Dual response surface methodology (Vining and Myers (1990)) has been successfully used as a cost-effective approach to improve the quality of products and processes since Taguchi (Tauchi (1985)) introduced the idea of robust parameter design on the quality improvement in the United States in mid-1980s. The original procedure is to use the mean and the standard deviation of the characteristic to form a dual response system in linear model structure, and to estimate the model coefficients using least squares methods. In this dissertation, a Bayesian hierarchical approach is proposed to model the dual response system so that the inherent hierarchical variance structure of the response can be modeled naturally. The Bayesian model is developed for both univariate and multivariate dual response surfaces, and for both fully replicated and partially replicated dual response surface designs. To evaluate its performance, the Bayesian method has been compared with the original method under a wide range of scenarios, and it shows higher efficiency and more robustness. In applications, the Bayesian approach retains all the advantages provided by the original dual response surface modelling method. Moreover, the Bayesian analysis allows inference on the uncertainty of the model parameters, and thus can give practitioners complete information on the distribution of the characteristic of interest. / Ph. D.

One single primary study is only a little piece of a bigger puzzle. Meta-analysis is the statistical combination of results from primary studies that address a similar question. The most general case is the random-effects model, in where it is assumed that for each study the vector of outcomes T_i~N(θ_i,Σ_i ) and that the vector of true-effects for each study is θ_i~N(θ,Ψ). Since each θ_i is a nuisance parameter, inferences are based on the marginal model T_i~N(θ,Σ_i+Ψ). The main goal of a meta-analysis is to obtain estimates of θ, the sampling error of this estimate and Ψ. Standard meta-analysis techniques assume that Σ_i is known and fixed, allowing the explicit modeling of its elements and the use of Generalized Least Squares as the method of estimation. Furthermore, one can construct the variance-covariance matrix of standard errors and build confidence intervals or ellipses for the vector of pooled estimates. In practice, each Σ_i is estimated from the data using a matrix function that depends on the unknown vector θ_i. Some alternative methods have been proposed in where explicit modeling of the elements of Σ_i is not needed. However, estimation of between-studies variability Ψ depends on the within-study variance Σ_i, as well as other factors, thus not modeling explicitly the elements of Σ_i and departure of a hierarchical structure has implications on the estimation of Ψ. In this dissertation, I develop an alternative model for random-effects meta-analysis based on the theory of hierarchical models. Motivated, primarily, by Hoaglin's article "We know less than we should about methods of meta-analysis", I take into consideration that each Σ_i is unknown and estimated by using a matrix function of the corresponding unknown vector θ_i. I propose an estimation method based on the Minimum Covariance Estimator and derive formulas for the expected marginal variance for two effect sizes, namely, Pearson's moment correlation and standardized mean difference. I show through simulation studies that the proposed model and estimation method give accurate results for both univariate and bivariate meta-analyses of these effect-sizes, and compare this new approach to the standard meta-analysis method.

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.

In this thesis, we study several different renormalization group (RG) methods, including the conventional Wilson renormalization group, Monte Carlo renormalization group (MCRG), exact renormalization group (ERG, or sometimes called functional RG), and tensor renormalization group (TRG). We use the two dimensional nearest neighbor Ising model to introduce many conventional yet important concepts. We then generalize the model to Dyson's hierarchical model (HM), which has rich phase properties depending on the strength of the interaction. The partition function zeros (Fisher zeros) of the HM model in the complex temperature plane is calculated and their connection with the complex RG flows is discussed. The two lattice matching method is used to construct both the complex RG flows and calculate the discrete β functions. The motivation of calculating the discrete β functions for various HM models is to test the matching method and to show how physically relevant fixed points emerge from the complex domain. We notice that the critical exponents calculated from the HM depend on the blocking parameter b. This motivated us to analyze the connection between the discrete and continuous RG transformation. We demonstrate numerical calculations of the ERG equations. We discuss the relation between Litim and Wilson-Polchinski equation and the effect of the cut-off functions in the ERG calculation. We then apply methods developed in the spin models to more complicated and more physically relevant lattice gauge theories and lattice quantum chromodynamics (QCD) like theories. Finite size scaling (FSS) technique is used to analyze the Binder cumulant of the SU(2) lattice gauge model. We calculate the critical exponent nu and omega of the model and show that it is in the same universality class as the three dimensional Ising model. Motivated by the walking technicolor theory, we study the strongly coupled gauge theories with conformal or near conformal properties. We compare the distribution of Fisher zeros for lattice gauge models with four and twelve light fermion flavors. We also briefly discuss the scaling of the zeros and its connection with the infrared fixed point (IRFP) and the mass anomalous dimension. Conventional numerical simulations suffer from the critical slowing down at the critical region, which prevents one from simulating large system. In order to reach the continuum limit in the lattice gauge theories, one needs either large volume or clever extrapolations. TRG is a new computational method that may calculate exponentially large system and works well even at the critical region. We formulate the TRG blocking procedure for the two dimensional O(2) (or XY ) and O(3) spin models and discuss possible applications and generalizations of the method to other spin and lattice gauge models. We start the thesis with the introduction and historical background of the RG in general.

