Applications of Geometry in Optimization and Statistical Estimation

Geometric properties of statistical models and their influence on statistical inference and asymptotic theory reveal the profound relationship between geometry and statistics. This thesis studies applications of convex and differential geometry to statistical inference, optimization and modelling. We, particularly, investigate how geometric understanding assists statisticians in dealing with non-standard inferential problems by developing novel theory and designing efficient computational algorithms. The thesis is organized in six chapters as it follows. Chapter 1 provides an abstract overview to a wide range of geometric tools, including affine, convex and differential geometry. It also provides the reader with a short literature review on the applications of geometry in statistical inference and exposes the geometric structure of commonly used statistical models. The contributions of this thesis are organized in the following four chapters, each of which is the focus of a submitted paper which is either accepted or under revision. Chapter 2 introduces a new parametrization to general family of mixture models of the exponential family. Despite the flexibility and popularity of mixture models, their associated parameter spaces are often difficult to represent due to fundamental identification problems. Other related problems include the difficulty of estimating the number of components, possible unboundedness and non-concavity of the log-likelihood function, non-finite Fisher information, and boundary problems giving rise to non-standard analysis. For instance, the order of a finite mixture is not well defined and often can not be estimated from a finite sample when components are not well separated, or some are not observed in the sample. We introduce a novel family of models, called the discrete mixture of local mixture models, which reparametrizes the space of general mixtures of the exponential family, in a way that the parameters are identifiable, interpretable, and, due to a tractable geometric structure, the space allows fast computational algorithms. This family also gives a well-defined characterization to the number of components problem. The component densities are flexible enough for fitting mixture models with unidentifiable components, and our proposed algorithm only includes the components for which there is enough information in the sample. Chapter 3 uses geometric concepts to characterize the parameter space of local mixture models (LMM), introduced in \cite{Marriott2002} as a local approximation to continuous mixture models. Although LMMs are shown to satisfy nice inferential properties, their parameter space is restricted by two types of boundaries, called the hard boundary and the soft boundary. The hard boundary guarantees that an LMM is a density function, while the soft boundary ensures that it behaves locally in a similar way to a mixture model. The boundaries are shown to have particular geometric structures that can be characterized by geometry of polytopes, ruled surface and developable surfaces. As working examples the LMM of a normal model and the LMM of a Poisson distribution are considered. The boundaries described in this chapter have both discrete aspects, (i.e. the ability to be approximated by polytopes), and smooth aspects (i.e. regions where the boundaries are exactly or approximately smooth). Chapter 4 uses the model space introduced in Chapter 2 for extending a prior model and defining a perturbation space in the Bayesian sensitivity analysis. This perturbation space is well-defined, tractable, and consistent with the elicited prior knowledge, the three properties that improve the methodology in \cite{Gustafson1996}. We study both local and global sensitivity in conjugate Bayesian models. In the local analysis the worst direction of sensitivity is obtained by maximizing the directional derivative of a functional between the perturbation space and the space of posterior expectations. For finding the maximum global sensitivity, however, two criteria are used; the divergence between posterior predictive distributions and the difference between posterior expectations. Both local and global analyses lead to optimization problems with a smooth boundary restriction. Chapter 5 studies Cox's proportional hazard model with an unobserved frailty for which no specific distribution is assumed. The likelihood function, which has a mixture structure with an unknown mixing distribution, is approximated by the model introduced in Chapter 2, which is always identifiable and estimable. The nuisance parameters in the approximating model, which represent the frailty distribution through its moments, lie in a convex space with a smooth boundary, characterized as a smooth manifold. Using differential geometric tools, a new algorithm is proposed for maximizing the likelihood function restricted by the smooth yet non-trivial boundary. The regression coefficients, the parameters of interest, are estimated in a two step optimization process, unlike the existed methodology in \cite{Klein1992} which assumes a gamma assumption and uses Expectation-Maximization approach. Simulation studies and data examples are also included, illustrating that the new methodology is promising as it returns small estimation bias; however, it produces larger standard deviation compared to the EM method. The larger standard deviation can be the result of using no information about the shape of the frailty model, while the EM model assumes the gamma model in advance; however, there are still ways to improve this methodology. Also, the simulation section and data analysis in this chapter is rather incomplete and more work needs to be done. Chapter 6 outlines a few topics as future directions and possible extensions to the methodologies developed in this thesis

Effect of Tracheostomy Timing On Outcomes in Patients With Traumatic Brain injury

Tracheostomy following severe traumatic brain injury (TBI) is common, yet the outcomes associated with tracheostomy timing are unclear. The objective of this study was to assess hospital outcomes of tracheostomy timing in TBI patients. We retrospectively analyzed data from the National Inpatient Sample database of adult patients aged ≥18 years with a primary diagnosis of TBI. Indexed hospitalizations of TBI patients who underwent either percutaneous or surgical tracheostomy between 1995 and 2015 in the United States were included. The interventional groups were 1) early tracheostomy (≤7 days) vs standard tracheostomy (8-14 days), vs late tracheostomy (≥15 days), and 2) tracheostomy vs no tracheostomy. Propensity score matching and conditional logistic regression models were used to analyze in-hospital mortality, length of hospitalization, and in-hospital complications among TBI patients in relation to tracheostomy timing. The risk of in-hospital mortality was 35% lower in patients who underwent tracheostomy vs those who did not (odds ratio 0.65

Influences of vaccination and public health strategies on COVID-19 dynamics in the United States: Evaluating policy impacts, behavioral responses, and variant proliferation

