Hierarchical logistic regression models have gained popularity in recent years as algorithms and computer software for fitting them improved. Little research exists to provide measures for assessing model fit in this area. We extend goodness-of-fit measures used in the standard logistic setting to the hierarchical case. Using simulation studies we examine the performance of unweighted sums of squares, Pearson residual and Hosmer-Lemeshow statistics at various levels of the hierarchical model. Our results suggest such statistics do not offer reasonable performance in the hierarchical logistic model in terms of Type I error rates. We also develop Kernel smoothed versions of the statistics using level one residuals and a modified unweighted sum of squares statistic based on residuals at higher levels. Performance of these statistics, using Type I error rates, is satisfactory. We also describe power studies suggesting that these statistics have limited power in certain hierarchical settings. Finally, we discuss limitations of this work and possible future research.

Trees have long been used as a flexible way to build regression and classification models for complex problems. They can accommodate nonlinear response-predictor relationships and even interactive intra-predictor relationships. Tree based models handle data sets with predictors of mixed types, both ordered and categorical, in a natural way. The tree based regression model can also be used as the base model to build additive models, among which the most prominent models are gradient boosting trees and random forests. Classical training algorithms for tree based models are deterministic greedy algorithms. These algorithms are fast to train, but they usually are not guaranteed to find an optimal tree. In this paper, we discuss a Bayesian approach to building tree based models. In Bayesian tree models, each tree is assigned a prior probability based on its structure, and standard Monte Carlo Markov Chain (MCMC) algorithms can be used to search through the posterior distribution. This thesis is aimed at improving the computational efficiency and performance of Bayesian tree based models. We consider introducing new proposal or “moves” in the MCMC algorithm to improve the efficiency of the algorithm. We use temperature based algorithms to help the MCMC algorithm get out of local optima and move towards the global optimum in the posterior distribution. Moreover, we develop semi-parametric Bayesian additive trees models where some predictors enter the model parametrically. The technical details about using parallel computing to shorten the computing time are also discussed in this thesis.

When categorical data are misplaced into the wrong category, we say the data is affected by misclassification. This is common for data collection. It is well-known that naive estimators of category probabilities and coefficients for regression that ignore misclassification can be biased. In this dissertation, we develop methods to provide improved estimators and confidence intervals for a proportion when only a misclassified proxy is observed, and provide improved estimators and confidence intervals for regression coefficients when only misclassified covariates are observed. Following the introduction and literature review, we develop two estimators for a proportion, one which reduces the bias, and one with smaller mean square error. Then we will give two methods to find a confidence interval for a proportion, one using optimization techniques, and the other one using Fieller’s method. After that, we will focus on developing methods to find corrected estimators for coefficients of regression with misclassified covariates, with or without perfectly measured covariates, and with a known estimated misclassification/reclassification model. These correction methods use the score function approach, regression calibration and a mixture model. We also use Fieller’s method to find a confidence interval for the slope of simple regression with misclassified binary covariates. Finally, we use simulation to demonstrate the performance of our proposed methods.

Abstract not available

We regard texture as a realization of a stochastic process defined on the square lattice. The model chosen is a Markov Random Field, which incorporates both local and global interactions, and it is a modification of the autobinomial model introduced by Besag (1974) and used by Cross and Jain (1983) for texture generation and synthesis. A Monte Carlo procedure called the Gibbs sampler is used to generate realizations from the model. Examples show how the parameters of the Markov random field control the strength, direction, and range of the clustering in the image. The problem of estimating the model parameters from a sample of independent realizations of the process is studied. The traditional maximum likelihood estimator is found to be consistent and asymptotically normal, but not computationally feasible. An alternative method of estimation, bivariate pseudolikelihood, is proposed. Although computationally intense, this method is much easier to implement than maximum likelihood. Consistency of the estimators is investigated under two different sets of assumptions. Experiments are performed to assess the accuracy of the estimates. In addition, the estimated parameters are used to generate images that are visually compared to those arising from the original model.

No description available.

In recent years there has been an explosion of network data from seemingly all corners of science – from computer science to engineering, from biology to physics, and from finance to sociology. We face analogues of many of the same fundamental types of problems encountered in a ‘Statistics 101’ course when analyzing network data. Despite roughly 20 years of research in the area, one of the fundamental capabilities that we still lack is quantifying uncertainty through propagation of network error. We conduct basic research laying statistical foundations for uncertainty quantification of this type, within a handful of key paradigms, focusing on problems ranging from epidemics to experiments on networks, when at least a few network replicates are available. Specifically, we study causal inference on noisy networks, and estimation of epidemic reproduction numbers in network-based and non-network-based settings. Ultimately, our work will bring critical insight into how ‘noise’ at the level of observed network connectivity impacts critical inferences and decisions derived from data in complex network systems. / 2022-04-07T00:00:00Z

Networks are quickly becoming one of the most common data types across diverse disciplines from the biological to the social sciences. Consequently, the study of networks as data objects is fundamental to developing statistical methodology for answering complex scientific questions. In this dissertation, we provide statistical solutions to three tasks related to multiple networks. We first consider the task of prediction given a collection of observed networks. In particular, we provide a Bayesian approach to performing classification, anomaly detection, and survival analysis with network inputs. Our methodology is based on encoding networks as pairwise differences in the kernel of a Gaussian process prior and we are motivated by the goal of predicting preterm delivery using individual microbiome networks. We next consider the task of exploring reaction space in high-throughput chemistry, where the inputs to a reaction are two or more molecules. Our goal is to create a workflow that facilitates quick, low-cost, and effective analysis of reactions. In order to operationalize this goal, we develop a statistical approach that breaks the analysis into several steps based on four unique challenges that we identify. Each of these challenges requires careful consideration in creating our analysis plan. For instance, to address the fact that reactions are run on multiwell plates, we formulate our proposal as a constrained optimization problem; then, we leverage the underlying structure by realizing a plate as a bipartite graph, which allows us to reformulate the problem as a maximal edge biclique problem. These solutions are necessary to optimally navigate a large reaction space given limited resources, which is critical in the application of reaction chemistry, for example, to drug discovery. The final task we consider is the recovery of a network given a sample of noisy unlabeled copies of the network. Toward this end, we make a connection between the noisy network literature and the correlated Erdős–Rényi graph model, which allows us to employ results from graph matching. Research on multiple unlabeled networks has otherwise been underdeveloped but is emerging in areas such as differential privacy and anonymized networks, as well as measurement error in network construction. / 2022-10-25T00:00:00Z

No description available.

abstract: One of the premier technologies for studying human brain functions is the event-related functional magnetic resonance imaging (fMRI). The main design issue for such experiments is to find the optimal sequence for mental stimuli. This optimal design sequence allows for collecting informative data to make precise statistical inferences about the inner workings of the brain. Unfortunately, this is not an easy task, especially when the error correlation of the response is unknown at the design stage. In the literature, the maximin approach was proposed to tackle this problem. However, this is an expensive and time-consuming method, especially when the correlated noise follows high-order autoregressive models. The main focus of this dissertation is to develop an efficient approach to reduce the amount of the computational resources needed to obtain A-optimal designs for event-related fMRI experiments. One proposed idea is to combine the Kriging approximation method, which is widely used in spatial statistics and computer experiments with a knowledge-based genetic algorithm. Through case studies, a demonstration is made to show that the new search method achieves similar design efficiencies as those attained by the traditional method, but the new method gives a significant reduction in computing time. Another useful strategy is also proposed to find such designs by considering only the boundary points of the parameter space of the correlation parameters. The usefulness of this strategy is also demonstrated via case studies. The first part of this dissertation focuses on finding optimal event-related designs for fMRI with simple trials when each stimulus consists of only one component (e.g., a picture). The study is then extended to the case of compound trials when stimuli of multiple components (e.g., a cue followed by a picture) are considered. / Dissertation/Thesis / Doctoral Dissertation Statistics 2019