In this paper response curves of patients with chronic myeloid leukemia are modeled using one and two-level censored models. Two-level models (also called mixed models) allow random effects and censored models are used to account for the large amount of values too small to be detected. The curves are observed from start of medication to a maximum of 36 months (9 measurement points). The data set is divided into two: "excellent responders" and "other". The "excellent responders" are modeled with a simple cubic censored model, and only one of the background variables measured at time zero ("blasts"), is found to be significant in explaining variation in the change curves, and even this with certain reservations. "Other" are modeled with a cubic two-level censored model and hemoglobin and eosinophile levels, as well as amount of blasts, are significant in explaining variation in this group.

The thesis deals with extreme value distributions and censored samples. Theoretical part describes a maximum likelihood method, types of censored samples and introduce a extreme value distributions. In the thesis are derived likelihood equations for censored samples from exponential, Weibull, lognormal, Gumbel and generalized extreme value distribution. For these distributions are also derived asymptotic interval estimates and is made simulation studies on the dependence of the parameter estimate on the percentage of censoring.

Open space provides a wide range of ecosystem services to communities. In growing communities, open space offers relief from congestion and other negative externalities associated with rapid development. To make effective policy and planning decisions pertaining to open space preservation, it is important to estimate monetary values of its benefits. In addition, assessing public opinions regarding open space provides information on demand and how residents value open space. This study estimated the monetary value of open space in Mississippi and Alabama Gulf Coast communities. The study also collected information on coastal residents’ attitudes towards open space, working waterfronts, and their willingness to support waterfront open space preservation monetarily. Two methodological approaches were employed to estimate the monetary value of waterfront open space: contingent valuation (CVM) and hedonic price (HPM) methods. Data were collected using a mail survey, the Multiple Listing Service (MLS), and publicly available data sources such as the U.S. Census. Data were analyzed using an interval regression, ordinary least squares, and geographically weighted regression (GWR) models. Mail survey results indicated that the majority of residents valued open space and were willing to pay from $80.52 to $162.14 per household as estimated by four different interval-censored econometric models. Respondent’s membership in groups promoting conservation goals, income, age, and residence duration were major factors associated with their willingness to pay. Results from the HPM indicated proximities to waterfronts, with the exception of bayous, were positively related to home prices, suggesting open space produced positive economic benefits. Findings from the HPM analysis using publicly available data were consistent and comparable with the results from the HPM that used MLS data. This similarity of results indicates the use of publicly available data is feasible in HPM analysis, which is important for broad applications of the method during city planning. In addition, GWR estimates provided site specific monetary values of waterfront open space benefits, which will be helpful for policymakers and city planners in developing site-specific conservation and preservation strategies. Findings can help formulate future decisions related to alternative development scenarios of coastal areas and conservation efforts to preserve open space.

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.

In this thesis, we provide a comprehensive overview of classical fractional derivatives and collect results on mapping properties. In particular, we discuss mapping properties e.g. we prove that the 𝛼 order fractional derivative maps the Sobolev space W_0^(p,s) to the fractional Sobolev-Slobodeckij space W^(p,s-α) for all 𝛼 < 𝑠 < 1. Further, we present several definitions of “Bernstein fractional derivatives” using the Bernstein function and in particular, we study the Bernstein censored fractional derivative by using the Picard method to get its inverse Bernstein censored fractional integral. Moreover, we use analytic tools to get the existence and uniqueness of the solution of the corresponding resolvent equation. Finally, we construct a stochastic process through Ikeda–Nagasawa–Watanabe (INW) piecing together procedure such that its generator is the Bernstein censored fractional derivative. Additionally, we show that this process gives a Feller semigroup.:Introduction 1 Basics 1.1 Some results in functional analysis 1.2 Fourier, Laplace and Mellin transforms 1.3 Regularly varying functions 1.4 Markov processes 1.5 Lévy processes and subordinators 2 Fractional derivatives and integrals 2.1 Classical fractional integrals and derivatives 2.2 Mapping properties of fractional integrals and derivatives 2.3 Bernstein functions 2.4 Fractional derivatives based on Bernstein Functions 2.5 Probabilistic interpretation of fractional derivatives 2.6 Fractional Laplace operator 3 Censored Bernstein fractional derivative and integral 3.1 Sonine pairs 3.2 Examples of Sonine pairs 3.3 Mapping properties of general fractional derivatives 3.4 Censored Bernstein fractional derivative and integral 3.5 Linear censored initial value problem 4 Censored process 4.1 Construction 4.2 Probabilistic representation 5 Application 5.1 Censored subordinator for a regularly varying kernel 5.2 Linear censored initial value problem for regularly varying kernels Bibliography