Background and Aim: The United States (US) government implemented interventions against COVID-19, but their effects on variant-related risks remain inconclusive. We aimed to assess the causal effects of vaccination rates, booster uptakes, face mask mandates, and public area mobility (societal behavioral factor) on early-stage COVID-19 case and death growth rates and identify the most effective public health response for controlling COVID-19 in the US. Materials and Methods: We performed retrospective analyses using four standard correlated random effects models, analyzing a robust panel dataset that encompasses 16,700 records across all fifty US states. Models 1 and 3 analyzed COVID-19 case rates and death growth rates, respectively, from January 2021 to November 2021. In contrast, using the data from August 2021 to November 2021, Models 2 and 4 assessed the effect of Delta variants and booster shots on COVID-19 case and death growth rates, respectively. Results: We found that face mask mandate (p < 0.01) and workplace mobility (p < 0.05) led to lower COVID-19 case growth rates. COVID-19 vaccination uptake rate reduced COVID-19 death growth rates (p < 0.01). Furthermore, contrary to Epsilon variant (p < 0.01), which contributed to reduced COVID-19 case growth rates, Delta variant led to significant increases in COVID-19 cases (p < 0.001). Conclusion: This study suggests that immediate public health interventions, like mask mandates, are crucial for crisis mitigation, while long-term solutions like vaccination effectively address pandemics. The findings of this study not only sheds light on the recent pandemic but also equips policy-makers and health professionals with tools and knowledge to tackle future public health emergencies more effectively

Vasopressor treatment and mortality following nontraumatic subarachnoid hemorrhage: a nationwide electronic health record analysis

OBJECTIVE: Subarachnoid hemorrhage (SAH) is a devastating cerebrovascular condition, not only due to the effect of initial hemorrhage, but also due to the complication of delayed cerebral ischemia (DCI). While hypertension facilitated by vasopressors is often initiated to prevent DCI, which vasopressor is most effective in improving outcomes is not known. The objective of this study was to determine associations between initial vasopressor choice and mortality in patients with nontraumatic SAH. METHODS: The authors conducted a retrospective cohort study using a large, national electronic medical record data set from 2000-2014 to identify patients with a new diagnosis of nontraumatic SAH (based on ICD-9 codes) who were treated with the vasopressors dopamine, phenylephrine, or norepinephrine. The relationship between the initial choice of vasopressor therapy and the primary outcome, which was defined as in-hospital death or discharge to hospice care, was examined. RESULTS: In total, 2634 patients were identified with nontraumatic SAH who were treated with a vasopressor. In this cohort, the average age was 56.5 years, 63.9% were female, and 36.5% of patients developed the primary outcome. The incidence of the primary outcome was higher in those initially treated with either norepinephrine (47.6%) or dopamine (50.6%) than with phenylephrine (24.5%). After adjusting for possible confounders using propensity score methods, the adjusted OR of the primary outcome was higher with dopamine (OR 2.19, 95% CI 1.70-2.81) and norepinephrine (OR 2.24, 95% CI 1.80-2.80) compared with phenylephrine. Sensitivity analyses using different variable selection procedures, causal inference models, and machine-learning methods confirmed the main findings. CONCLUSIONS: In patients with nontraumatic SAH, phenylephrine was significantly associated with reduced mortality in SAH patients compared to dopamine or norepinephrine. Prospective randomized clinical studies are warranted to confirm this finding

Impaired health-related quality of life in idiopathic inflammatory myopathies:a cross-sectional analysis from the COVAD-2 e-survey

ObjectivesTo investigate health-related quality of life in patients with idiopathic inflammatory myopathies (IIMs) compared with those with non-IIM autoimmune rheumatic diseases (AIRDs), non-rheumatic autoimmune diseases (nrAIDs), and without autoimmune diseases (controls), using Patient-Reported Outcome Measurement Information System (PROMIS) instrument data obtained from the second COVID-19 vaccination in autoimmune disease (COVAD-2) e-survey database.MethodsDemographics, diagnosis, comorbidities, disease activity, treatments, and PROMIS instrument data were analysed. Primary outcomes were PROMIS Global Physical Health (GPH) and Global Mental Health (GMH) scores. Factors affecting GPH and GMH scores in IIMs were identified using multivariable regression analysis.ResultsWe analysed responses from 1582 IIMs, 4700 non-IIM AIRDs, 545 nrAIDs, and 3675 controls gathered until May 23, 2022. GPH median (IQR) scores were the lowest in IIMs and non-IIM AIRDs (13 [10–15] IIMs vs.s 13 [11–15] non-IIM AIRDs vs.s 15 [13–17] nrAIDs vs.s 17 [15–18] controls, p &lt; 0.001). GMH median (IQR) scores in IIMs were also significantly lower compared with those without autoimmune diseases (13 [10–15] IIMs vs.s 15 [13–17] controls, p &lt; 0.001). Inclusion body myositis, comorbidities, active disease, and glucocorticoid use were the determinants of lower GPH scores, whereas overlap myositis, interstitial lung disease, depression, active disease, lower PROMIS Physical Function-10a, and higher PROMIS Fatigue-4a scores were associated with lower GMH scores in IIMs.ConclusionBoth physical and mental health are significantly impaired in IIMs, particularly in those with comorbidities and increased fatigue, emphasizing the importance of patient-reported experiences and optimized multidisciplinary care to enhance well-being in people with IIMs