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Multiply Robust Empirical Likelihood Inference For Missing Data And Causal Inference Problems
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Book Synopsis Multiply Robust Empirical Likelihood Inference for Missing Data and Causal Inference Problems by : Shixiao Zhang
Download or read book Multiply Robust Empirical Likelihood Inference for Missing Data and Causal Inference Problems written by Shixiao Zhang and published by . This book was released on 2019 with total page 119 pages. Available in PDF, EPUB and Kindle. Book excerpt: Missing data are ubiquitous in many social and medical studies. A naive complete-case (CC) analysis by simply ignoring the missing data commonly leads to invalid inferential results. This thesis aims to develop statistical methods addressing important issues concerning both missing data and casual inference problems. One of the major explored concepts in this thesis is multiple robustness, where multiple working models can be properly accommodated and thus to improve robustness against possible model misspecification. Chapter 1 serves as a brief introduction to missing data problems and causal inference. In this Chapter, we highlight two major statistical concepts we will repeatedly adopt in subsequent chapters, namely, empirical likelihood and calibration. We also describe some of the problems that will be investigated in this thesis. There exists extensive literature of using calibration methods with empirical likelihood in missing data and causal inference. However, researchers among different areas may not realize the conceptual similarities and connections with one another. In Chapter 2, we provide a brief literature review of calibration methods, aiming to address some of the desirable properties one can entertain by using calibration methods. In Chapter 3, we consider a simple scenario of estimating the means of some response variables that are subject to missingness. A crucial first step is to determine if the data are missing completely at random (MCAR), in which case a complete-case analysis would suffice. We propose a unified approach to testing MCAR and the subsequent estimation. Upon rejecting MCAR, the same set of weights used for testing can then be used for estimation. The resulting estimators are consistent if the missingness of each response variable depends only on a set of fully observed auxiliary variables and the true outcome regression model is among the user-specified functions for deriving the weights. The proposed testing procedure is compared with existing alternative methods which do not provide a method for subsequent estimation once the MCAR is rejected. In Chapter 4, we consider the widely adopted pretest-posttest studies in causal inference. The proposed test extends the existing methods for randomized trials to observational studies. We propose a dual method to testing and estimation of the average treatment effect (ATE). We also consider the potential outcomes are subject to missing at random (MAR). The proposed approach postulates multiple models for the propensity score of treatment assignment, the missingness probability and the outcome regression. The calibrated empirical probabilities are constructed through maximizing the empirical likelihood function subject to constraints deducted from carefully chosen population moment conditions. The proposed method is in a two-step fashion where the first step is to obtain the preliminary calibration weights that are asymptotically equivalent to the true propensity score of treatment assignment. Then the second step is to form a set of weights incorporating the estimated propensity score and multiple models for the missingness probability and the outcome regression. The proposed EL ratio test is valid and the resulting estimator is also consistent if one of the multiple models for the propensity score as well as one of the multiple models for the missingness probability or the outcome regression models are correctly specified. Chapter 5 extends Chapter 4's results to testing the equality of the cumulative distribution functions of the potential outcomes between the two intervention groups. We propose an empirical likelihood based Mann-Whitney test and an empirical likelihood ratio test which are multiply robust in the same sense as the multiply robust estimator and the empirical likelihood ratio test for the average treatment effect in Chapter 4. We conclude this thesis in Chapter 6 with some additional remarks on major results presented in the thesis along with several interesting topics worthy of further exploration in the future.
