In this dissertation, robust and efficient alternatives to quasi-likelihood estimation and likelihood ratio tests are developed for discrete generalized linear models. The estimation method considered is a penalized minimum Hellinger distance procedure that generalizes a procedure developed by Harris and Basu for estimating parameters of a single discrete probability distribution from a random sample. A bootstrap algorithm is proposed to select the weight of the penalty term. Simulations are carried out to compare the new estimators with quasi-likelihood estimation. The robustness of the estimation procedure is demonstrated by simulation work and by Hapel's α-influence curve. Penalized minimum Hellinger deviance tests for goodness-of-fit and for testing nested linear hypotheses are proposed and simulated. A nonparametric bootstrap algorithm is proposed to obtain critical values for the testing procedure.

As data collection and modeling improve, ecologists increasingly discover that interspecies dynamics greatly affect the success of individual species. Models accounting for the dynamics of multiple species are becoming more important. In this work, we explore the relationship between mountain pine beetle (MPB, Dendroctonus ponderosae Hopkins) and two mutualistic fungi, Grosmannia clavigera and Ophiostoma montium. These species are involved in a multipartite symbiosis, critical to the survival of MPB, in which each species benefits. Extensive phenological modeling has been done to determine how temperature affects the timing of life events and cold-weather mortality of MPB. The fungi have also been closely studied to determine how they interact with MPB and how they differ in terms of virulence, response to temperature, and nutritional benefits to developing beetles. Overall, researchers consider G. clavigera to be the superior mutualist. Beetles developing near G. clavigera are larger, produce more brood, and have higher survival rates. Regarding temperature preferences, G. clavigera is considered “cool-loving,” growing at cooler temperatures than O. montium. These findings lead researchers to wonder 1) why has G. clavigera not displaced iv O. montium from the mutualism (if it is the superior mutualist) and 2) what will happen to the MPB-fungus mutualism in the face of a warming climate. In this work we present two models connecting fungal growth in a tree to predictions of MPB emergence: a stochastic, individual-based model and a deterministic, tree-based model. We begin by exploring whether variability in temperature can act as a stabilizing mechanism and find that temperature variability due to MPB periodically transitioning between different thermal environments is the most likely explanation for the continued presence of both fungi in the mutualism. Using the second model, we parameterize and validate the model using attack and emergence observations of MPB and the fungi they are carrying. In the process, we test several submodels to learn more about specific MPB-fungi interactions. Finally, utilizing information from previous fungal growth experiments, we test and parameterize several growth rate curves using Bayesian techniques to determine whether the inclusion of prior knowledge can lead to more realistic fits.

In evolutionary theory the existence of self-sacrificing cooperative traits poses a problem that has engendered decades of debate. The principal theories of the evolution of altruism are inclusive fitness, reciprocal altruism, and multilevel selection. To provide a framework for the unification o f these apparently disparate theories, this dissertation identifies two fundamental conditions required for the evolution of altruism: 1) non-zero-sum fitness benefits for cooperation and 2) positive assortment among altruistic behaviors. I demonstrate the underlying similarities in these three theories in the following two ways. First, I show that the game-theoretic model of the prisoner’s dilemm a (PD) is inherent to all three theories. While the PD has been used extensively to model reciprocal altruism, I demonstrate that the n-player PD captures fundamental aspects o f multilevel selection and inclusive fitness in that NPD model parameters relate simply to Simpson’s paradox, the Price covariance equation, and Hamilton’s rule. The tension between hierarchical levels that defines a PD reflects the tension between Abstract levels o f selection that is explicit in multilevel selection theory, and im plicit in the other two theories. Second, Ham ilton’s rule from inclusive fitness theory applies to the other theories. As mentioned, I demonstrate that this rule relates to multilevel selection via the NPD. I also show that Queller’s generalization of Hamilton’s rule applies to the conditional strategies of reciprocal altmism. This challenges the selfish-gene viewpoint by highlighting the fact that it is the phenotypes o f others, not their genotypes, that is critical to the evolution o f altruism. I integrate the PD and H am ilton’s rule as follows: the evolution o f altruism in general involves PD situations in which Hamilton’s rule specifies the necessary relationship between 1) the degree of non-zero-sumness within the PD and 2) the degree of positive assortment among altruistic behaviors. Additional contributions of this research include a demonstration that randomly formed associations can provide the necessary positive assortment for strong altruism to evolve, the development of a new selection decomposition that is symmetrical to the Price equation, and a game-theoretic analysis showing the essential similarity of weak and strong altruism under selection.

