We propose and discuss two distinct and separate innovative Bayesian models. In the first model, we propose a replacement for standard statistical methodologies for longitudinal sexual behavior data. HIV intervention trials generally collect sexual behavior data repeatedly over time and involve multiple outcomes including the number of partners which are nested in subjects and the number of protected and unprotected sex acts with each partner which are inherently nested within partners. The data is further complicated by characteristics of both partners and acts. Partners can be HIV$^+$ or HIV$^-$ while sex acts can be protected or unprotected. Properly modeling these outcomes and distinguishing these characteristics is critical. Here we use a multilevel multivariate Bayesian model for modeling sexual behavior outcomes. The proposed model accounts for the full complexity of sexual behavior allowing for simultaneous modeling of the number of partners and the number of sex acts with each partner, differentiation of behavior by partner serostatus, accounting for study eligibility criterions associated with the outcome of interest, and correlations between observations with the same subject, observations with the same partner, and observations across time. We further show that the proposed model can be used to quantify and draw inference on seroadaptive behaviors. Seroadaptive behaviors describe behaviors that vary based on the HIV status of partners with the goal of reducing the risk of transmission. The model is used to analyze data from the Healthy Living Project.

In the second half of this thesis, we explore a novel extension to the Dirichlet process mixture (DPM) model to accommodate longitudinal data. Longitudinal data is characterized by two features. First, the data are a function of time implying dependence between sampling densities across time. Second, the \emph{same} subjects are repeatedly measured over time. The standard DPM model is a nonparametric Bayesian model that naturally clusters similar observations together and assigns a single value to each cluster. It can be used to model an unknown density but addresses neither of these two features in longitudinal data. A number of current extensions of the DPM model can accommodate dependent distributions which could be used to model the sampling distributions at each time point addressing the first feature. However, assumptions in these extensions imply these models do not take advantage of the second feature of longitudinal data where the \emph{same} subjects are followed over time. To account for both features, we propose the cluster memory Dirichlet process mixture (cmDPM) model extending the DPM model to properly accommodate longitudinal data. In the cmDPM model, subjects are modeled as a DPM model at baseline. Cluster assignments at future time points depend on where the subject was previously clustered. Each subject may retain their cluster from the previous time point with some nonzero probability. This implies that at later times, subjects are no longer exchangeable and their observed values depend on their previous clustering history. Clusters that are retained over time evolve through a time dependent process. The cmDPM model extends the DPM to use both the information of where the subject was previously clustered and the value assigned to that cluster to model subject data at the current time point.

Meta-analysis (MA) combines multiple studies to estimate a quantity of interest. Some existing MA models have shortcomings in the form of 1) inappropriate inference targets, 2) strong assumptions about how studies are sampled, and 3) prior distributions for variance parameters with inadequate shrinkage.

In Chapter 2 we build a three-random effect (3RE) Bayesian random effects MA model for observational contingency table data as an extension of the standard two-random effect (2RE) model. We add a random effect for the log-odds of having a risk factor with random effects for the log-odds of an event and for the log-odds ratio of the event for those with or without the risk factor. The 3RE model allows for calculation of more statistics than the 2RE model, and we define a novel estimand for statistics calculated from contingency tables -- the expected value of a statistic for a new study given the hyperparameters. The new estimand shows less bias and higher 95% credible interval coverage as compared with a naive plug-in estimator. We apply the model to a dataset of studies on patients presenting to the emergency department with syncope. In Chapter 3 we propose a new approach to combining multiple selection models for publication bias using Bayesian stacking of posterior distributions. We demonstrate the effectiveness of stacking selection models through simulations and real datasets that exhibit symptoms of publication bias.

Chapter 4 proposes a new class of prior distributions for the covariance matrix of random effects in MA. The new priors allow random effects variances to be shrunk towards zero and for shrinkage of correlations between random effects. We show through both synthetic and real data examples that the new prior distributions lead to less diffuse posterior distributions and shorter 95% credible intervals in a 3RE MA model for observational data and an arm-based network meta-analysis (AB-NMA) model for randomized controlled trials.

