I consider statistical modelling of data gathered by photographic identification in mark-recapture studies and propose a new method that incorporates the inherent uncertainty of photographic identification in the estimation of abundance, survival and recruitment. A hierarchical model is proposed which accepts scores assigned to pairs of photographs by pattern recognition algorithms as data and allows for uncertainty in matching photographs based on these scores. The new models incorporate latent capture histories that are treated as unknown random variables informed by the data, contrasting past models having the capture histories being fixed. The methods properly account for uncertainty in the matching process and avoid the need for researchers to confirm matches visually, which may be a time consuming and error prone process. Through simulation and application to data obtained from a photographic identification study of whale sharks I show that the proposed method produces estimates that are similar to when the true matching nature of the photographic pairs is known. I then extend the method to incorporate auxiliary information to predetermine matches and non-matches between pairs of photographs in order to reduce computation time when fitting the model. Additionally, methods previously applied to record linkage problems in survey statistics are borrowed to predetermine matches and non-matches based on scores that are deemed extreme. I fit the new models in the Bayesian paradigm via Markov Chain Monte Carlo and custom code that is available by request.

Well-preserved echinoderm faunas are rare in the fossil record, and when uncovered, understanding their occurrence can be useful in interpreting other faunas. In this study, two such faunas of the same age from separate localities in the shallow-marine Ramey Creek Member of the Slade Formation in the Upper Mississippian (Chesterian) rocks of eastern Kentucky are examined. Of the more than 5,000 fossil specimens from both localities, only 9–34 percent were echinoderms from 3–5 classes. Nine non-echinoderm (8 invertebrate and one vertebrate) classes occurred at both localities, but of these, bryozoans, brachiopods and sponges dominated. To understand the attributes of both localities (Valley Stone and 213 quarries), the geologic and structural settings, lithofacies and depositional environments, as well as faunal makeup and abundances (diversity, evenness, density), were compared and contrasted. Faunas from the Valley Stone Quarry were located on an uplifted fault block in more shallow, open-marine waters with higher energies. As indicated by four distinct lithofacies, the depositional setting was more extensive and varied with interspersed shoals and basins that could accommodate a greater richness (65 species), even though organism densities and abundance were less. In contrast, fauna from the 213 Quarry were located on a downdropped fault block in a more localized, deeper, storm-shelf setting, characterized by a single lithofacies. Although organism density and abundance were nearly twice as high as that at the Valley Stone Quarry, species richness was lower (45 species), and only one species, a bryozoan, predominated. Overall, echinoderm classes, species and individuals were more abundant at the Valley Stone Quarry, and I suggest that this is related to the shallower and more varied depositional environments that developed in response to presence on the shallow, uplifted fault block. This suggests the importance of regional features like faults in controlling environments and organism distribution through time. Although the faunas were originally collected for their echinoderm-dominated “crinoid gardens,” in fact, echinoderms were in the minority, and bryozoans and brachiopods predominated in the communities. Hence, the communities might better be described as bryozoan “thickets” and brachiopod “pavements.”

Data arising from Cluster Randomized Trials (CRTs) and longitudinal studies are correlated and generalized estimating equations (GEE) are a popular analysis method for correlated data. Previous research has shown that analyses using GEE could result in liberal inference due to the use of the empirical sandwich covariance matrix estimator, which can yield negatively biased standard error estimates when the number of clusters or subjects is not large. Many techniques have been presented to correct this negative bias; However, use of these corrections can still result in biased standard error estimates and thus test sizes that are not consistently at their nominal level. Therefore, there is a need for an improved correction such that nominal type I error rates will consistently result. First, GEEs are becoming a popular choice for the analysis of data arising from CRTs. We study the use of recently developed corrections for empirical standard error estimation and the use of a combination of two popular corrections. In an extensive simulation study, we find that nominal type I error rates can be consistently attained when using an average of two popular corrections developed by Mancl and DeRouen (2001, Biometrics 57, 126-134) and Kauermann and Carroll (2001, Journal of the American Statistical Association 96, 1387-1396) (AVG MD KC). Use of this new correction was found to notably outperform the use of previously recommended corrections. Second, data arising from longitudinal studies are also commonly analyzed with GEE. We conduct a simulation study, finding two methods to attain nominal type I error rates more consistently than other methods in a variety of settings: First, a recently proposed method by Westgate and Burchett (2016, Statistics in Medicine 35, 3733-3744) that specifies both a covariance estimator and degrees of freedom, and second, AVG MD KC with degrees of freedom equaling the number of subjects minus the number of parameters in the marginal model. Finally, stepped wedge trials are an increasingly popular alternative to traditional parallel cluster randomized trials. Such trials often utilize a small number of clusters and numerous time intervals, and these components must be considered when choosing an analysis method. A generalized linear mixed model containing a random intercept and fixed time and intervention covariates is the most common analysis approach. However, the sole use of a random intercept applies assumptions that will be violated in practice. We show, using an extensive simulation study based on a motivating example and a more general design, alternative analysis methods are preferable for maintaining the validity of inference in small-sample stepped wedge trials with binary outcomes. First, we show the use of generalized estimating equations, with an appropriate bias correction and a degrees of freedom adjustment dependent on the study setting type, will result in nominal type I error rates. Second, we show the use of a cluster-level summary linear mixed model can also achieve nominal type I error rates for equal cluster size settings.