Book Synopsis Statistical Inferences for Missing Data/causal Inferences Based on Modified Empirical Likelihood by : Sima Sharghi
Download or read book Statistical Inferences for Missing Data/causal Inferences Based on Modified Empirical Likelihood written by Sima Sharghi and published by . This book was released on 2021 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation we first modify profile empirical likelihood function conditioned on complete data to estimate the population mean in presence of missing values in the response variable. Also in Chapter 3 under the counterfactual potential outcome by Rubin (1974, 1976, 1977), we propose some methods to estimate causal effect. This dissertation specifically expands upon the work of Qin and Zhang (2007), as they fail to address two main shortcomings of their empirical likelihood utilization. The first flaw is when the estimation fails to exist. The second flaw is under- coverage probability of the confidence region. Both of these two flaws get exacerbated when the sample size is small.In Chapter 2, we modify the associated empirical likelihood function to obtain consistent estimators which address each of the shortcomings. Our adjusted-empirical-likelihood-based consistent estimator, using similar strategy to Chen et al. (2008), adds a point to the convex hull of the data to ensure the algorithm converges. Furthermore, inspired by Jing et al.2017, we propose a quadratic transformation to the associated empirical likelihood ratio test statistic to yield a consistent estimator with greater coverage probability.In Chapter 3 using the techniques developed in Chapter 2, adjusted empirical likelihood causal effect estimator which is consistent is developed.In Chapter 2 simulation study for estimating the mean response under the presence of missing values, both of our proposed estimators show competitive results compared with other historical method. These modified estimators generally outperform historical estimators in terms of RMSE and coverage probability. Chapter 3 simulations exhibit that the consistent adjusted empirical likelihood causal effect estimator is competitive compared to the historical methods.Along the way, we also propose a weighted adjusted empirical likelihood for both estimating the mean response, and causal effect, which is proved to be consistent under the presence of missing values in the response variable. This estimator exhibits competitive results compared with the empirical likelihood estimator proposed by Qin and Zhang (2007).
Book Synopsis Empirical Likelihood Methods in Missing Response Problems and Causal Interference by : Kaili Ren
Download or read book Empirical Likelihood Methods in Missing Response Problems and Causal Interference written by Kaili Ren and published by . This book was released on 2016 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: This manuscript contains three topics in missing data problems and causal inference. First, we propose an empirical likelihood estimator as an alternative to Qin and Zhang (2007) in missing response problems under MAR assumption. A likelihood-based method is used to obtain the mean propensity score instead of a moment-based method. Our proposed estimator shares the double-robustness property and achieves the semiparametric efficiency lower bound when the regression model and the propensity score model are both correctly specified. Our proposed estimator has better performance when the propensity score is correctly specified. In addition, we extend our proposed method to the estimation of ATE in observational causal inferences. By utilizing the proposed method on a dataset from the CORAL clinical trial, we study the causal effect of cigarette smoking on renal function in patients with ARAS. The higher cystatin C and lower CKD-EPI GFR for smokers demonstrate the negative effect of smoking on renal function in patients with ARAS. Second, we explore a more efficient approach in missing response problems under MAR assumption. Instead of using one propensity score model and one working regression model, we postulate multiple working regression and propensity score models. Moreover, rather than maximizing the conditional likelihood, we maximize the full likelihood under constraints with respect to the postulated parametric functions. Our proposed estimator is consistent if one of the propensity scores is correctly specified and it achieves the semiparametric efficiency lower bound when one of the working regression models is correctly specified as well. This estimator is more efficient than other current estimators when one of the propensity scores is correctly specified. Finally, I propose empirical likelihood confidence intervals in missing data problems, which make very weak distribution assumptions. We show that the -2 empirical log-likelihood ratio function follows a scaled chi-squared distribution if either the working propensity score or the working regression model is correctly specified. If the two models are both correctly specified, the -2 empirical log-likelihood ratio function follows a chi-squared distribution. Empirical likelihood confidence intervals perform better than Wald confidence intervals of the AIPW estimator, when sample size is small and distribution of the response is highly skewed. In addition, empirical likelihood confidence intervals for ATE can also be built in causal inference.
