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This dissertation focuses on developing statistical methods for semiparametric inference and its applications. Semiparametric theory provides statistical tools that are flexible and robust to model misspecification. Utilizing the theory, this work proposes robust estimation approaches that are applicable to several scenarios with mild conditions, and establishes their asymptotic properties for inference. Chapter 1 provides a brief review of the literature related to this work. It first introduces the concept of semiparametric models and the efficiency bound. It further discusses two nonparametric techniques employed in the following chapters, kernel regression and B-spline approximation. The chapter then addresses the concept of dataset shift. In Chapter 2, novel estimators of causal effects for categorical and continuous treatments are proposed by using an optimal covariate balancing strategy for inverse probability weighting. The resulting estimators are shown to be consistent for causal contrasts and asymptotically normal, when either the model explaining the treatment assignment is correctly specified, or the correct set of bases for the outcome models has been chosen and the assignment model is sufficiently rich. Asymptotic results are complemented with simulations illustrating the finite sample properties. A data analysis suggests a nonlinear effect of BMI on self-reported health decline among the elderly. In Chapter 3, we consider a semiparametric generalized linear model and study estimation of both marginal mean effects and marginal quantile effects in this model. We propose an approximate maximum likelihood estimator and rigorously establish the consistency, the asymptotic normality, and the semiparametric efficiency of our method in both the marginal mean effect and the marginal quantile effect estimation. Simulation studies are conducted to illustrate the finite sample performance, and we apply the new tool to analyze non-labor income data and discover a new interesting predictor. In Chapter 4, we propose a procedure to select the best training subsample for a classification model. Identifying patient's disease status from electronic health records (EHR) is a frequently encountered task in EHR related research. However, assessing patient's phenotype is costly and labor intensive, hence a proper selection of EHR as a training set is desired. We propose a procedure to tailor the training subsample for a classification model minimizing its mean squared error (MSE). We provide theoretical justification on its optimality in terms of MSE. The performance gain from our method is illustrated through simulation and a real data example, and is found often satisfactory under criteria beyond mean squared error. In Chapter 5, we study label shift assumption and propose robust estimators for quantities of interest. In studies ranging from clinical medicine to policy research, the quantity of interest is often sought for a population from which only partial data is available, based on complete data from a related but different population. In this work, we consider this setting under the so-called label shift assumption. We propose an estimation procedure that only needs standard nonparametric techniques to approximate a conditional expectation, while by no means needs estimates for other model components. We develop the large sample theory for the proposed estimator, and examine its finite-sample performance through simulation studies, as well as an application to the MIMIC-III database.
A closed-form formula is derived for the limiting variance and the plug-in estimator is shown to be consistent. We demonstrate the unified approach through the special cases of left truncation, length-bias, the case-cohort design and variants thereof. Simulation studies and applications to real data sets are also presented.
This book compiles and presents new developments in statistical causal inference. The accompanying data and computer programs are publicly available so readers may replicate the model development and data analysis presented in each chapter. In this way, methodology is taught so that readers may implement it directly. The book brings together experts engaged in causal inference research to present and discuss recent issues in causal inference methodological development. This is also a timely look at causal inference applied to scenarios that range from clinical trials to mediation and public health research more broadly. In an academic setting, this book will serve as a reference and guide to a course in causal inference at the graduate level (Master's or Doctorate). It is particularly relevant for students pursuing degrees in statistics, biostatistics, and computational biology. Researchers and data analysts in public health and biomedical research will also find this book to be an important reference.