The two sample t-test is the test usually taught in introductory statistics courses to test for the equality of means of two populations. However, the t-test is not the only test available to compare the means of two populations. The randomization test is being incorporated into some introductory courses. There is also the bootstrap test. It is also not uncommon to decide the equality of the means based on confidence intervals for the means of these two populations. Are all those methods equally powerful? Can the idea of non-overlapping t confidence intervals be extended to bootstrap confidence intervals? The powers of seven alternative ways of comparing two population means are analyzed using small samples with data coming from distributions with different degrees of skewness and kurtosis. The analysis is done using simulation; programs in GAUSS were especially written for this purpose.

The probit function is the inverse of the cumulative distribution function associated with the standard normal distribution. It is of great utility in statistical modelling. The Carleman embedding technique has been shown to be effective in solving first order and, less efficiently, second order nonlinear differential equations. In this thesis, we show that solutions to the second order nonlinear differential equation for the probit function can be approximated efficiently using a variant of the Carleman embedding technique.

It is our intention to derive a methodology for which to model discrete paired longitudinal data. Through the use of transition matrices and maximum likelihood estimation techniques by means of software, we develop a way to model the progression of such data. We provide an example by applying this method to the Wisconsin Epidemiological Study of Diabetic Retinopathy data set. The data set is comprised of individuals, all diabetics, who have had their eyes examined for diabetic retinopathy. The eyes are treated as paired data, and we have the results of the examination at the four unequally spaced time points spanning over a fourteen year duration.

Percentiles are used everyday in descriptive statistics and data analysis. In real life, many quantities are normally distributed and normal percentiles are often used to describe those quantities. In life sciences, distributions like exponential, uniform, Weibull and many others are used to model rates, claims, pensions etc. The need to compare two or more independent populations can arise in data analysis. The ratio of percentiles is just one of the many ways of comparing populations. This thesis constructs a large sample confidence interval for the ratio of percentiles whose underlying distributions are known. A simulation study is conducted to evaluate the coverage probability of the proposed interval method. The distributions that are considered in this thesis are the normal, uniform and exponential distributions.

The study sample was comprised of 18,953 subjects aged 18 years and older who participated in the Third National Health and Nutrition Examination Survey. Preference for diet soda over regular soda was higher in females as compared to males and in Caucasians as compared to other races. Mean alcohol consumption was almost three times higher in males than in females. Undiagnosed hypertension was more common in males than in females, in 65-90 than in 50-64 year olds, and in people with less than high school education compared to those with higher education. Diastolic pressure was correlated with the level of consumption of diet soda, coffee, tea, and alcohol. Diet soda and alcohol consumption had a statistically significant positive relationship with hypertension even after adjusting for demographic variables and body mass index. The study provides useful information of the patterns of beverage uses and the prevalence of hypertension in the United States.

In many cases of crimes involving a firearm, police investigators need to know how far the firearm was held from the victim when it was discharged. Knowing this distance, vital questions regarding the re-construction of the crime scene can be known. Often, the original firearm used in commission of a suspected crime is not available for testing or is damaged. Crime laboratories require the original firearm in order to conduct distance determination tests. However, no empirical research has ever been conducted to determine if same make and model firearms produce different results in distance determination testing. It was the purpose of this study to determine if there are significant differences between the same make and model of firearms in distance determination testing. The findings indicate no significant differences; furthermore they imply that if the original firearm is not available, another firearm of the same make and model may be used.

Surveys ordinarily ask questions in an ordinal scale and often result in missing data. We suggest a regression based technique for imputing missing ordinal data. Multilevel cumulative logit model was used with an assumption that observed responses of certain key variables can serve as covariate in predicting missing item responses of an ordinal variable. Individual predicted probabilities at each response level were obtained. Average individual predicted probabilities for each response level were used to randomly impute the missing responses using a uniform distribution. Finally, likelihood ratio chi square statistics was used to compare the imputed and observed distributions. Two other forms of multiple imputation algorithms were performed for comparison. Performance of our imputation technique was comparable to other 2 established algorithms. Our method being simpler does not involve any complex algorithms and with further research can potentially be used as an imputation technique for missing ordinal variables.