I consider the type of statistical experiment commonly referred to as adaptive trials, in which the experimenter interacts with an individual or a set of individuals, and, sequentially, over time steps $t=1,\ldots,T$, observes a vector of measurements $L_1(t)$ on the individual or individuals, then assigns treatment vector $A(t)$, and then observes a post-treatment vector of measurements $L_2(t)$. In an adaptive trial, the experimenter can update the treatment distribution at $t$ based on previous observations.

This very general formulation covers many common settings such as dynamic treatment regimes, the stochastic contextual bandit model, the Markov Decision Process model in reinforcement learning. I consider two related types of learning tasks: causal inference from data collected under an adaptive trial, and sequential decision making with the objective of either maximizing the sample efficiency for an estimation task, or of minimizing some form of cumulative regret.

My primary concerns are to develop statistical methods and algorithms that use statistical models that assume no more than is known from domain knowledge (and therefore are nonparametric), and that are as sample efficient as possible.

We consider estimation of causal effects when treatment assignment is potentially subject to unmeasured confounding, but a valid instrumental variable is available. Moreover, our models capture treatment effect heterogeneity, and we allow conditioning on an arbitrary subset of baseline covariates in estimating causal effects. We develop detailed methodology to estimate several types of quantities of interest: 1) the dose-response curve, where our parameter of interest is the projection unto a finite-dimensional working model; 2) the mean outcome under an optimal treatment regime, subject to a cost constraint; and 3) the mean outcome under an optimal intent-to-treat regime, subject to a cost constraint, in which an optimal intervention is done on the instrumental variable. These quantities have a central role for calculating and evaluating individualized treatment regimes. We use semiparametric modeling throughout and make minimal assumptions. Our estimate of the dose-response curve allows treatment to be continuous and makes slightly weaker assumptions than previous research. This work is the first to estimate the effect of an optimal treatment regime in the instrumental variables setting. For each of our parameters of interest, we establish identifiability, derive the efficient influence curve, and develop a new targeted minimum loss-based estimator (TMLE). In accordance with the TMLE methodology, these substitution estimators are asymptotically efficient and double robust. Detailed simulations confirm these desirable properties, and that our estimators can greatly outperform standard approaches. We also apply our estimator to a real dataset to estimate the effect of parents' education on their infant's health.

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 the syntax for specifying

longitudinal data structures, allowing for

simulations of network-based processes that evolve in time (e.g.,

contagion in epidemic modeling).

We provide various examples of simulation studies that involve units

connected by various network models. These simulations were designed to

mimic the types of studies one might conduct in real life with the aim

of answering specific causal public health questions.

We also demonstrate one application of these new tools by conducting a

simulation study that compares the performance of three estimators of

the counterfactual mean outcome in a network-dependent data setting.

Finally, we describe a simulation study with longitudinal data that mimics

a spread of HIV epidemic over time for highly inter-connected communities.

Intensive longitudinal data, defined as time-varying data collected frequently over time, holds immense promise to advance many healthcare and public health concerns. High quality use of time-series primarily depends on the efficient use of causal inference methodology, online machine learning and sequential decision-making. Today, causal inference is central to thestudy of the most impactful scientific questions. For long data streams and elaborate dependence, online estimation has become a paramount technique for learning in real-time, enabling estimation despite high computational cost. Finally, sequential decision-making is essential for practitioners and policy makers to learn when, in what context, and what exposures to assign to each person with the objective of optimizing desired outcome. For example, one might need to decide, with some confidence, when to stop administrating treatment if sufficient benefit is not observed, and what is the best alternative based on patient’s current characteristics and adherence. Investing in the development of methodological approaches for intensive longitudinal data is paramount for advancement of fields such as precision health, where we use data to learn which components of successful strategies are essential to their success, and how best to tailor personalized exposures to meet the specific needs and contexts of individuals, clinics, and communities. Careful considerations and new, modern statistical methods are necessary in order to establish causality, deal with dependence (across time and samples), and estimate relevant parts of the process without imposing unnecessary assumptions.

