UC San Diego

In Chapter 1, we consider the problem of detecting a sparse mixture as studied by Ingster (1997) and Donoho and Jin (2004). We consider a wide array of base distributions. In particular, we study the situation when the base distribution has polynomial tails, a situation that has not received much attention in the literature. Perhaps surprisingly, we find that in the context of such a power-law distribution, the higher criticism does not achieve the detection boundary. However, the scan statistic does.

In Chapter 2, we derive the large-sample distribution of several variants of the scan statistic applied to a point process on an interval, which can be applied to detect the presence of an anomalous interval with any length. The main ingredients in the proof are Kolmogorov's theorem, a Poisson approximation, and recent technical results by \cite{kabluchko2014limiting}.

In Chapter 3, we consider causal inference in survival analysis in the presence of unmeasured confounders. Instrumental variable is an essential tool for addressing unmeasured confounding in observational studies. Two stage predictor substitution (2SPS) estimator and two stage residual inclusion(2SRI) are two commonly used approaches in applying instrumental variables. Recently 2SPS was studied under the additive hazards model in the presence of competing risks of time-to-events data, where linearity was assumed for the relationship between the treatment and the instrument variable. This assumption may not be the most appropriate when we have binary treatments. We consider the 2SRI estimator under the additive hazards model for general survival data and in the presence of competing risks, which allows generalized linear models for the relation between the treatment and the instrumental variable. We derive the asymptotic properties including a closed-form asymptotic variance estimate for the 2SRI estimator. We carry out numerical studies in finite samples, and apply our methodology to the linked Surveillance, Epidemiology and End Results (SEER) - Medicare database comparing radical prostatectomy versus conservative treatment in early-stage prostate cancer patients.

In Chapter 4, we investigate the causal effects of etanercept (trade name Enbrel) on birth defects, a pharmaceutical that treats autoimmune diseases and recently went through the US FDA revised labeling for use in pregnancy, as the proportion of liveborn infants with major birth defects was higher for women exposed to etanercept compared to diseased etanercept unexposed women. An outstanding problem, which was not addressed in the data analysis leading up to the FDA relabeling, is the missing birth defect outcomes due to spontaneous abortion since, in accepted standard practice an infant or a fetus is assumed not to be malformed unless a defect is found. This led to likely bias (and missing not at random) because, according to the theory of ``terathanasia'', a defected fetus is more likely to be spontaneously aborted. In addition, the previous analysis stratified on live birth against spontaneous abortion, which was itself a post-exposure variable showing higher rate of spontaneous abortion in the unexposed women, hence did not lead to causal interpretation of the stratified results. We aim to estimate and provide inference for the causal parameters of scientific interest, including the principal effects, making use of the missing data mechanism informed by terathanasia. During the process we also deal with complications in the data including left truncation, observational nature, and rare events. We report our findings which not only provide a more in-depth analysis than previously done on etanercept, but also shed light on how similar studies on causal effects of medication (or vaccine, other substances etc.) during pregnancy may be analyzed.