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The overall theme of this thesis focuses on the joint modeling of longitudinal covariates and a censored survival outcome, where a survival outcome is modeled using a conditional quantile regression. In traditional joint modeling approaches, a survival outcome is usually parametrically modeled as a Cox regression. Censored quantile regressions can model a survival outcome without pre-specifying a parametric likelihood function or assuming a proportional hazard ratio. Existing censored quantile methods are mostly limited to fixed cross-sectional covariates, while in many longitudinal studies, researchers wish to investigate the associations between longitudinal covariates and a survival outcome. The first part considers the problem of joint modeling with a survival outcome under a mixture of censoring: left censoring, interval censoring or right censoring. We pose a linear mixed effect model for a longitudinal covariate and a conditional quantile regression for a censored survival outcome, assuming that a longitudinal covariate and a survival outcome are conditional independent on individual level random effects. We propose a Gibbs sampling approach as an extension of a censored quantile based data augmentation algorithm, to allow for a longitudinal covariate process. We also propose an iterative algorithm that alternately updates individual level random effects and model parameters, where a censored survival outcome is treated in the way of re-weighting. Both of our methods are illustrated by the application to the LEGACY Girls cohort Study to understand the influence of individual genetic profiles on the pubertal development (i.e., the onset of breast development) while adjusting for BMI growth trajectories. The second part considers the problem of joint modelling with a random right censoring survival outcome. We pose a linear mixed effect model for a longitudinal covariate and a conditional quantile regression for a censored survival outcome, assuming that a longitudinal covariate and a survival outcome are conditional independent on individual level random effects. We propose a Gibbs sampling approach as an extension of a censored quantile based data augmentation algorithm, to allow for a longitudinal covariate process. Theoretical properties for the resulting parameter estimates are established. We also propose an iterative algorithm that alternately updates individual level random effects and model parameters, where a censored survival outcome is treated in the way of re-weighting. Both of our methods are illustrated by the application to Mayo Clinic Primary Biliary Cholangitis Data to assess the effect of drug D-penicilamine on risk of liver transplantation or death, while controlling for age at registration and serBilir marker.

The goal of Low Impact Development (LID) is to mimic the pre-development hydrologic regime of a catchment through infiltration, filtration, storage, evaporation, and detention of post-development runoff using small-scale hydrologic controls close to the source. A LID facility located in Northern Virginia was examined for pollutant removal and hydrologic performance. The treatment train included four in-line grass swales followed by a bioretention cell with a gravel base. The facility retained 85% of the rainfall. Influent and effluent pollutant loads were calculated using three common substitution methods for datasets censored by values below the analytical detection limit. The Summation of Loads (SOL) method was used to facilitate understanding of how data censoring affected performance results when substitution methods were used. The SOL analysis showed positive removal performance for most nutrient species, sediment, oxygen demanding substances, selected trace metals and total petroleum hydrocarbons. Negative performance was observed for oxidized nitrogen, total dissolved solids and oil & grease. LID facility influent and effluent loads were also compared using the Effluent Probability Method (EPM). The EPM analysis showed statistically significant (p d 0.05) pollutant load removal performance over the entire range of sampled events for total suspended solids, total phosphorus, total nitrogen, total Kjeldahl nitrogen, ammonia nitrogen, chemical oxygen demand, copper, zinc and alkalinity. EPM analysis did not show significant removals of oxidized nitrogen, total dissolved solids, orthophosphate phosphorus and hardness. / Master of Science