This dissertation focuses on development of robust non/semi-parametric methods for complex parameters involving time-series data. Divided into five chapters, content discussed entails new methodological approaches for (1) online ensemble machine learning and (2) (causal) sequential decision-making in time-dependent settings. Common themes through-out the chapters deal with (1) different dependence structures (time and/or network) in realistic statistical models and (2) leveraging fully personalized target parameters (single time-series, “N-of-1” approaches) vs. relying on multiple samples. Ideas of studying asymptotics in time, samples or both are also explored.

We start with the idea of developing a “N-of-1” online ensemble machine learning algorithm in Chapter 1, denoted Personalized Online Super Learner. In particular, Chapter 1 studies an Online Super Learner which learns relevant parts of the likelihood while taking into account the amount of data collected (including dynamic enrollment), stationarity of the time-series, and the mutual characteristics/network of a group of trajectories. Further exploring the “N-of-1” paradigm, we propose a causal approach which assigns treatment conditional on the current context of the patient in Chapters 2 and 3, defined as the conditional (or context-specific) causal effects. Let Y (t) denote the outcome, and Co(t) a fixed dimensional context at time t. A “N-of-1” statistical approach answers the following question: “Averaged over times t, given Co(t), what is the distribution of Y (t+s) had we intervened on treatment nodes between t and t + s, s > 0, on sample i?”. In Chapter 2, we propose a time-varying effect of interventions on multiple repeated nodes via context-specific average treatment effect in observation settings, and study the theoretical properties of the proposed estimator. In Chapter 3, we propose a method that learns an optimal treatment allocation for a single individual, adapting the randomization mechanism for future time-point experiments. We demonstrate that one can learn the optimal context defined rule based on a single sample, and thereby adjust the design at any point t with valid inference for the mean target parameter.

The high intensity exposure adaptation available in time-series data proves tremendous potential for infectious disease surveillance and control. For instance, due to the highly dynamic nature of most epidemic diseases, surveillance methods must adapt quickly in order to target individuals at the highest risk of infection. Instead of only considering dependence through time, sequential decision-making for infectious disease must account for the overall status of the epidemic in the population, including multiple trajectories with possible network dependence. In Chapter 4, we describe an adaptive surveillance design which optimizes testing allocation among a class of testing schemes based on the current status of the epidemic. While Chapter 4 focuses on adaptive monitoring for a closed community, the statistical problem is addressed within a model for the data-generating distribution that is completely nonparametric. As such, it represents a first step towards an important goal of developing adaptive sequential design for infectious disease surveillance in the general population underno assumptions on the dependence structure.

Finally, in order to develop data-driven, effective sequential interventions, it is crucial to learn new policies (series of treatment decisions) using existing data, and understand their long-term efficacy. In Chapter 5, we propose and analyze a novel double robust estimator for the “off-policy” evaluation problem in reinforcement learning. We show empirically that our estimator uniformly wins over existing off-policy evaluation methods, and characterize the asymptotic distribution and rate of convergence for the proposed estimator.

In most randomized controlled trials (RCTs), investigators typically rely on estimators of causal effects that do not exploit the information in the many baseline covariates that are routinely collected in addition to treatment and the outcome. Ignoring these covariates can lead to a significant loss is estimation efficiency and thus power. Statisticians have underscored the gain in efficiency that can be achieved from covariate adjustment in RCTs with a focus on problems involving linear models. Despite recent theoretical advances, there has been a reluctance to adjust for covariates based on two primary reasons; 1) covariate-adjusted estimates based on non-linear regression models have been shown to be less precise than unadjusted methods, and, 2) concern over the opportunity to manipulate the model selection process for covariate adjustment in order to obtain favorable results. This dissertation describes statistical approaches for covariate adjustment in RCTs using targeted maximum likelihood methodology for estimation of causal effects with binary and right-censored survival outcomes.