Genetic data analysis has been capturing a lot of attentions for understanding the mechanism of the development and progressing of diseases like cancers, and is crucial in discovering genetic markers and treatment targets in medical research. This dissertation focuses on several important issues in genetic data analysis, graphical network modeling, feature selection, and covariance estimation. First, we develop a gene network modeling method for discrete gene expression data, produced by technologies such as serial analysis of gene expression and RNA sequencing experiment, which generate counts of mRNA transcripts in cell samples. We propose a generalized linear model to fit the discrete gene expression data and assume that the log ratios of the mean expression levels follow a Gaussian distribution. We derive the gene network structures by selecting covariance matrices of the Gaussian distribution with a hyper-inverse Wishart prior. We incorporate prior network models based on Gene Ontology information, which avails existing biological information on the genes of interest. Next, we consider a variable selection problem, where the variables have natural grouping structures, with application to analysis of chromosomal copy number data. The chromosomal copy number data are produced by molecular inversion probes experiments which measure probe-specific copy number changes. We propose a novel Bayesian variable selection method, the hierarchical structured variable se- lection (HSVS) method, which accounts for the natural gene and probe-within-gene architecture to identify important genes and probes associated with clinically relevant outcomes. We propose the HSVS model for grouped variable selection, where simultaneous selection of both groups and within-group variables is of interest. The HSVS model utilizes a discrete mixture prior distribution for group selection and group-specific Bayesian lasso hierarchies for variable selection within groups. We further provide methods for accounting for serial correlations within groups that incorporate Bayesian fused lasso methods for within-group selection. Finally, we propose a Bayesian method of estimating high-dimensional covariance matrices that can be decomposed into a low rank and sparse component. This covariance structure has a wide range of applications including factor analytical model and random effects model. We model the covariance matrices with the decomposition structure by representing the covariance model in the form of a factor analytic model where the number of latent factors is unknown. We introduce binary indicators for estimating the rank of the low rank component combined with a Bayesian graphical lasso method for estimating the sparse component. We further extend our method to a graphical factor analytic model where the graphical model of the residuals is of interest. We achieve sparse estimation of the inverse covariance of the residuals in the graphical factor model by employing a hyper-inverse Wishart prior method for a decomposable graph and a Bayesian graphical lasso method for an unrestricted graph.

This dissertation presents a Bayesian hierarchical model to combine two-resolution metrology data for inspecting the geometric quality of manufactured parts. The high- resolution data points are scarce, and thus scatter over the surface being measured, while the low-resolution data are pervasive, but less accurate or less precise. Combining the two datasets could supposedly make a better prediction of the geometric surface of a manufactured part than using a single dataset. One challenge in combining the metrology datasets is the misalignment which exists between the low- and high-resolution data points. This dissertation attempts to provide a Bayesian hierarchical model that can handle such misaligned datasets, and includes the following components: (a) a Gaussian process for modeling metrology data at the low-resolution level; (b) a heuristic matching and alignment method that produces a pool of candidate matches and transformations between the two datasets; (c) a linkage model, conditioned on a given match and its associated transformation, that connects a high-resolution data point to a set of low-resolution data points in its neighborhood and makes a combined prediction; and finally (d) Bayesian model averaging of the predictive models in (c) over the pool of candidate matches found in (b). This Bayesian model averaging procedure assigns weights to different matches according to how much they support the observed data, and then produces the final combined prediction of the surface based on the data of both resolutions. The proposed method improves upon the methods of using a single dataset as well as a combined prediction without addressing the misalignment problem. This dissertation demonstrates the improvements over alternative methods using both simulated data and the datasets from a milled sine-wave part, measured by two coordinate measuring machines of different resolutions, respectively.