Finite Mixture model has been studied for a long time, however, traditional methods assume that the variables are measured without error. Mixtures-of-regression model with measurement error imposes challenges to the statisticians, since both the mixture structure and the existence of measurement error can lead to inconsistent estimate for the regression coefficients. In order to solve the inconsistency, We propose series of methods to estimate the mixture likelihood of the mixtures-of-regressions model when there is measurement error, both in the responses and predictors. Different estimators of the parameters are derived and compared with respect to their relative efficiencies. The simulation results show that the proposed estimation methods work well and improve the estimating process.

Sea level rise (SLR) varies depending on location. It is therefore important to local residents, businesses and government to analyze SLR locally. Further, because of increasing ice melt and other effects of climate change, rates of SLR may change. It is therefore also important to evaluate rates of change of SLR, which we call sea level acceleration (SLA) or deceleration. The present thesis will review the annual average sea level data compiled at the Key West tidal gauge in Key West, Florida. We use a multi-model approach that compares the results of various models on that data set. The goal is to determine if there is a consistent result that can be ascertained from the various models. Generally, all the models reveal a clear upward trend of SLR. Further, the models provide evidence that the trend has increased over the last 8-10 years, i.e., that there is SLA.

The purpose of this research is to investigate the performance of some ridge regression estimators for the logistic regression model in the presence of moderate to high correlation among the explanatory variables. As a performance criterion, we use the mean square error (MSE), the mean absolute percentage error (MAPE), the magnitude of bias, and the percentage of times the ridge regression estimator produces a higher MSE than the maximum likelihood estimator. A Monto Carlo simulation study has been executed to compare the performance of the ridge regression estimators under different experimental conditions. The degree of correlation, sample size, number of independent variables, and log odds ratio has been varied in the design of experiment. Simulation results show that under certain conditions, the ridge regression estimators outperform the maximum likelihood estimator. Moreover, an empirical data analysis supports the main findings of this study. This thesis proposed and recommended some good ridge regression estimators of the logistic regression model for the practitioners in the field of health, physical and social sciences.

The estimation of the Pareto distribution can be computationally expensive and the method is badly biased. In this work, an improved Least Squares derivation is used and the estimation will be less biased. Numerical examples and figures are provided so that one may observe the solution more clearly. Furthermore, by varying the different methods of estimation, a comparing of the estimators of the parameters is given. The improved Least Squares derivation is confidently employed for it is economic and efficient.

Sixteen "model building" and "model selection" procedures commonly encountered in industry, all of which were initially alleged to be capable of identifying the best model from the collection of 2k possible linear models corresponding to a given set of k predictors in a multiple linear regression analysis, were individually summarized and subsequently evaluated by considering their comparative advantages and limitations from both a theoretical and a practical standpoint. It was found that none of the proposed procedures were absolutely infallible and that several were actually unsuitable. However, it was also found that most of these techniques could still be profitably employed by the analyst, and specific directional guidelines were recommended for their implementation in a proper analysis. Furthermore, the specific role of the analyst in a multiple linear regression application was clearly defined in a practical sense.

The purpose of this thesis is to analyze various aspects of quality of life and to determine if there is a difference in quality of life offered by a conventional ileostomy versus a continent ileostomy. An instrument was developed to measure several factors thought to influence quality of life as well as several structural/demographic variables. This instrument was designed for persons with a conventional ileostomy and was modified for persons who had undergone conversion surgery from conventional to continent ileostomy. Analysis of variance was performed to determine differences in quality of life for persons with a conventional, conversion, or original continent ileostomy. In addition to an overall quality of life measure, measures for specific areas: self esteem, family relationships, marriage relationships and a composite measure, were tested. No difference was detected for the three types of ileostomy for these variables. Analysis of variance was also performed on variables measuring specific aspects of life such as social activities and travel. This identified several differences in the ileostomy types which the analysis of the more general variables failed to detect. Those who had conversion surgery from conventional to continent ileostomies answered each question twice, comparing life with no ileostomy to life with a conventional, then comparing life with a conventional ileostomy to life with a continent. Three analyses were performed on the resulting data: sign test, chi-square test, and Fisher's exact test. The use of these three tests showed differences in results concerning quality of life and differences in the statistical power of the tests. Both aspects are discussed. Significant improvement in quality of life for almost every aspect tested was seen for this group. Finally, principal component analysis was applied to the set of variables measuring specific aspects of quality of life and several new variables developed from the resulting factors. Analysis of variance was performed on these, as well as the original quality of life measures to determine which of the structural/demographic variables had an effect on quality of life.

Two methods of analyzing multi-dimensional frequency data are detailed. The Second Order Exponential (SOE) model is applicable for dichotomous classifications. The distribution has two sets of parameters, ϴi's and ϴj's. The ϴi's are interpreted as the log of the odds of the marginal probabilities if no two factor relationships exist. Or if all ϴij are not zero, then the ϴi's are analogous to a main effect in a 2m factorial analysis, (m = number of factors or classifications). The ϴif's may be interpreted as a measure and direction of the two factor relationships. These ϴij are analogous to partial or adjusted phi-coefficients. The second method discussed assumes a multinomial distribution and the statistics are developed from an Information Theoretic Approach. Each hypothesis is tested using twice the minimum discrimination information statistic (m.d.i.s), 2I. From the null hypothesis it is possible to estimate unique cell probabilities by an iterative metod. Then 2 is equal to 2 (sample frequencies) log (sample frequencies) - 2 (expected frequencies) log (expected frequencies). (141 pages)