This dissertation is concerned with application of robust semi-parametric methods to problems of estimation in network-dependent data and the conduct of large-scale simulation studies for causal inference research in epidemiological and medical data. Specifically, Chapter 1 presents a modern semi-parametric approach to estimation of causal effects in a population connected by a single social network. The connectivity of the population units will typically imply that the observed data on these units is no longer independent and identically distributed. Moreover, such social settings typically result in highly dimensional data. This chapter contributes to current statistical methodology by presenting an approach that allows valid estimation and inference and addresses the statistical issues specific to such networked population datasets. The framework of semi-parametric estimation, called the targeted maximum likelihood estimation (TMLE), is presented. This framework improves upon the existing methods by offering robustness, weakened sensitivity to near positivity violations, as well as the ability to deal with high-dimensionality issues of social network data. In particular, this approach relies on the accurate reflection of the background knowledge available for a given scientific problem, allowing estimation and inference without having to make unrealistic assumptions about the structure of the data. In addition, this chapter generalizes previous work describing estimation of complex causal parameters, such as the direct treatment effects under interference and the causal effects of interventions on social network structure. Although the past decade has produced many contributions towards estimation of causal effects in social network settings, there has been considerably less research on the topic of variance estimation for such highly-dependent data. This chapter presents an approach to constructing valid inference, providing a variance estimator that is scalable to very large datasets with highly-connected observations. The efficient open-source software implementation of these methods also accompanies this chapter. Chapter 2 presents open-source software tools for conduct of reproducible simulation studies for complex parameters that emerge from application of causal inference methods in epidemiological and medical research. This simulation software is build on the framework of non-parametric structural equation modeling. This chapter also studies simulation-based testing of statistical methods in causal inference for longitudinal data with time-varying exposure and confounding. It contributes to existing literature by presenting a unified syntax for non-parametrically defining complex causal parameters, which can be used as the model-free and agnostic gold standard for comparison of different statistical methods for causal inference. For instance, this chapter provides various examples of specification and evaluation of causal parameters that arise naturally in longitudinal causal effect analyses when using marginal structural models (MSMs). The application of these newly developed software tools to replication of several previously published simulation studies in causal inference are also described. Chapter 3 builds on the work described in Chapter 2 and addresses the issue of dependent data simulation for causal inference research in social network data. In particular, it provides a model-free approach to test the validity of various estimation procedures in simulated network-settings. This chapter first outlines a non-parametric causal model for units connected by a network and provides various applied examples of simulations with social network data. This chapter also showcases a possible application of the highly scalable open-source software implementation of the semi-parametric estimation methods described in Chapter 1. In particular, a large scale social network simulation study is described, and the performance of three dependent-data estimators from Chapter 1 is examined. This simulation study also examines the problem of inference for network-dependent data, specifically, by comparing the performance of the dependent-data TMLE variance estimator from Chapter 1 to the true TMLE variance derived from simulations. Finally, Chapter 3 concludes with a simulation study of an HIV epidemic described in terms of a longitudinal process which evolves over a static network in discrete time-steps among several highly inter-connected communities. The abstracts of the three works which make up this dissertation are reproduced below. Chapter 1: This chapter describes the robust semi-parametric approach towards estimation and inference for the sample average treatment-specific mean in observational settings where data are collected on a single network of connected units (e.g., in the presence of interference or spillover). Despite recent advances, many of the currently used statistical methods rely on assumption of a specific parametric model for the outcome, even though some of the most important statistical assumptions required by these models are most likely violated in the observational network data settings, resulting in invalid and anti-conservative statistical inference. In this chapter, we rely on the recent methodological advances for the targeted maximum likelihood estimation (TMLE) for data collected on a single population of causally connected units, to describe an estimation approach that permits for more realistic classes of data-generative models and provides valid statistical inference in the context of such network-dependent data. The approach is applied to an