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Author: Edward H. Kennedy
Publisher:
ISBN:
Category :
Languages : en
Pages : 248
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Book Description
Semiparametric doubly robust methods for causal inference help protect against bias due to model misspecification, while also reducing sensitivity to the curse of dimensionality (e.g., when high-dimensional covariate adjustment is necessary). However, doubly robust methods have not yet been developed in numerous important settings. In particular, standard semiparametric theory mostly only considers independent and identically distributed samples and smooth parameters that can be estimated at classical root-n rates. In this dissertation we extend this theory and develop novel methodology for three settings outside these bounds: (1) matched cohort studies, (2) nonparametric dose-response estimation, and (3) complex high-dimensional effects with continuous instrumental variables. After giving an introduction in Chapter 1, we show in Chapter 2 that, for matched cohort studies, efficient and doubly robust estimators of effects on the treated are computationally equivalent to standard estimators that ignore the non-standard sampling. We also show that matched cohort studies are often more efficient than random sampling for estimating effects on the treated, and derive the optimal number of matches for given matching variables. We apply our methods in a study of the effect of hysterectomy on the risk of cardiovascular disease. In Chapter 3 we develop a novel approach for causal dose-response curve estimation that is doubly robust without requiring any parametric assumptions, and which naturally incorporates general off-the-shelf machine learning. We derive asymptotic properties for a kernel-based version of our approach and propose a data-driven method for bandwidth selection. The methods are used to study the effect of hospital nurse staffing on excess readmissions penalties. In Chapter 4 we develop novel estimators of the local instrumental variable curve, which represents the treatment effect among compliers who would take treatment when the instrument passes some threshold. