Broadly, this dissertation can be divided into two parts. Part I addresses various issues in the analysis of medical and health services data from observational studies. Part II addresses computational and methodological challenges in the analysis of complex computer simulations.

Health care costs in the United States continue to rise, yet there is little evidence that this increased spending has yielded better health outcomes. There remain a plethora of challenges in improving health outcomes through health care delivery innovations and an increased understanding of the comparative effectiveness of various treatment, intervention, and surgical options for patients. The ideal approach to address these challenges is to conduct large randomized controlled trials, however this is rarely possible due to their cost, long-term nature, and the difficulties in their implementation. Researchers have thus turned to increasingly abundant and complex data sources from observational studies to answer these pressing medical and health care questions. Observational studies, however, present their own unique statistical challenges. Understanding causality is nearly always the key focus of medical research, yet determining cause from observational studies remains elusive. Another key challenge in observational studies with large and diverse populations is in addressing patient heterogeneity. The primary focus of this Part I is in exploring and addressing these issues in the following topics: • Instrumental variable estimation in the semiparametric accelerated failure time model. • Risk modeling for heterogeneous populations: - Dealing with high-dimensionality via hierarchical variable importance in hospitalsystem risk modeling - Flexible modeling and regression-guided visualization with semiparametric sufficient dimension reduction.

Studying complex physical phenomena through the use of simulated computer codes has resulted in vast improvements in our knowledge of processes in engineering, physics, astrophysics, climatology, and many more fields. Using these simulated codes, researchers seek to understand the relationship between input conditions and an output of interest. For many physical processes this input and output relationship is often represented by a highly complex and nonlinear function. Often simulation codes are highly computationally expensive, and thus the field of computer experiments has focused on using statistical techniques to emulate the functional relationship between input and output by constructing a surrogate model for the simulation codes. Constructing accurate emulators is a serious challenge, especially for physical processes that have many input conditions. Part II of this dissertation focuses on the following two topics broadly concerned with improving predictive performance of emulators of complex phenomena: • Improving computational stability of gradient-enhanced emulation • Flexible emulation using deep neural networks.