Book Synopsis Empirical Likelihood by : Art B. Owen
Download or read book Empirical Likelihood written by Art B. Owen and published by CRC Press. This book was released on 2001-05-18 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: Empirical likelihood provides inferences whose validity does not depend on specifying a parametric model for the data. Because it uses a likelihood, the method has certain inherent advantages over resampling methods: it uses the data to determine the shape of the confidence regions, and it makes it easy to combined data from multiple sources. It al
Book Synopsis Empirical Likelihood Inference for Two-sample Problems by : Ying Yan
Download or read book Empirical Likelihood Inference for Two-sample Problems written by Ying Yan and published by . This book was released on 2010 with total page 40 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we are interested in empirical likelihood (EL) methods for two-sample problems, with focus on the difference of the two population means. A weighted empirical likelihood method (WEL) for two-sample problems is developed. We also consider a scenario where sample data on auxiliary variables are fully observed for both samples but values of the response variable are subject to missingness. We develop an adjusted empirical likelihood method for inference of the difference of the two population means for this scenario where missing values are handled by a regression imputation method. Bootstrap calibration for WEL is also developed. Simulation studies are conducted to evaluate the performance of naive EL, WEL and WEL with bootstrap calibration (BWEL) with comparison to the usual two-sample t-test in terms of power of the tests and coverage accuracies. Simulation for the adjusted EL for the linear regression model with missing data is also conducted.
Book Synopsis Causal Inference with Covariate Balance Optimization by : Yuying Xie
Download or read book Causal Inference with Covariate Balance Optimization written by Yuying Xie and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Causal inference is a popular problem in biostatistics, economics, and health science studies. The goal of this thesis is to develop new methods for the estimation of causal effects using propensity scores or inverse probability weights where weights are chosen in such a way to achieve balance in covariates across the treatment groups. In Chapter 1, we introduce Neyman-Rubin Causal framework and causal inference with propensity scores. The importance of covariate balancing in causal inference is furthered discussed in this chapter. Besides, some general definitions and notations for causal inference are provided with many other popular propensity score approaches or weighting techniques in Chapter 2. In Chapter 3, we describe a new model averaging approach to propensity score estimation in which parametric and nonparametric estimates are combined to achieve covariate balance. Simulation studies are conducted across different scenarios varying in the degree of interactions and nonlinearity in the treatment model. The results show that the proposed method produces less bias and smaller standard errors than existing approaches. They also show that a model averaging approach with the objective of minimizing the average Kolmogorov-Smirnov statistic leads to the best performance. The proposed approach is applied to a real data set in evaluating the causal effect of formula or mixed feeding versus exclusive breastfeeding in the first month of life on a child's BMI Z-score at age 4. The data analysis shows that formula or mixed feeding is more likely to lead to obesity at age 4, compared to exclusive breastfeeding. In Chapter 4, we propose using kernel distance to measure balance across different treatment groups and propose a new propensity score estimator by setting the kernel distance to be zero. Compared to other balance measures, such as absolute standardized mean difference (ASMD) and Kolmogorov Smirnov (KS) statistic, kernel distance is one of the best bias indicators in estimating the causal effect. That is, the balance metric based on kernel distance is shown to have the strongest correlation with the absolute bias in estimating the causal effect, compared to several commonly used balance metrics. The kernel distance constraints are solved by generalized method of moments. Simulation studies are conducted across different scenarios varying in the degree of nonlinearity in both the propensity score model and outcome model. The proposed approach produces smaller mean squared error in estimating causal treatment effects than many existing approaches including the well-known covariate balance propensity score (CBPS) approach when the propensity score model is misspecified. An application to data from the International Tobacco Control (ITC) policy evaluation project is provided. Often interest lies in the estimation of quantiles other than the average causal effect. Other quantities such as quantiles or the quantile treatment effect may be of interest. In Chapter 5, we propose a multiply robust method for estimating marginal quantiles of potential outcomes by achieving mean balance in (1) the propensity score, and (2) the conditional distributions of potential outcomes. An empirical likelihood or entropy measure can be utilized instead of using inverse probability weighting. Simulation studies are conducted across different scenarios of correctness in both the propensity score models and outcome models. Our estimator is consistent if any of the models are correctly specified.
