This book discusses Bayesian dose-response modeling in small samples applied to two different settings. The first setting is early phase clinical trials, and the second setting is toxicology studies in cancer risk assessment. In early phase clinical trials, experimental units are humans who are actual patients. Prior to a clinical trial, opinions from multiple subject area experts are generally more informative than the opinion of a single expert, but we may face a dilemma when they have disagreeing prior opinions. In this regard, we consider compromising the disagreement and compare two different approaches for making a decision. In addition to combining multiple opinions, we also address balancing two levels of ethics in early phase clinical trials. The first level is individual-level ethics which reflects the perspective of trial participants. The second level is population-level ethics which reflects the perspective of future patients. We extensively compare two existing statistical methods which focus on each perspective and propose a new method which balances the two conflicting perspectives. In toxicology studies, experimental units are living animals. Here we focus on a potential non-monotonic dose-response relationship which is known as hormesis. Briefly, hormesis is a phenomenon which can be characterized by a beneficial effect at low doses and a harmful effect at high doses. In cancer risk assessments, the estimation of a parameter, which is known as a benchmark dose, can be highly sensitive to a class of assumptions, monotonicity or hormesis. In this regard, we propose a robust approach which considers both monotonicity and hormesis as a possibility. In addition, We discuss statistical hypothesis testing for hormesis and consider various experimental designs for detecting hormesis based on Bayesian decision theory. Past experiments have not been optimally designed for testing for hormesis, and some Bayesian optimal designs may not be optimal under a wrong parametric assumption. In this regard, we consider a robust experimental design which does not require any parametric assumption.

In epidemiologic studies, researchers are commonly interested in quantifying geospatial effects on the incidence of disease to illustrate health disparities potentially attributable to environmental, demographic, and/or socioeconomic factors. The current research we focus on the development of advanced statistical modeling methods to determine how the pattern of disease outcome changes over geographical location.

We first extend and illustrate the utility of MapGAM, a user-friendly R package that provides a unified framework for estimating, predicting and drawing inference on covariate-adjusted spatial effects using individual-level data. The package also facilitates visualization of spatial effects via automated mapping procedures. MapGAM estimates covariate-adjusted spatial associations with a univariate outcome using generalized additive models that include a non-parametric bivariate smooth term of geolocation parameters. Estimation and mapping methods are implemented for continuous, discrete, and right-censored survival data.

Next, we note that smoothing approaches commonly implemented in generalized additive models and used in spatial analyses assume the amount of smoothing to be equal across geographical regions. The result is that some regions tend to be under-smoothed, while others are over-smoothed. We extend the work of Yue, etc. (2010) in the context of brain imaging analysis and propose a hierarchical Bayesian adaptive thin-plate spline that allows for spatial smoothing of continuous, binary and count outcomes along with the ability to adjust for potential confounding factors. The proposed method allows for the amount of smoothing to flexibly vary depending on the local extent of spatial effect by using nonstationary spatial Gaussian Markov random fields. Performance of the approach is evaluated via simulation and the proposed methodology is applied to an epidemiologic study investigating spatial heterogeneity in the risk of preterm birth among Massachusetts residents.

Finally, we further extend the proposed adaptive smoothing techniques to the case of right-censored survival outcome data. Simulation is used to compare the performance of the proposed method to commonly used non-adaptive smoothing method, and the approach is applied to data from an epidemiology study seeking to quantify spatial variability in the survival times of advanced-stage ovarian cancer patients in California.

In quantifying heterogeneity in time-to-event data, potential biomarker information and demographic status reflecting key pathophysiology at the individual level are increasingly available. Typical analysis of time-to-event data relies on observed event data. To ensure adequate statistical power, longer follow-up periods or additional participants are required, adding enormously to study costs. Incorporating biomarker information can reduce the cost of data collection and maximize statistical power.

There are, however, several drawbacks to the direct incorporation of biomarkers for identification and analysis. Specifically, most models assume time-invariant effects, independent censoring, and complete information. Little work has been done to assess the robustness of the currently used models beyond these assumptions.

The Cox proportional hazards model is the most commonly used model for time-to-event data. This model compares observed covariates to the weighted average of covariates in the risk set. The Cox model consistently estimates a weighted time-averaged effect under proportional hazards. However, in the field of Alzheimer's disease, the proportional hazard model often fails as the covariate effect diminishes due to disease progression. Directly applying the model ignores the potential changes in the underlying effect. In addition, the assumption of independent censoring may fail to acknowledge possible associations between biomarkers and the missing data mechanism. As a result, the observed event in later follow-up times may be underrepresented relative to the overall population due to censoring. Currently, there is no existing research to address the above issues simultaneously while being robust to possible violations of assumptions and adaptive to various types of biomarker information.