<p>By concentrating materials and increasing the speed with which rainfall is conveyed off of the landscape, nearly all forms of land use change lead to predictable shifts in the hydrologic, thermal, and chemical regimes of receiving waters that can lead to the local extirpation of sensitive aquatic biota. In Central Appalachian river networks, alkaline mine drainage (AlkMD) derived from mountaintop removal mining for coal (MTM) noticeably simplifies macroinvertebrate communities. In this dissertation, I have used this distinct chemical regime shift as a platform to move beyond current understanding of chemical pollution in river networks. In Chapter Two, I applied a new model, the Hierarchical Diversity Decision Framework (HiDDeF) to a macroinvertebrate dataset along a gradient of AlkMD. By using this new modeling tool, I showed that current AlkMD water quality standards allow one-quarter of regional macroinvertebrates to decline to half of their maximum abundances. In Chapter Three, I conducted a field study in the Mud River, WV to understand how AlkMD influences patterns in aquatic insect production. This work revealed roughly 3-fold declines in annual production of sensitive taxa throughout the year in reaches affected by AlkMD. These declines were more severe during summer base flow when pollutant concentrations were higher, thereby preventing sensitive organisms from completing their life cycles. Finally, in Chapter Four I described the idea of chemical fragmentation in river networks by performing a geospatial analysis of chemical pollution in Central Appalachia. In this work I showed that the ~30% of headwaters that remain after MTM intensification over the last four decades support ~10% of macroinvertebrates not found in mined reaches. Collectively my work moves beyond the simple tools used to understand the static, local consequences of chemical pollution in freshwater ecosystems.</p> / Dissertation

Pollution studies have sought to understand the relationships between adverse health effects and harmful exposures. Many environmental health studies are predicated on the idea that each exposure has both acute and long term health effects that need to be accurately mapped. Considerable work has been done linking air pollution to deleterious health outcomes but the underlying biological pathways and contributing sources remain difficult to identify. There are many statistical issues that arise in the exploration of these longitudinal study designs such as understanding pathways of effects, addressing missing data, and assessing the health effects of multipollutant mixtures. To this end this dissertation aims to address the afore mentioned statistical issues. Our first contribution investigates the mechanistic pathways between air pollutants and measures of cardiac electrical instability. The methods from chapter 1 propose a path analysis that would allow for the estimation of health effects according to multiple paths using structural equation models. Our second contribution recognizes that panel studies suffer from attrition over time and the loss of data can affect the analysis. Methods from Chapter 2 extend current regression calibration approaches by imputing missing data through the use of moving averages and assumed correlation structures. Our last contribution explores the use of factor analysis and two-stage hierarchical regression which are two commonly used approaches in the analysis of multipollutant mixtures. The methods from Chapter 3 attempt to compare the performance of these two existing methodologies for estimating health effects from multipollutant sources.

Many prior distributions are suggested for variance parameters in the hierarchical model. The “Non-informative” interval of the conjugate inverse-gamma prior might cause problems. I consider three priors – conjugate inverse-gamma, log-normal and truncated normal for the variance parameters and do the numerical analysis on Gelman’s 8-schools data. Then with the posterior draws, I compare the Bayesian credible intervals of parameters using the three priors. I use predictive distributions to do predictions and then discuss the differences of the three priors suggested. / text

Hierarchical models provide a flexible way of modeling complex behavior. However, the complicated interdependencies among the parameters in the hierarchy make training such models difficult. MCMC methods have been widely used for this purpose, but can often only approximate the necessary distributions. Acceptance-rejection sampling allows for perfect simulation from these often unnormalized distributions by drawing from another distribution over the same support. The efficacy of acceptance-rejection sampling is explored through application to a small dataset which has been widely used for evaluating different methods for inference on hierarchical models. A particular algorithm is developed to draw variates from the posterior distribution. The algorithm is both verified and validated, and then finally applied to the given data, with comparisons to the results of different methods.