Book Synopsis Semiparametric and Robust Methods for Complex Parameters in Causal Inference by : Wenjing Zheng
Download or read book Semiparametric and Robust Methods for Complex Parameters in Causal Inference written by Wenjing Zheng and published by . This book was released on 2014 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book In All Likelihood written by Yudi Pawitan and published by Oxford University Press. This book was released on 2001-06-21 with total page 552 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text concentrates on what can be achieved using the likelihood/Fisherian methods of taking into account uncertainty when studying a statistical problem. It takes the concept of the likelihood as the best method for unifying the demands of statistical modeling and theory of inference. Every likelihood concept is illustrated with realistic examples ranging from a simple comparison of two accident rates to complex studies that require generalized linear or semiparametric modeling. The emphasis is on likelihood not as just a device used to produce an estimate, but as an important tool for modeling.
Download or read book Causal inference written by K. J. Rothman and published by Kenneth Rothman. This book was released on 1988 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Statistical Inference as Severe Testing by : Deborah G. Mayo
Download or read book Statistical Inference as Severe Testing written by Deborah G. Mayo and published by Cambridge University Press. This book was released on 2018-09-20 with total page 503 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mounting failures of replication in social and biological sciences give a new urgency to critically appraising proposed reforms. This book pulls back the cover on disagreements between experts charged with restoring integrity to science. It denies two pervasive views of the role of probability in inference: to assign degrees of belief, and to control error rates in a long run. If statistical consumers are unaware of assumptions behind rival evidence reforms, they can't scrutinize the consequences that affect them (in personalized medicine, psychology, etc.). The book sets sail with a simple tool: if little has been done to rule out flaws in inferring a claim, then it has not passed a severe test. Many methods advocated by data experts do not stand up to severe scrutiny and are in tension with successful strategies for blocking or accounting for cherry picking and selective reporting. Through a series of excursions and exhibits, the philosophy and history of inductive inference come alive. Philosophical tools are put to work to solve problems about science and pseudoscience, induction and falsification.
Book Synopsis The Prevention and Treatment of Missing Data in Clinical Trials by : National Research Council
Download or read book The Prevention and Treatment of Missing Data in Clinical Trials written by National Research Council and published by National Academies Press. This book was released on 2010-12-21 with total page 163 pages. Available in PDF, EPUB and Kindle. Book excerpt: Randomized clinical trials are the primary tool for evaluating new medical interventions. Randomization provides for a fair comparison between treatment and control groups, balancing out, on average, distributions of known and unknown factors among the participants. Unfortunately, these studies often lack a substantial percentage of data. This missing data reduces the benefit provided by the randomization and introduces potential biases in the comparison of the treatment groups. Missing data can arise for a variety of reasons, including the inability or unwillingness of participants to meet appointments for evaluation. And in some studies, some or all of data collection ceases when participants discontinue study treatment. Existing guidelines for the design and conduct of clinical trials, and the analysis of the resulting data, provide only limited advice on how to handle missing data. Thus, approaches to the analysis of data with an appreciable amount of missing values tend to be ad hoc and variable. The Prevention and Treatment of Missing Data in Clinical Trials concludes that a more principled approach to design and analysis in the presence of missing data is both needed and possible. Such an approach needs to focus on two critical elements: (1) careful design and conduct to limit the amount and impact of missing data and (2) analysis that makes full use of information on all randomized participants and is based on careful attention to the assumptions about the nature of the missing data underlying estimates of treatment effects. In addition to the highest priority recommendations, the book offers more detailed recommendations on the conduct of clinical trials and techniques for analysis of trial data.