Chapter 2 provides the targeted maximum likelihood approach to covariate adjustment in RCTs with binary outcomes, focusing on the estimation of the risk difference, relative risk and odds ratio. In such trials, investigators generally rely on the unadjusted estimate as the literature indicates that covariate-adjusted estimates based on logistic regression models are less efficient. The crucial step that has been missing when adjusting for covariates is that one must integrate/average the adjusted estimate over those covariates in order to obtain the population-level effect. Chapter 2 shows that covariate adjustment

in RCTs using logistic regression models can be mapped, by averaging over the covariate(s), to obtain a fully robust and efficient estimator of the marginal effect, which equals a targeted maximum likelihood estimator. Simulation studies are provided that demonstrate that this targeted maximum likelihood method increases efficiency and power over the unadjusted method, particularly for smaller sample sizes, even when the regression model is misspecified.

Chapter 3 applies the methodology presented in Chapter 3 to a sampled RCT dataset with a binary outcome to further explore the origin of the gains in efficiency and provide a criterion for determining whether a gain in efficiency can be achieved with covariate adjustment over the unadjusted method. This chapter demonstrates through simulation studies and the data analysis that not only is the relation between $R^2$ and efficiency gain important, but also the presence of empirical confounding. Based on the results of these studies, a complete strategy for analyzing these type of data is formalized that provides a robust method for covariate adjustment while protecting investigators from misuse of these methods for obtaining favorable inference.

Chapters 4 and 5 focus on estimation of causal effects with right-censored survival outcomes. Time-to-event outcomes are naturally subject to right-censoring due to early patient withdrawals. In chapter 4, the targeted maximum likelihood methodology is applied to the estimation of treatment specific survival at a fixed end-point in time. In chapter 5, the same methodology is applied to provide a competitor to the logrank test. The proposed covariate adjusted estimators, under no or uninformative censoring, do not require any additional parametric modeling assumptions, and under informative censoring, are consistent under consistent estimation of the censoring mechanism or the conditional hazard for survival. These targeted maximum likelihood estimators have two important advantages over the Kaplan-Meier and logrank approaches; 1) they exploit covariates to improve efficiency, and 2) they are consistent in the presence of informative censoring. These properties are demonstrated through simulation studies.

Chapter 6 concludes with a summary of the preceding chapters and a discussion of future research directions.

Robust inference of a low-dimensional parameter in a large semi-parametric model relies on external estimators of infinite-dimensional features of the distribution of the data. Typically, only one of the latter is optimized for the sake of constructing a well behaved estimator of the low-dimensional parameter of interest. Optimizing more than one of them for the sake of achieving a better bias-variance trade-off in the estimation of the parameter of interest is the core idea driving the general template of the collaborative targeted minimum loss-based estimation (C-TMLE) procedure. In this dissertation, we first resolves the computational issue in the widely-used greedy variable selection C-TMLE. Then we further investigate how to extend the discrete, variable selection C-TMLE for a more general model selection purpose.

Chapter 1 begins by introducing the framework of causal inference in observational studies. We introduce the non-parametric structural equation model for modeling the data generating distribution. We briefly review the targeted minimum loss-based estimation (TMLE). We also introduce the general template of C-TMLE and its greedy-search variable selection version.

In chapter 2, we propose the template for scalable variable selection C-TMLEs to overcome the computational burden in the greedy variable selection C-TMLE. The original instantiation of the C-TMLE template can be presented as a greedy forward stepwise C-TMLE algorithm. It does not scale well when the number $p$ of covariates increases drastically. This motivates the introduction of a novel instantiation of the C-TMLE template where the covariates are pre-ordered. Its time complexity is $\mathcal{O}(p)$ as opposed to the original $\mathcal{O}(p^2)$, a remarkable gain. We propose two pre-ordering strategies and suggest a rule of thumb to develop other meaningful strategies. Because it is usually unclear a priori which pre-ordering strategy to choose, we also introduce another instantiation called SL-C-TMLE algorithm that enables the data-driven choice of the better pre-ordering strategy given the problem at hand. Its time complexity is $\mathcal{O}(p)$ as well. The computational burden and relative performance of these algorithms were compared in simulation studies involving fully synthetic data or partially synthetic data based on a real world large electronic health database; and in analyses of three real, large electronic health databases. In all analyses involving electronic health databases, the greedy C-TMLE algorithm is unacceptably slow. Simulation studies seem to indicate that our scalable C-TMLE and SL-C-TMLE algorithms work well.