observational setting with a single time point stochastic intervention. We start by assuming that the true observed data-generating distribution belongs to a large class of semi-parametric statistical models. We then impose some restrictions on the possible set of the data-generative distributions that may belong to our statistical model. For example, we assume that the dependence among units can be fully described by the known network, and that the dependence on other units can be summarized via some known (but otherwise arbitrary) summary measures. We show that under our modeling assumptions, our estimand is equivalent to an estimand in a hypothetical IID data distribution, where the latter distribution is a function of the observed network data-generating distribution. With this key insight in mind, we show that the TMLE for our estimand in dependent network data can be described as a certain IID data TMLE algorithm, also resulting in a new simplified approach to conducting statistical inference. We demonstrate the validity of our approach in a network simulation study. We also extend prior work on dependent-data TMLE towards estimation of novel causal parameters, e.g., the unit-specific direct treatment effects under interference and the effects of interventions that modify the initial network structure. Chapter 2: This chapter introduces the \pkg{simcausal} \proglang{R} package - an open-source software tool for specification and simulation of complex longitudinal data structures that are based on non-parametric structural equation models. The package aims to provide a flexible tool for simplifying the conduct of transparent and reproducible simulation studies, with a particular emphasis on the types of data and interventions frequently encountered in real-world causal inference problems, such as, observational data with time-dependent confounding, selection bias, and random monitoring processes. The package interface allows for concise expression of complex functional dependencies between a large number of nodes, where each node may represent a measurement at a specific time point. The package allows for specification and simulation of counterfactual data under various user-specified interventions (e.g., static, dynamic, deterministic, or stochastic). In particular, the interventions may represent exposures to treatment regimens, the occurrence or non-occurrence of right-censoring events, or of clinical monitoring events. Finally, the package enables the computation of a selected set of user-specified features of the distribution of the counterfactual data that represent common causal quantities of interest, such as, treatment-specific means, the average treatment effects and coefficients from working marginal structural models. The applicability of \pkg{simcausal} is demonstrated by replicating the results of two published simulation studies. Chapter 3: The past decade has seen an increasing body of literature devoted to the estimation of causal effects in network-dependent data. However, the validity of many classical statistical methods in such data is often questioned. There is an emerging need for objective and practical ways to assess which causal methodologies might be applicable and valid in such novel network-based datasets. In this chapter we describe a set of tools implemented as part of the \pkg{simcausal} \proglang{R} package that allow simulating data based on the non-parametric structural equation model for connected units. We also provide examples of how these simulations may be applied to evaluation of different statistical methods for estimation of causal effects in such data. In particular, these simulation tools are targeted to the types of data and interventions frequently encountered in real-world causal inference research in social networks, such as, observational studies with spill-over or interference. We developed a novel \proglang{R} language interface which simplifies the specification of network-based functional relationships between connected units. Moreover, this network-based syntax can be combined with.
This book deals with the mathematical aspects of survival analysis and reliability as well as other topics, reflecting recent developments in the following areas: applications in epidemiology; probabilistic and statistical models and methods in reliability; models and methods in survival analysis, longevity, aging, and degradation; accelerated life models; quality of life; new statistical challenges in genomics. The work will be useful to a broad interdisciplinary readership of researchers and practitioners in applied probability and statistics, industrial statistics, biomedicine, biostatistics, and engineering.
During the last two decades, many areas of statistical inference have experienced phenomenal growth. This book presents a timely analysis and overview of some of these new developments and a contemporary outlook on the various frontiers of statistics.Eminent leaders in the field have contributed 16 review articles and 6 research articles covering areas including semi-parametric models, data analytical nonparametric methods, statistical learning, network tomography, longitudinal data analysis, financial econometrics, time series, bootstrap and other re-sampling methodologies, statistical computing, generalized nonlinear regression and mixed effects models, martingale transform tests for model diagnostics, robust multivariate analysis, single index models and wavelets.This volume is dedicated to Prof. Peter J Bickel in honor of his 65th birthday. The first article of this volume summarizes some of Prof. Bickel's distinguished contributions.
This dissertation focuses on developing robust semiparametric methods for complex parameters that emerge at the interface of causal inference and biostatistics, with applications to epidemiological and medical research. Specifically, it address three important topics: Part I (chapter 1) presents a framework to construct and analyze group sequential covariate-adjusted response-adaptive (CARA) randomized controlled trials (RCTs) that admits the use of data-adaptive approaches in constructing the randomization schemes and in estimating the conditional response model. This framework adds to the existing literature on CARA RCTs by allowing flexible options in both their design and analysis. Part II (chapters 2 and 3) concerns two parameters that arise in longitudinal causal effect analysis using marginal structural models (MSMs). Chapter 2 presents a targeted maximum likelihood estimator (TMLE) for the the dynamic MSM for the hazard function. This estimator improves upon the existing inverse probability weighted (IPW) estimators by providing efficiency gain and robustness protection against model misspecification. Chap- ter 3 addresses the issue of effect modification (in a MSM) by an effect modifier that is post exposure. This parameter is particularly relevant if an effect modifier of interest is missing at random; or if one wishes to evaluate the effect modification of a second-line-treatment by a post first-line-treatment variable, where assignment of the first-line-treatment shares common determinants with the outcome of interest. We also present a TMLE for this parameter. Part III (chapters 4 and 5) addresses semiparametric inference for mediation analysis. Chapter 4 presents a TMLE estimator for the natural direct and indirect effects in a one-time point setting; it improves upon existing estimators by offering robustness, weakened sensitivity to near positivity violations, and potential applications to situations with high-dimensional mediators. Chapter 5 studies longitudinal mediation analysis with time-varying exposure and mediators. In it, we propose a reformulation of the mediation problem in terms of stochastic interventions, establish an identification formula for the mediation functional, and present a TMLE for this parameter. This chapter contributes to existing literature by presenting a nonparametrically defined parameter of interest in longitudinal mediation and a multiply robust and efficient estimator for it. Chapter 1: An adaptive trial design allows pre-specified modifications to some aspects of the on-going trial based on analysis of the accruing data, while preserving the validity and integrity of the trial. This flexibility potentially translates into more efficient studies (e.g. shorter duration, fewer subjects) or greater chance of answering clinical questions of interest (e.g. detecting a treatment effect if one exists, broader does-response information, etc). In an adaptive CARA RCT, the treatment randomization schemes are allowed to depend on the patient's pre-treatment covariates, and the investigators have the opportunity to adjust these schemes during the course of the trial based on accruing information, including previous responses, in order to meet some pre-specified objectives. In a group-sequential CARA RCT, such adjustments take place at interim time points given by sequential inclusion of blocks of c patients, where c ≥ 1 is a pre-specified integer. In this chapter, we present a novel group-sequential CARA RCT design and corresponding analytical procedure that admits the use of flexible approaches in constructing randomization schemes and a wide range of data-adaptive techniques in estimating the conditional response model. Under the proposed framework, the sequence of randomization schemes is group-sequentially determined, using the accruing data, by targeting a formal, user- specified optimal randomization design. The parameter of interest is nonparametrically defined and is estimated using the paradigm of targeted minimum loss estimation. We establish that under appropriate empirical process conditions, the resulting sequence of randomization schemes converges to a fixed design, and the proposed estimator is consistent and asymptotically Gaussian, with an asymptotic variance that is estimable from data, thus giving rise to valid confidence intervals of given asymptotic levels. To illustrate the pro- posed framework, we consider LASSO regression in estimating the conditional outcome given treatment and baseline covariates. The asymptotic results ensue under minimal condition on the growth of the dimension of the regression coefficients and mild conditions on the complexity of the classes of randomization schemes. Chapter 2: In many applications, one is often interested in the effect of a longitudinal exposure on a time-to-event process. In particular, consider a study where subjects are followed over time; in addition to their baseline covariates, at various time points we also record their time-varying exposure of interest, time-varying covariates, and indicators for the event of interest (say death). Time varying confounding is ubiquitous in these situations: the exposure of interest depends on past covariates that confound the effect of the exposure on the outcome of interest, in turn exposure affects future confounders; right censoring may also be present in a study of this nature, often in response to past covariates and exposure. One way to assess the comparative effect of different regimens of interest is to study the hazard as a function of such regimens. The features of this hazard are often encoded in a marginal structural model. This chapter builds upon the work of Petersen, Schwab, Gruber, Blaser, Schomaker, and van der Laan (2014) to present a targeted maximum likelihood estimator for the marginal structural model for the hazard function under longitudinal dynamic interventions. The