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.

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.

In studying the progression of a disease and to better predict time to death (survival data), investigators often collect repeated measures over time (longitudinal data) and are interested in testing the association between risk factors, including collected repeated measures, and time to death. One such example is testing the association between the biomarker serum albumin that is measured repeatedly on end-stage renal disease (ESRD) patients. A modeling framework that is capable of modeling longitudinal and survival outcomes simultaneously is called a joint longitudinal-survival model.

Joint longitudinal-survival models have received a great deal of attention over the past years where many different joint models have been proposed. Joint models commonly make parametric assumptions on either the functional form of the repeated measures or on the distribution of survival times. In this dissertation we are interested in joint models that are robust to common parametric and semi-parameteric survival and longitudinal assumptions. We propose a flexible Bayesian joint longitudinal-survival framework that avoids common parametric and semi-parameteric assumptions. More specifically, our modeling framework incorporates a flexible longitudinal component by utilizing Gaussian process (GP) technique. This technique avoids any explicit functional assumption on the trajectory of the repeated measures. Our modeling framework also uses Dirichlet process (DP) prior to avoid explicit distributional assumptions on survival times.

We further extend our framework to modeling multiple longitudinal processes simultaneously. We propose a multivariate joint longitudinal-survival technique to jointly model the association between multiple longitudinal processes with survival outcomes. Our proposed technique is capable of taking correlation between longitudinal processes into account. This is particularly useful when observed measures from different longitudinal processes are taken at different frequencies. That means, some longitudinal processes are observed less frequently compared to other longitudinal processes. By jointly modeling these processes, one can take the correlation between the processes into account, and hence, better estimate the trajectory of the processes including those less frequent ones.

Our proposed joint modeling frameworks use Dirichlet process techniques. Therefore, understanding parameter estimation in these models is vital. Using synthetic longitudinal and survival data, we compare parameter estimation under DPM models as opposed to commonly used parametric techniques. We are particularly interested in evaluation of the performance of the model in parameter estimation when a population consists of sub-populations with latent features that are different across subgroups. We propose a Dirichlet process mixture survival model that is capable of detecting latent subpopulations characterized by differing baseline risks for mortality. Our proposed technique is particularly useful when interest lies in estimation of the conditional effect of covariates as opposed to estimates that are marginalized across all subpopulations.

Throughout, our work is motivated by data on patients with end stage renal disease (ESRD), a condition where the kidneys are no longer capable of cleaning blood sufficiently enough to sustain life. In this context a modeling framework capable of finding mortality-related biomarkers, which are measured longitudinally over time, can significantly help physicians and practitioners to lower mortality among these patients.

Analysis of time-to-event data using Cox's proportional hazards model is ubiquitous in scientific research. Most commonly, a sample is taken from the population of interest and covariate information is collected on everyone. If the event of interest is rare and it is difficult or not feasible to collect full covariate information for all study participants, the nested case-control design reduces costs with minimal impact on inferential precision. However, no work has been done to investigate the performance of the nested case-control design under model mis-specification. In this dissertation we show that under model mis-specification the quantity being estimated under the nested case-control design will depend not only on the censoring distribution, but also on the number of controls sampled at each event time. This is true in the case of a binary covariate when the proportional hazards assumption is not satisfied, and in the case of a continuous covariate where the functional form is mis-specified. We propose several estimators that allow us to recover the statistic that would have been computed under the full cohort data as well as a censoring-robust estimator. We also investigate the performance of time-dependent receiver operating characteristic curves under the full cohort and nested case-control sampling scenarios. We show that if the risk score model is mis-specified, estimates of the area under the curve will also depend on the censoring distribution and we propose the use of censoring-robust risk scores that allow us to recover censoring-independent area under

There is a need to develop robust and efficient versions of commonly used biostatistical methods in order to maintain valid statistical inference while remaining comprehensible to other investigators and recognizable to regulatory agencies. In this dissertation, we will explore the intersection of robustness and efficiency in realistic and complex biostatistical settings through two clinically-motivated examples. The first relates to the validation trial of a novel bleed severity classification system, where a kappa statistic was used to assess inter- and intra-rater reliability. We investigate how clustering and within-category item non-exchangeability seen in the trial affect the analytic variance estimate of kappa, and propose a new variance estimator to correct for variance inflation. We also explore how such a trial could be made more efficient with the use of group sequential design methodology. We show that implementation of traditional boundary estimation methods on kappa yield approximately correct operating characteristics, and discuss study design considerations particular to kappa. The second example relates to the discovery of novel biomarkers in Alzheimer’s disease. New biomarkers are often explored within observational study frameworks and while there is no artificial intervention being tested, efficient procedures – such as group sequential design – are still desirable to assess the viability of potentially large numbers of prospective biomarker candidates using scarce or difficult-to-obtain samples. We examine how failure to incorporate confounding mechanisms when designing group sequential observational studies affects operating characteristics, and develop algorithms and processes to correct them.