two modified moment estimators for parameters of the negative binomial distribution (NBD). Several parametric forms have been suggested for the NBD. The estimation problems vary according to the form which is used. In particular, the form proposed by Anscombe [Biometrika, 37 (1950), pp. 358-382), with parameters λ and α, has received wide attention and was selected for study in this investigation. In Anscombe's parametric form, the mean of the NBD is λ and the variance is λ + λ²/α. While the parameter λ is universally estimated by the sample mean, many different methods of estimation for α have been attempted. Among these, the maximum likelihood estimator α* and the simple moment estimator â are most often used. However, α* is quite difficult to obtain numerically and often this computation requires the use of an electronic computer. In addition, â, while not difficult to compute, is often inefficient. For these reasons, it was felt that a study of the two modified moment estimators â₁ and â₂, suggested by Shenton and Wallington [Moment Estimators and Modified Moment Estimators with Special Reference to the Negative Binomial Distribution (unpublished)], was needed. In the text, the method of obtaining modified moment estimators in general is given in detail. The application of this method to the NBD is discussed and, in particular, the derivations of â₁ and â₂ are presented. Since orthogonal statistics play an important part in this work, their definition and applications are reviewed. In order to evaluate the small sample properties of â₁ and â₂, asymptotic expansions, in powers of 1/n, of their biases, variances, covariance determinants, and higher moments were determined numerically in the parameter space (1 ≤ α ≤ 100, 1 ≤ λ ≤ 100), through terms to n⁻⁴. The computational method for this work is described in detail. Tables and charts which display the nature of the expansions are given in the text. The results show that the behavior patterns of the moment expansions for â₁ and â₂ are somewhat similar to those for â and α*. For both â₁ and â₂, the n⁻⁴ term contributes heavily in all the expansions when α > λ. Thus, as with the other estimators, a first term approximation would not suffice for the properties of â₁ and â₂. Further, the results give evidence that â₁ and â₂ are highly efficient for most α and λ, and, in some regions of the parameter space, have less bias than α* and â. Some experimental data was fitted to the NBD using the estimators â₁, â₂, â, and α*. In all of the examples given, the modified moment estimators provided a better fit of the data than did the simple moment estimator and, in one instance, a better fit than was obtained by the maximum likelihood estimator. / Ph. D.

This thesis essentially takes two separate paths to solve the same problem, namely that of obtaining an estimator, a parameter of the negative binomial distribution, for which we can show that such properties as bias and variance of this estimator are "better" than corresponding properties of the simple moment estimator, the latter being the estimator which is used most often in practice. We first consider two moment estimators involving the third sample moment. In the case of both of these estimators, for a restricted range of the parameters and of sample size, these estimators are not an improvement over the simple moment estimator. In fact, for the range considered, the bias and variance of the simple moment estimator was always smaller. We then considered an estimator which was defined as the simple moment estimator for part of the sample space and defined as a constant elsewhere. This was primarily done to remove a "singularity" in the moment estimator. It was felt that this singularity was causing the large bias and variance which seemed to exist for certain values of the parameters. For n=100, the bias and variance were approximated in a range of interest of the parameters. The results indicate an improvement over the sample moment estimator. / Ph. D.