In chapter 3, we extend C-TMLE to a more general model selection problem: we apply C-TMLE to select from a set of continuously-indexed nuisance parameter (the propensity score, PS) estimators. The propensity score models have traditionally been selected based on the goodness-of-fit for the treatment mechanism itself, without consideration of the causal parameter of interest. In contrast, the C-TMLE takes into account information on the causal parameter of interest when selecting a PS model. This ``collaborative learning'' considers variable associations with both treatment and outcome when selecting a PS model in order to minimize a bias-variance trade off in the estimated treatment effect. In this study, we introduce a novel approach for collaborative model selection when using the LASSO estimator for PS estimation in high-dimensional covariate settings. To demonstrate the importance of selecting the PS model collaboratively, we designed quasi-experiments based on a real electronic healthcare database, where only the potential outcomes were manually generated, and the treatment and baseline covariates remained unchanged. Results showed that the C-TMLE algorithm outperformed other competing estimators for both point estimation and confidence interval coverage. In addition, the PS model selected by C-TMLE could be applied to other PS-based estimators, which also resulted in substantive improvement for both point estimation and confidence interval coverage. We illustrate the discussed concepts through an empirical example comparing the effects of non-selective Nonsteroidal anti-inflammatory drugs with selective COX-2 inhibitors on gastrointestinal complications in a population of Medicare beneficiaries.

In chapter 4, we propose using C-TMLE to adaptively truncated the propensity score when there exist practical positivity violations. The positivity assumption, or the experimental treatment assignment (ETA) assumption, is important for identifiability in causal inference. Even if the positivity assumption holds, practical violations of this assumption may jeopardize the finite sample performance of the causal estimator. One of the consequences of practical violations of the positivity assumption is extreme values in the estimated propensity score. A common practice to address this issue is truncating the PS estimate when constructing PS-based estimators. In this study, we propose a novel adaptive truncation method, Positivity-C-TMLE, based on the C-TMLE methodology. We further show how to construct a robust confidence interval by a targeted variance estimator. We demonstrate the outstanding performance of our novel approach in a variety of simulations by comparing it with other commonly studied estimators, for both point estimation and confidence interval coverage. Results show that by adaptively truncating the estimated PS with a more targeted objective function, the Positivity-C-TMLE estimator achieves the best performance for both point estimation and confidence interval coverage among all estimators considered.

The code for all the variations of C-TMLE in this dissertation are publicly available in the \emph{ctmle} R package.

Collaborative targeted maximum likelihood estimation is an extension to targeted maximum likelihood estimation (TMLE), first introduced by van der Laan and Rubin in 2006. TMLE is an efficient, double robust, semi-parametric methodology for estimating a pathwise differential parameter of a statistical distribution given censored data. The TMLE procedure involves a parametric fluctuation of an initial estimate of the relevant factor of the density of the observed data, Q, that involves estimating the nuisance portion of the likelihood---censoring mechanism, g. DR estimators are consistent when at least one of these is estimated consistently. Though the best approach to nuisance parameter estimation is a current topic of debate in the literature, methods typically rely on maximizing a likelihood for g, however, by establishing the collaborative double robustness of the efficient influence curve, van der Laan and Gruber (2010) provides a theoretical justification for moving away from the practice of external nuisance parameter estimation. That paper also presents the collaborative TMLE (C-TMLE), and provides an algorithm for constructing the estimator.