Given the increasing availability of repeated biomarker measurements, there is a growing interest in jointly modeling longitudinal and time-to-event data to maximize statistical information on the association between potential biomarkers and disease progression. Within longitudinal data, it is often observed that subgroups exist among populations, where the underlying development can be clustered into different patterns. These clustering patterns offer valuable insights into the natural history of diseases, including their progression, risk factors, and potential causes. In practice, longitudinal data frequently encounter missing values. While typical clustering methods handles intermittent missing values through imputation methods, in disease studies, monotone missingness often occurs in longitudinal data when measurements are lost after a specific time due to terminal events and censoring. Existing imputation methods may introduce bias when attempting to recover trajectories of similar lengths. However, there has been little examination of the currently used methods on longitudinal clustering with monotone missingness.

This dissertation aims to develop a flexible and robust statistical model to evaluate predictors of time-to-event data. The resulting estimator will be consistent and robust under violated assumptions and repeated measures. In Chapter 3, we proposed a reweighted censoring robust estimator using censoring weights and conditional covariate variance. In Chapter 4, we introduced an robust variance estimator based on the influence function for the estimator proposed in Chapter 3. In Chapter 5, we investigated the performance of the current longitudinal clustering method under the monotone missing censoring mechanism, and proposed a shape-based longitudinal partial mapping clustering method to complement the estimator proposed in Chapter 3.

Epidemiologists frequently aim to quantify geospatial heterogeneity in disease occurrence to identify relevant hidden health disparities. With the growing prevalence of individual-level point-referenced data, generalized additive models (GAMs) are becoming increasingly popular to map geospatial disease risk patterns while adjusting for confounding effects when the study is a cross-sectional one with an exponential family response. In the meanwhile, local regression smoothers are frequently adopted for spatial effects estimation in GAM framework by researchers partially due to their intuitive ideas and adaptation to changing population density.

However, studies with records over a (potentially long) period of time, including those with repeated measurements on subjects, commonly come into play nowadays. For these studies, traditional GAMs could be problematic. Firstly, since data could be recorded over a period of time while spatial risk patterns should not be assumed to be invariant in many cases, statistical tools to access time-varying spatial effects are required. On the other hand, if the study is longitudinally designed, traditional GAMs could lead to incorrect inference due to their incapability of accomodating within-individual correlation.

This dissertation work sought to develop statistical methodologies to address these problems under the GAM framework with kernel smoothers, using local regression smoothers in particular. In Chapter 3, we proposed GAMs with stratified kernel smoothers that could be applied for time-specific spatial effects modeling. Based on the new class of GAMs, we further designed a hypothesis testing procedure to formally detect temporal heterogeneity of spatial effects. In Chapter 4 and 5, we incorporated random effects, as well as kernel smoothers, into GAM, resulting in a class of generalized additive mixed models (GAMMs) with kernel smoothers. We further elaborated the novel fitting and inference procedures for the proposed models.

Relevant empirical results showed the utility and advantages in model fitting under some fairly designed scenarios, with comparison to classic models. We further applied our proposed methods in a study on birth defects in Massachusetts in Chapter 3 and a study on residents' serum PFOA concentration in Lubeck, WV, and Little Hocking, OH region.

Prediction of clinical outcomes is of primary scientific interest in many public health studies. While prediction assessment tools have been developed and refined for simple random samples and characterized by non-missing outcome data, modern epidemiologic datasets bring unique challenges to the assessment and selection of prediction models. These challenges include high-dimensional predictor spaces, non-random sampling, and censored and longitudinal outcomes. In this dissertation, we consider prediction assessment that incorporates these complexities. We first consider prediction in a high-dimensional predictor space setting with non-random sampling of study participants. We empirically illustrate the limitations of the weighted ridge regression estimator in this case and propose a novel estimator that allows for the adjustment of sampling weights in the ridge regression penalty structure to provide more generalizable predictions. We then consider an analytic estimate of the out-of-sample prediction error for regression based censored survival models, where performance is measured via the Brier score. We derive an analytic estimate of out-of-sample error under single population models with uncensored data and extend this to propose an algorithm for prediction assessment under Cox regression with covariate adjustment. Finally we consider the marginal prediction of a longitudinal process impacted by both differential follow-up and non-random sampling. We derive a cluster-wise re-sampling framework that incorporates unbalanced sampling at the subject level and within-subject correlation. We empirically illustrate the utility of the proposed framework via simulation. Throughout, the methods are applied to data predicting cognition in the setting of Alzheimer’s disease and to the prediction of access graft failure among hemodialysis patients.

The research presented in this thesis focuses on the analysis of data arising from matched case-control designs, with particular emphasis on case-crossover designs. We begin by providing a scientific example that motivates the research presented, highlighting statistical issues raised in addressing the scientific goal of study. We provide background and notation that lays the foundation for the remainder of the dissertation. The occurrence of repeated events per patient or cluster and an imbalance in cluster sizes poses statistical challenges in the analysis of case-crossover studies (or more generally in matched case-control studies). We begin with a background of existing methods, then focus on methods to estimate association parameters in matched cases control designs while accounting for within-subject correlation in the data. The methods discussed assume the willingness to break the individual matched case-control bonds within matched sets, thereby accounting for within-subject correlation directly in the estimation procedure. It is illustrated that existing estimation procedures can result in severe bias depending upon the number of repeated events per patient/cluster and the magnitude of covariate effect on the response.