Book Synopsis Empirical Likelihood and Bootstrap Inference with Constraints by : Chunlin Wang
Download or read book Empirical Likelihood and Bootstrap Inference with Constraints written by Chunlin Wang and published by . This book was released on 2017 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt: Empirical likelihood and the bootstrap play influential roles in contemporary statistics. This thesis studies two distinct statistical inference problems, referred to as Part I and Part II, related to the empirical likelihood and bootstrap, respectively. Part I of this thesis concerns making statistical inferences on multiple groups of samples that contain excess zero observations. A unique feature of the target populations is that the distribution of each group is characterized by a non-standard mixture of a singular distribution at zero and a skewed nonnegative component. In Part I of this thesis, we propose modelling the nonnegative components using a semiparametric, multiple-sample, density ratio model (DRM). Under this semiparametric setup, we can efficiently utilize information from the combined samples even with unspecified underlying distributions. We first study the question of testing homogeneity of multiple nonnegative distributions when there is an excess of zeros in the data, under the proposed semiparametric setup. We develop a new empirical likelihood ratio (ELR) test for homogeneity and show that this ELR has a $\chi^2$-type limiting distribution under the homogeneous null hypothesis. A nonparametric bootstrap procedure is proposed to calibrate the finite-sample distribution of the ELR. The consistency of this bootstrap procedure is established under both the null and alternative hypotheses. Simulation studies show that the bootstrap ELR test has an accurate nominal type I error, is robust to changes of underlying distributions, is competitive to, and sometimes more powerful than, several popular one- and two-part tests. A real data example is used to illustrate the advantages of the proposed test. We next investigate the problem of comparing the means of multiple nonnegative distributions, with excess zero observations, under the proposed semiparametric setup. We develop a unified inference framework based on our new ELR statistic, and show that this ELR has a $\chi^2$-type limiting distribution under a general null hypothesis. This allows us to construct a new test for mean equality. Simulation results show favourable performance of the proposed ELR test compared with other existing tests for mean equality, especially when the correctly specified basis function in the DRM is the logarithm function. A real data set is analyzed to illustrate the advantages of the proposed method. In Part II of this thesis, we investigate the asymptotic behaviour of, the commonly used, bootstrap percentile confidence intervals when the parameters are subject to inequality constraints. We concentrate on the important one- and two-sample problems with data generated from distributions in the natural exponential family. Our attention is focused on quantifying asymptotic coverage probabilities of the percentile confidence intervals based on bootstrapping maximum likelihood estimators. We propose a novel local framework to study the subtle asymptotic behaviour of bootstrap percentile confidence intervals when the true parameter values are close to the boundary. Under this framework, we discover that when the true parameter is on, or close to, the restriction boundary, the local asymptotic coverage probabilities can always exceed the nominal level in the one-sample case; however, they can be, surprisingly, both under and over the nominal level in the two-sample case. The results provide theoretical justification and guidance on applying the bootstrap percentile method to constrained inference problems. The two individual parts of this thesis are connected by being referred to as {\em constrained statistical inference}. Specifically, in Part I, the semiparametric density ratio model uses an exponential tilting constraint, which is a type of equality constraint, on the parameter space. In Part II, we deal with inequality constraints, such as a boundary or ordering constraints, on the parameter space. For both parts, an important regularity condition in traditional likelihood inference, that parameters should be interior points of the parameter space, is violated. Therefore, the respective inference procedures involve non-standard asymptotics that create new technical challenges.