proposed estimator is efficient and doubly robust, hence offers an improvement over the incumbent IPW estimator. Chapter 3: A crucial component of comparative effectiveness research is evaluating the modification of an exposure's effect by a given set of baseline covariates (effect modifiers). In complex longitudinal settings where time-varying confounding exists, this effect modification analysis is often performed using a marginal structural model. Generally, the conditioning effect modifiers in a MSM are cast as variables of the observed past. Yet, in some applications the effect modifiers of interest are in fact counterfactual. For in- stance, for a specific value of the first-line treatment, one may wish to evaluate the effect modification of a second-line-treatment by a post first-line-treatment variable, wherein the first-line-treatment assignment shares common determinants with the outcome of interest. In this case a simple stratification on the first-line treatment will only yield effect modification over a subpopulation given by said determinants. Hence, the wished parameter of interest should be formulated in terms of randomization on first-line treatment as well. In another example, the effect modifiers may be subject to missingness, which may depend on other baseline confounders; a simple complete-case analysis may introduce selection bias due to the high correlation of these confounders with the missingness of the effect modifier. In this case, one would formulate the wish parameter of interest in terms of an intervention on missingness. We call these counterfactual effect modifiers. In such situations, analysis by stratification alone may harbor selection bias. In this chapter, we investigate MSM defined by counterfactual effect modifiers. Firstly, we determine the identification of the causal dose-response curve and MSM parameters in this setting. Secondly, we establish the semiparametric efficiency theory for these statistical parameters, and present a substitution-based, semiparametric efficient and doubly robust estimator us- ing the targeted maximum likelihood estimation methodology. However, as we shall see, due to the form of the efficient influence curve, the implementation of this estimator may prove arduous in applications where the effect modifier is high dimensional. To address this problem, our third contribution is a projected influence curve (and the corresponding TMLE estimator), which retains most of the robustness of its efficient peer and can be easily implemented in applications where the use of the efficient influence curve becomes taxing. In addition to these two robust estimators, we also present an IPW estimator, and a non-targeted G-computation estimator. Chapter 4: In many causal inference problems, one is interested in the direct causal effect of an exposure on an outcome of interest that is not mediated by certain intermediate variables. Robins and Greenland (1992) and Pearl (2001) formalized the definition of two types of direct effects (natural and controlled) under the counterfactual framework. The efficient influence curves (under a nonparametric model) for the various natural effect parameters and their general robustness conditions, as well as an estimating equation based estimator using the efficient influence curve, are provided in Tchetgen Tchetgen and Shpitser (2011a). In this chapter, we apply the targeted maximum likelihood frame- work to construct a semiparametric efficient, multiply robust, substitution estimator for the natural direct effect which satisfies the efficient influence curve equation derived in Tchetgen Tchetgen and Shpitser (2011a). We note that the robustness conditions in Tchetgen Tchetgen and Shpitser (2011a) may be weakened, thereby placing less reliance on the estimation of the mediator density. More.
Putting a particular emphasis on nonparametric methods that rely on modern empirical process techniques, the author contributes to the theory of static and time-varying stochastic models for multivariate association based on the concept of copulas. These functions enable a profound understanding of multivariate association, which is pivotal for judging whether a large set of risky assets entails diversification effects or aggravates risk from an entrepreneurial point of view. Since serial dependence is a stylized fact of financial time series, an asymptotic theory for estimating the structure of association in this context is developed under weak assumptions. A new measure of multivariate association, based on a notion of distance to stochastic independence, is introduced. Asymptotic results as well as hypothesis tests are established which are directly applicable to important types of multivariate financial time series. To ensure that risk management properly captures the current structure of association, it is crucial to assess the constancy of the structure. Therefore, nonparametric tests for a constant copula with either a specified or unspecified change point (candidate) are derived. The thesis concludes with a study of characterizations of association between non-continuous random variables.
Copulas are mathematical objects that fully capture the dependence structure among random variables and hence offer great flexibility in building multivariate stochastic models. Since their introduction in the early 50's, copulas have gained considerable popularity in several fields of applied mathematics, such as finance, insurance and reliability theory. Today, they represent a well-recognized tool for market and credit models, aggregation of risks, portfolio selection, etc. This book is divided into two main parts: Part I - "Surveys" contains 11 chapters that provide an up-to-date account of essential aspects of copula models. Part II - "Contributions" collects the extended versions of 6 talks selected from papers presented at the workshop in Warsaw.