This dissertation explores collaborative double robustness to provide an understanding of the requirements for effective nuisance parameter estimation that are the foundation of C-TMLE, and presents the forward-selection C-TMLE algorithm. This approach to targeted maximum likelihood estimation is especially useful when the true censoring mechanism is unknown, and adjusting for a large number of correlated confounders leads to highly variable estimates. It is also particularly effective when there is sparsity in the data that renders a statistical parameter borderline identifiable. Variations on the C-TMLE estimator are presented, and their behavior is compared and contrasted with several TMLEs and other estimators in the literature using simulated and real data. There is an emphasis on finite sample performance under sparsity and model misspecification, and the incorporation of data-adaptive techniques while still preserving influence-curve based inference. A method for ensuring that TMLEs respect known global constraints on the model is presented, and a guide to the R package tmle, for TMLE estimation of binary point treatment effects is provided in the Appendix.

An active area of research interest is the inference of ecological models of complex microbial communities. Inferring such ecological models entails understanding the interactions between microbes and how they affect each other’s growth. This dissertation employs a statistical perspective to contribute further to the knowledge currently addressing this problem.

Part I explains how high-throughput droplet-based microfluidics technology can be used to screen for microbial interactions. An explicit, statistical framework is motivated and developed that can guide the analysis of data from such experiments. Chapter 1 investigates the specific questions that need to be answered to study microbial interactions. Chapter 1 explains why high-throughput droplet-based microfluidics technology overcomes previous limitations to answering these questions. It is shown how answering these questions can be recast as the statistical problem of estimating a network with mixed-sign edge weights. Chapter 2 investigates how to approach these questions using statistical models. Chapter 2 explains that the data from the noisy dynamical systems corresponding to each droplet can be understood as censored observations of a multivariate Markov process. The statistical understanding of a droplet’s initial state is identified as crucial to overcoming the main limitation of these experiments, the uncontrolled assignment of microbes to droplets.

Part II explains how it might be possible to predict, based on the experimental setup, how much data will be produced to infer given microbial interactions. Running the experiment once without incubating the droplets turns out to be necessary to make such pre- dictions. Chapter 3 investigates which statistical (working) models can be used to describe a droplet’s initial state. Chapter 3 shows that specific assumptions justify a default working model. New working models are derived by relaxing each of these assumptions. Chapter 4 investigates whether the failure of any of these assumptions leads to substantially new behav- ior. Chapter 4 demonstrates that log likelihood ratios can be used to answer this question. Failure of the sampling without replacement assumption turns out to have negligible effects in practice, but failures of the other assumptions could be important. Chapter 5 investigates how failures of the relevant assumptions affect the targeted estimands that enable the pre- diction of how much data will be produced to infer given microbial interactions. Chapter 5 confirms that more severe failures of these assumptions lead to more severe discrepancies with the predictions derived from the default working model. The nature of the effect depends on the chosen grouping of droplets defining the targeted estimands. Chapter 6 investigates how to estimate failures of the relevant assumptions from the data produced by unincubated droplets. Chapter 6 presents both plugin and maximum likelihood estimators for doing so. Failures of these assumptions are shown to be understandable non-parametrically.

Part III demonstrates the feasibility of inferring microbial interactions from the data produced by these experiments. Relevant ideas from the microbiological and ecological literature are recast into an explicit, statistical framework. Chapter 7 investigates how a particular measure of relative fitness can be recast into the statistical framework of average treatment effects. Chapter 7 explains how violations of positivity assumptions are inevitable for this problem, making controlling for confounding difficult. Explicit assumptions are given under which the estimands are identifiable from the observed data produced by incubated droplets, even though initial states of the droplets are not directly observed. Chapter 8 investigates how comparisons of ecological interactions can be recast into the statistical framework of loss functions for signed networks. Chapter 8 explains how avoiding unexpected behavior requires loss functions for signed networks to satisfy what is called herein “the double penalization principle”. Starting from loss functions of unsigned networks, several examples of loss functions for signed networks are derived that satisfy this property.

Future work will explore further choices that can be made when modelling this problem, how to connect these ideas to more sophisticated statistical methodologies, and the biological interpretation of the results of applying these ideas to real experimental datasets. This work demonstrates the plausibility of characterizing microbial interactions using high-throughput droplet-based microfluidics technologies, and hopefully will guide the analysis of data produced by such experiments in the future.

