Theory and Methods for Spatiotemporal Point Processes

In this dissertation we study several related problems under the general umbrella of spatiotemporal counting/point processes. We study counting processes in two distinct contexts. First, we study a clinical problem of assessing medical response or non-response to a treatment on the basis of known biomarker values. This problem commonly occurs in oncology, where measures of objective response are critical to drug approval, but may require long periods of total follow-up with limited cohorts. In this context it is critical for researchers to be able to efficiently estimate biomarker cutoff values as soon as possible. However, as trials take long periods of time there will generally be substantial incomplete data at interim analysis. We find that conventional estimators for handling missing data may reduce bias relative to conventional estimators in realistic scenarios. We provide recommendations that applied researchers use a variety of estimators and use careful sensitivity analyses to guide conclusions drawn from them. Second, we study point processes in a spatial context, where we develop new methods for utilizing point process valued data as covariates in spatial regression. We illustrate the new method through several simulation scenarios and an analysis of the impact of contributory sources of air pollution in western Pennsylvania. Finally, we study a problem of causal inference with a point process valued outcome: severe asthma exacerbation events in western Pennsylvania. We develop causal estimands and associated estimators to study the relationship between unconventional natural gas production and severe asthma in western Pennsylvania. As part of this work we develop a formal theory for incorporation of mechanistic considerations in causal inference. Contribution to Public Health: This dissertation contributes to public health in multiple ways. In the clinical trials portion we assess the potential of common statistical methods for bias reduction in clinical trials at interim analysis, an important objective for drug development. The spatial epidemiology work contributes to a growing literature assessing the impact of industrial activity and air pollution on human health. The theoretical work contributes to public health by providing foundations for causal interpretation of observational studies and enabling new research designs.