We develop models for longitudinal count data with a large number of zeros, a feature known as zero-inflation. Familiar distributions for modeling count data (Poisson, binomial, negative binomial) often do not account for the observed frequency of zeros. Further, in longitudinal data, the same subjects are repeatedly measured over time inducing correlation between sets of measurements on the same individual. Modeling of longitudinal data that does not account for this correlation can give rise to misleading inferences. This dissertation develops three classes of models for longitudinal count data: (i) a Bayesian longitudinal hurdle model for data with prespecified measurement times, (ii) a Bayesian longitudinal hurdle model for data with varying measurement times, and (iii) a multivariate longitudinal zero-inflated Poisson model. Approach (i) is an analysis of the number of days of heaving drinking in a study of screening, brief intervention, and referral to treatment (SBIRT), and approaches (ii) and (iii) are motivated by analyses of the Linking Inmates to Care (LINK LA) study. Building on two-part models that predict non-zero versus zero outcomes while incorporating assumptions about the distribution of non-zero outcomes, the newly developed methods use mixed-effect modeling strategies to account for irregular measurement times and correlated patterns in count data beyond those reflected in random intercept models. The superiority of the proposed methods over random intercept models is established using goodness-of-fit metrics that consider the number of model parameters, and the appeal of modeling multiple count outcomes simultaneously is reflected in Bayesian credible intervals that point to non-zero correlations among the respective count outcomes. We build upon previous longitudinal zero-inflated and hurdle models by introducing time varying random effects in the count models with random effects distributed a priori as multivariate normal with a parameterized covariance matrix. We propose several covariance models, which improve fit over random intercept models in both the SBIRT and LINK LA data. We introduce latent time varying main and random effects to allow count rates and zero probabilities to change with time since intervention and include exposure offsets to account for varying times over which counts are recorded. Finally, for use with multivariate data, we propose a multivariate longitudinal zero-inflated Poisson model for observations with varying exposure, which we use to simultaneously model three different kinds of doctor visits recorded in the LINK LA study.

Topic models are probabilistic models used to abstract topical information from collections of text documents. Documents are modeled as probability distributions over latent topics and topics are modeled as probability distributions over words. In a single collection of documents, topics are global, that is, they are shared across the multiple documents in the collection. A nested document collection has documents that are nested inside a higher order structure, for example, stories in a book, articles in a journal, or web pages in a web site. Regular topic models ignore the nesting and treat all documents as distinct. In contrast, a nested document collection, such as web pages nested in web sites, web pages of the same web sites share similarities with each other that they do not share with web pages of other web sites. Regular topic models allow inferences about each web page individually; they are not suited for making inferences about an entire web site.

We propose hierarchical local topic models that place a hierarchical prior on web page topic distributions and explicitly model local topics, or topics that are unique to one web site. The hierarchical prior asserts that web page topic distributions vary around their web site topic distribution and that web site topic distributions vary around a global topic distribution.Explicitly modeling local topics reduces the number of global topics needed, identifies the local topics and their owning web site, and lets us adjust inferences about how topics are covered.

We propose hierarchical topic presence models that place a sparsity inducing prior on topic distributions; they let us model the presence of topics in web sites, web pages, or both web sites and web pages. Topic presence in a web site can be modeled with logistic regression, as a function of covariates.

We apply hierarchical topic presence models to identify health topics in United States county health department web sites, estimate the percent of web sites that cover particular health topics, and identify demographic predictors of topic presence for human immunodeficiency virus (HIV) and opioid use disorder (OUD) topics.

Illicit drug use and concomitant problems such as high incarceration rates pose tremendous challenges to those directly involved and to society as a whole. In recent years, many medical, public health, and social science researchers have conducted studies to characterize the nature of these problems and look for potential solutions. These studies often record events such as subjects relapsing into drug use following periods of abstinence, and interest centers on finding or assessing the impact of risk factors for event occurrence. While analyses of these studies have often employed advanced statistical techniques borrowed from the social science literature, there have been few applications of modern Bayesian techniques or advanced event time models. In this dissertation, we develop a general Bayesian event history model with supporting computing tools, and we apply this work in a substance abuse context.

Because the field of survival and event history analysis is so broad, we first provide an extensive review of concepts and literature relevant to our subsequent methodological work. We then describe in detail our general Bayesian event history model. This model combines a number of features which are frequently omitted from analyses because available event time analysis software does not allow their simultaneous use. These features include simultaneous semiparametric incorporation of multiple continuous time-varying covariates, multiple event types or competing risks, and recurrent at-risk episodes. We provide novel Markov chain Monte Carlo (MCMC) algorithms for obtaining posterior inferences from this model, evaluate their performance, and apply them to data on recurrent episodes of cocaine use in a population of illicit drug users.

Next, we extend our general Bayesian event history model to a full multistate model for histories in which subjects pass in between discrete states. In these histories, the state transitions are the events of interest, and the numbers and types of covariates and possible transition events depend on the subject's current state. We apply this multistate model to lifetime histories of cocaine use and incarceration following first use of cocaine. Finally, we conclude the dissertation with a description of our software implementation of our models and a discussion of future projects.

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