Then, methods are discussed where it is no longer acceptable to break the matched case-control bonds. These methods employ substantially different weighting methods to obtain parameter estimates, and the resulting estimand consistently estimated by each procedure is investigated. We focus on the scenario of varying matched set sizes (varying cluster sizes), where effect modification exists across clusters. It is shown that currently implemented frequentist methods for analyzing case-crossover data with unbalanced cluster sizes force one to choose between weighting schemes that estimate marginal or conditionally-weighted covariate effects.

In order to directly model and contrast marginal and subject-specific estimates of association in matched case-control studies, a novel method for obtaining estimates is developed. The proposed methodology allows for simultaneous estimation of both marginal and subject-specific covariate effects by implementing a semi-parametric Bayesian hierarchical framework.

Throughout, the utility of the resulting methodology is illustrated using data obtained from a case-crossover study of children sampled from Orange County, CA seeking to quantify the effect of air pollution exposure on the risk of asthma-related hospital encounters.

Electronic health records (EHRs) have become a powerful resource for studying health outcomes. In time-to-event settings, EHRs are usually subject to interval censoring on the true time of the event. It is common to consider the outcome to be the time of diagnosis. Standard survival analysis tools for right-censored data, such as the Cox proportional hazards model, are commonly used to estimate covariate associations with the time-to-event in such settings. Patients may, however, have access to multiple health care providers across different systems. If patients seek care from external health systems (a phenomenon we call system migration), the diagnosis times within the observed system may be erroneously prolonged. No work has considered the performance of the Cox model under system migration. In this dissertation, we show that system migration related to the outcome of interest results in biased estimates of hazard ratios from the Cox model. We develop an extension to the Cox model that adjusts for system migration by 1) estimating the probability of system migration for each patient and 2) uses multiple imputation to adjust diagnosis times for patients identified as migrating across systems. A vital part of this method involves developing a prediction model from patient-specific system usage patterns for estimating the probability of system migration. To improve prediction assessment, we develop an estimator for time-dependent sensitivity and specificity in the recurrent event setting with unbalanced data across subpopulations. Finally, we consider the choice of time scales for assessing the relative risk of disease diagnosis. We compare the appropriateness of two commonly assumed time scales that define risk sets in the Cox model: the age time scale and the time-on-study time scale. Previous research has suggested that the age time scale, corresponding to birth as a time origin, is most appropriate for epidemiological studies. However, simulation studies have suggested that the time-on-study time scale with covariate adjustment for baseline age is more robust to misspecification of the time scale. We investigate the performance of the Cox model under each time scale under varying degrees of model misspecification and further assess the robustness of each approach when modeling time-varying covariates.

Assessing the impact of complex interventions on measurable health outcomes is a growing concern in health care and health policy. Interrupted time series (ITS) designs borrow from traditional case-crossover designs and function as quasi-experimental methodology that enables researchers to retrospectively analyze the impact of an intervention. Statistical models used to analyze ITS data a priori restrict the interruption's effect to a predetermined time point or censor data for which the intervention effects may not be fully realized, and neglect changes in the temporal dependence and variability. In addition, current methods limit the analysis to one hospital unit or entity and are not well specified for discrete outcomes (e.g., patient falls). This dissertation develops novel ITS methods based on segmented regression that address the aforementioned limitations.

We propose the 'Robust-ITS' model, a single-unit model able to estimate (rather than merely assume) the lagged effect of an intervention on a health outcome. Robust-ITS accounts for plausible differences in the mean, temporal dependence and variability of an outcome pre- and post-intervention. Next, we develop the 'Robust Multiple ITS' model as an extension of Robust-ITS for multi-unit data. Alongside Robust Multiple ITS, we propose the 'supremum Wald test', able to formally test for the existence of a change point across unit specific mean functions. Lastly, we present the 'Generalized Robust ITS' model, appropriate for outcomes whose underlying distribution belongs to the family of exponential distributions. Generalized Robust ITS expands the available methodology to adequately model multi-unit binary, count and rate ITS. The methods proposed allow researchers to test for the existence of and estimate the change point, borrow information across units in multi-unit settings, and test for differences in the mean function and correlation structure pre- and post-intervention. Throughout, the methodology is illustrated by analyzing patient centered data from a hospital that implemented and evaluated a new care delivery model in multiple units.

Most statistical methods assume that samples are representative of a target population of interest, but this assumption is commonly violated in biomedical applications with human volunteers. Study participants self-selection into a sample can cause it to be unrepresentative which in turn leads to sampling bias. When analyzing data from a biased sample, the sampling scheme must be accounted for to obtain inference and predictions that generalize to the target population. In this dissertation, we discuss an approach for addressing sampling bias by estimating sampling weights using an auxiliary data set. We then apply estimated sampling weights in the field of causal inference. We assess the impact on bias and variance of estimated causal effects when sampling weights are included or omitted when estimating propensity scores and propensity-adjusted causal effects. Lastly, we quantify the impact of sampling bias on estimates of the prediction error for a target population and compare estimation methods.