This thesis discusses the Super Learner and Targeted Maximum Likelihood Estimation (TMLE) for longitudinal data structures in nonparametric statistical models. It focuses specifically on time-dependent data structures where the outcome of interest may be described as a counting process. A Super Learner for the conditional intensity of the counting process is proposed based on the minimization of squared error and negative Bernoulli loglikelihood risks. An analytic comparison of the oracle inequality for the cross validation selector of the squared error and negative Bernoulli loglikelihood risks is provided. TMLE is extended to enforce a calibration property, defined as an implicit constraint, on the Super Learner. Tradeoffs between calibration and risk minimization are explored through simulation. The final chapter discusses and implements a recently developed TMLE for the intervention-specific marginal mean in general longitudinal data structures. A modification of this general TMLE algorithm template is implemented to respect model constraints for the estimation of the cumulative event probability in survival data with time-dependent covariates and informative right-censoring. Each chapter is supplemented with practical examples of these estimators using data from the the Kaiser Permanente ATRIA-1 cohort study of adults with atrial fibrillation. The primary appendix presents a new Super Learner software implementation as a SAS (Statistical Analysis System) macro for data-adaptive machine learning. SAS macro code for the TMLEs are provided in secondary appendices.

Targeted maximum likelihood estimator (and semiparametric efficient estimators in general) involves deriving the efficient influence function of target parameters and adjusting an estimate of the data distribution towards the target estimand. This adjustment step requires fitting a least favorable submodel on the initial estimator with the same dimensionality of the parameter, which can become unstable for high-dimensional target parameters.

Another direction that will vastly improve the credibility of these semiparametric estimators is to improve the finite-sample coverage of confidence intervals.

In this dissertation, we first study the robust estimation of high-dimensional target parameters. Then we investigate how to perform finite sample inference in a large semi-parametric model. We also build an estimator that is simultaneously efficient for a large family of target parameters by undersmoothing a single regression.

In Chapter 1, we propose using universal least favorable submodel to robustly estimate high-dimensional target parameters, with applications to survival analysis. We establish a novel connection between a universal least favorable submodel and moving along a sparse local least favorable submodel, and demonstrate the extensions in survival analysis when the whole survival curve needs to be nonparametrically estimated and given statistical inference. We assess the finite sample performance in both a simulation study and an observational study on monoclonal gammopathy.

In Chapter 2, we theoretically develop and extend nonparametric bootstrap inference for the targeted maximum likelihood estimator (TMLE). We establish a formal theorem showing that the nonparametric bootstrap is an asymptotically valid procedure for finite sample TMLE inference using highly-adaptive LASSO (HAL) as the nuisance parameter estimator and demonstrate superior coverage than existing influence-function-based methods. This article explores the problem of applying semiparametric models and machine learning algorithms to small datasets and still have honest causal and statistical inference. Prior to this work, one either has to run nonparametric bootstrap by assuming small parametric models or do estimation in a large semiparametric model where the nonparametric bootstrap has no theoretical guarantee. We propose an effective tuning parameter selection method that optimizes confidence interval coverage (rather than estimation precision) which shows good coverage even for non-doubly robust causal parameters.

In Chapter 3, we propose two efficient estimators based on highly-adaptive LASSO (HAL): targeted HAL and undersmoothed HAL.

Using undersmoothed HAL to estimate the likelihood gives us an efficient estimator for a large family of target parameters. The key is to propose a strategy to choose the tuning parameter that results in a sectional variation norm larger than the one selected using cross-validation. In this chapter, we propose a `multi-task tuning' method that can be generally applied to a wide range of target parameters.

The second method called targeted HAL solves the efficient score equations by including an additional covariate into the LASSO design matrix that targets the statistical parameter of interest.

We provide examples of our methods for estimating the average treatment effect and illustrate using two simulations where one favors inverse probability weighting methods (such as estimating equations and TMLE) and another challenging design where there is practical violation of the positivity assumption.

We demonstrate the outstanding performance of the undersmoothed HAL in both scenarios.

We also show theoretical results that shed light on why undersmoothed HAL is performing well in data generating distributions where positivity assumption is violated.