<p dir="ltr">This dissertation advances the field of causal inference by developing methodologies in the face of assumption violations. Traditional causal inference methodologies hinge on a core set of assumptions, which are often violated in the complex landscape of modern experiments and observational studies. This dissertation proposes novel methodologies designed to address the challenges posed by single or multiple assumption violations. By applying these innovative approaches to real-world datasets, this research uncovers valuable insights that were previously inaccessible with existing methods. </p><p><br></p><p dir="ltr">First, three significant sources of complications in causal inference that are increasingly of interest are interference among individuals, nonadherence of individuals to their assigned treatments, and unintended missing outcomes. Interference exists if the outcome of an individual depends not only on its assigned treatment, but also on the assigned treatments for other units. It commonly arises when limited controls are placed on the interactions of individuals with one another during the course of an experiment. Treatment nonadherence frequently occurs in human subject experiments, as it can be unethical to force an individual to take their assigned treatment. Clinical trials, in particular, typically have subjects that do not adhere to their assigned treatments due to adverse side effects or intercurrent events. Missing values also commonly occur in clinical studies. For example, some patients may drop out of the study due to the side effects of the treatment. Failing to account for these considerations will generally yield unstable and biased inferences on treatment effects even in randomized experiments, but existing methodologies lack the ability to address all these challenges simultaneously. We propose a novel Bayesian methodology to fill this gap. </p><p><br></p><p dir="ltr">My subsequent research further addresses one of the limitations of the first project: a set of assumptions about interference structures that may be too restrictive in some practical settings. We introduce a concept of the ``degree of interference" (DoI), a latent variable capturing the interference structure. This concept allows for handling arbitrary, unknown interference structures to facilitate inference on causal estimands. </p><p><br></p><p dir="ltr">While randomized experiments offer a solid foundation for valid causal analysis, people are also interested in conducting causal inference using observational data due to the cost and difficulty of randomized experiments and the wide availability of observational data. Nonetheless, using observational data to infer causality requires us to rely on additional assumptions. A central assumption is that of \emph{ignorability}, which posits that the treatment is randomly assigned based on the variables (covariates) included in the dataset. While crucial, this assumption is often debatable, especially when treatments are assigned sequentially to optimize future outcomes. For instance, marketers typically adjust subsequent promotions based on responses to earlier ones and speculate on how customers might have reacted to alternative past promotions. This speculative behavior introduces latent confounders, which must be carefully addressed to prevent biased conclusions. </p><p dir="ltr">In the third project, we investigate these issues by studying sequences of promotional emails sent by a US retailer. We develop a novel Bayesian approach for causal inference from longitudinal observational data that accommodates noncompliance and latent sequential confounding. </p><p><br></p><p dir="ltr">Finally, we formulate the causal inference problem for the privatized data. In the era of digital expansion, the secure handling of sensitive data poses an intricate challenge that significantly influences research, policy-making, and technological innovation. As the collection of sensitive data becomes more widespread across academic, governmental, and corporate sectors, addressing the complex balance between making data accessible and safeguarding private information requires the development of sophisticated methods for analysis and reporting, which must include stringent privacy protections. Currently, the gold standard for maintaining this balance is Differential privacy. </p><p dir="ltr">Local differential privacy is a differential privacy paradigm in which individuals first apply a privacy mechanism to their data (often by adding noise) before transmitting the result to a curator. The noise for privacy results in additional bias and variance in their analyses. Thus, it is of great importance for analysts to incorporate the privacy noise into valid inference.</p><p dir="ltr">In this final project, we develop methodologies to infer causal effects from locally privatized data under randomized experiments. We present frequentist and Bayesian approaches and discuss the statistical properties of the estimators, such as consistency and optimality under various privacy scenarios.</p>

To obtain a better understanding of the pharmacokinetic and/or pharmacodynamic characteristics of an investigated treatment, clinical data is often analysed with nonlinear mixed effects modelling. The developed models can be used to design future clinical trials or to guide individualised drug treatment. Missing data is a frequently encountered problem in analyses of clinical data, and to not venture the predictability of the developed model, it is of great importance that the method chosen to handle the missing data is adequate for its purpose. The overall aim of this thesis was to develop methods for handling missing data in the context of nonlinear mixed effects models and to compare strategies for handling missing data in order to provide guidance for efficient handling and consequences of inappropriate handling of missing data. In accordance with missing data theory, all missing data can be divided into three categories; missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR). When data are MCAR, the underlying missing data mechanism does not depend on any observed or unobserved data; when data are MAR, the underlying missing data mechanism depends on observed data but not on unobserved data; when data are MNAR, the underlying missing data mechanism depends on the unobserved data itself. Strategies and methods for handling missing observation data and missing covariate data were evaluated. These evaluations showed that the most frequently used estimation algorithm in nonlinear mixed effects modelling (first-order conditional estimation), resulted in biased parameter estimates independent on missing data mechanism. However, expectation maximization (EM) algorithms (e.g. importance sampling) resulted in unbiased and precise parameter estimates as long as data were MCAR or MAR. When the observation data are MNAR, a proper method for handling the missing data has to be applied to obtain unbiased and precise parameter estimates, independent on estimation algorithm. The evaluation of different methods for handling missing covariate data showed that a correctly implemented multiple imputations method and full maximum likelihood modelling methods resulted in unbiased and precise parameter estimates when covariate data were MCAR or MAR. When the covariate data were MNAR, the only method resulting in unbiased and precise parameter estimates was a full maximum likelihood modelling method where an extra parameter was estimated, correcting for the unknown missing data mechanism's dependence on the missing data. This thesis presents new insight to the dynamics of missing data in nonlinear mixed effects modelling. Strategies for handling different types of missing data have been developed and compared in order to provide guidance for efficient handling and consequences of inappropriate handling of missing data.