Book Synopsis Causal Inference with Measurement Error by : Di Shu
Download or read book Causal Inference with Measurement Error written by Di Shu and published by . This book was released on 2018 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: Causal inference methods have been widely used in biomedical sciences and social sciences, among many others. With different assumptions, various methods have been proposed to conduct causal inference with interpretable results. The validity of most existing methods, if not all, relies on a crucial condition: all the variables need to be precisely measured. This condition, however, is commonly violated. In many applications, the collected data are not precisely measured and are subject to measurement error. Ignoring measurement error effects can lead to severely biased results and misleading conclusions. In order to obtain reliable inference results, measurement error effects should be carefully addressed. Outside the context of causal inference, research on measurement error problems has been extensive and a large body of methods have been developed. In the paradigm of causal inference, however, there is limited research on measurement error problems, although an increasing, but still scarce, literature has emerged. Certainly, this is an area that deserves in-depth investigation. Motivated by this, this thesis focuses on causal inference with measurement error. We investigate the impact of measurement error and propose methods which correct for measurement error effects for several useful settings. This thesis consists of nine chapters. As a preliminary, Chapter 1 gives an introduction to causal inference, measurement error and other features such as missing data, as well as an overview of existing methods on causal inference with measurement error. In this chapter we also describe the problems of our interest that will be investigated in depth in subsequent chapters. Chapter 2 considers estimation of the causal odds ratio, the causal risk ratio and the causal risk difference in the presence of measurement error in confounders, possibly time-varying. By adapting two correction methods for measurement error effects applicable for the noncausal context, we propose valid methods which consistently estimate the causal effect measures for settings with error-prone confounders. Furthermore, we develop a linear combination based method to construct estimators with improved asymptotic efficiency. Chapter 3 focuses on the inverse-probability-of-treatment weighted (IPTW) estimation of causal parameters under marginal structural models with error-contaminated and time-varying confounders. To account for bias due to imprecise measurements, we develop several correction methods. Both the so-called stabilized and unstabilized weighting strategies are covered in the development. In Chapter 4, measurement error in outcomes is of concern. For settings of inverse probability weighting (IPW) estimation, we study the impact of measurement error for both continuous and binary outcome variables and reveal interesting consequences of the naive analysis which ignores measurement error. When a continuous outcome variable is mismeasured under an additive measurement error model, the naive analysis may still yield a consistent estimator; when the outcome is binary, we derive the asymptotic bias in a closed-form. Furthermore, we develop consistent estimation procedures for practical scenarios where either validation data or replicates are available. With validation data, we propose an efficient method. To provide protection against model misspecification, we further develop a doubly robust estimator which is consistent even when one of the treatment model and the outcome model is misspecified. In Chapter 5, the research problem of interest is to deal with measurement error generated from more than one sources. We study the IPW estimation for settings with mismeasured covariates and misclassified outcomes. To correct for measurement error and misclassification effects simultaneously, we develop two estimation methods to facilitate different forms of the treatment model. Our discussion covers a broad scope of treatment models including typically assumed logistic regression models as well as general treatment assignment mechanisms. Chapters 2-5 emphasize addressing measurement error effects on causal inference. In applications, we may be further challenged by additional data features. For instance, missing values frequently occur in the data collection process in addition to measurement error. In Chapter 6, we investigate the problem for which both missingness and misclassification may be present in the binary outcome variable. We particularly consider the IPW estimation and derive the asymptotic biases of three types of naive analysis which ignore either missingness or misclassification or both. We develop valid estimation methods to correct for missingness and misclassification effects simultaneously. To provide protection against misspecification, we further propose a doubly robust correction method. Doubly robust estimators developed in Chapter 6 offer us a viable way to address issues of model misspecification and they can be easily applied for practical problems. However, such an appealing property does not say that doubly robust estimators have no weakness. When both the treatment model and the outcome model are misspecified, such estimators will not necessarily be consistent. Driven by this consideration, in Chapter 7, we propose new estimation methods to correct for effects of misclassification and/or missingness in outcomes. Differing from the doubly robust estimators which are constructed based on a single treatment model and a single outcome model, the new methods are developed by considering a set of treatment models and a set of outcome models. Such enlargements of the associated models enable us to construct consistent estimators which will enjoy the so-called multiple robustness, a property that has been discussed in the literature of missing data. To expedite the application of our developed methods, we implement the proposed methods in Chapter 4 and develop an R package for general users. The details are included in Chapter 8. The thesis concludes with a discussion in Chapter 9.
Book Synopsis Semiparametric Theory and Missing Data by : Anastasios Tsiatis
Download or read book Semiparametric Theory and Missing Data written by Anastasios Tsiatis and published by Springer Science & Business Media. This book was released on 2007-01-15 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book summarizes current knowledge regarding the theory of estimation for semiparametric models with missing data, in an organized and comprehensive manner. It starts with the study of semiparametric methods when there are no missing data. The description of the theory of estimation for semiparametric models is both rigorous and intuitive, relying on geometric ideas to reinforce the intuition and understanding of the theory. These methods are then applied to problems with missing, censored, and coarsened data with the goal of deriving estimators that are as robust and efficient as possible.
Book Synopsis Causal Inference in Statistics by : Judea Pearl
Download or read book Causal Inference in Statistics written by Judea Pearl and published by John Wiley & Sons. This book was released on 2016-01-25 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: CAUSAL INFERENCE IN STATISTICS A Primer Causality is central to the understanding and use of data. Without an understanding of cause–effect relationships, we cannot use data to answer questions as basic as "Does this treatment harm or help patients?" But though hundreds of introductory texts are available on statistical methods of data analysis, until now, no beginner-level book has been written about the exploding arsenal of methods that can tease causal information from data. Causal Inference in Statistics fills that gap. Using simple examples and plain language, the book lays out how to define causal parameters; the assumptions necessary to estimate causal parameters in a variety of situations; how to express those assumptions mathematically; whether those assumptions have testable implications; how to predict the effects of interventions; and how to reason counterfactually. These are the foundational tools that any student of statistics needs to acquire in order to use statistical methods to answer causal questions of interest. This book is accessible to anyone with an interest in interpreting data, from undergraduates, professors, researchers, or to the interested layperson. Examples are drawn from a wide variety of fields, including medicine, public policy, and law; a brief introduction to probability and statistics is provided for the uninitiated; and each chapter comes with study questions to reinforce the readers understanding.
Book Synopsis Empirical Likelihood Inference for the Accelerated Failure Time Model Via Kendall Estimating Equation by : Yinghua Lu
Download or read book Empirical Likelihood Inference for the Accelerated Failure Time Model Via Kendall Estimating Equation written by Yinghua Lu and published by . This book was released on 2009 with total page 55 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we study two methods for inference of parameters in the accelerated failure time model with right censoring data. One is the Wald-type method, which involves parameter estimation. The other one is empirical likelihood method, which is based on the asymptotic distribution of likelihood ratio. We employ a monotone censored data version of Kendall estimating equation, and construct confidence intervals from both methods. In the simulation studies, we compare the empirical likelihood (EL) and the Wald-type procedure in terms of coverage accuracy and average length of confidence intervals. It is concluded that the empirical likelihood method has a better performance. We also compare the EL for Kendall's rank regression estimator with the EL for other well known estimators and find advantages of the EL for Kendall estimator for small size sample. Finally, a real clinical trial data is used for the purpose of illustration.
Book Synopsis Elements of Causal Inference by : Jonas Peters
Download or read book Elements of Causal Inference written by Jonas Peters and published by MIT Press. This book was released on 2017-11-29 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: A concise and self-contained introduction to causal inference, increasingly important in data science and machine learning. The mathematization of causality is a relatively recent development, and has become increasingly important in data science and machine learning. This book offers a self-contained and concise introduction to causal models and how to learn them from data. After explaining the need for causal models and discussing some of the principles underlying causal inference, the book teaches readers how to use causal models: how to compute intervention distributions, how to infer causal models from observational and interventional data, and how causal ideas could be exploited for classical machine learning problems. All of these topics are discussed first in terms of two variables and then in the more general multivariate case. The bivariate case turns out to be a particularly hard problem for causal learning because there are no conditional independences as used by classical methods for solving multivariate cases. The authors consider analyzing statistical asymmetries between cause and effect to be highly instructive, and they report on their decade of intensive research into this problem. The book is accessible to readers with a background in machine learning or statistics, and can be used in graduate courses or as a reference for researchers. The text includes code snippets that can be copied and pasted, exercises, and an appendix with a summary of the most important technical concepts.
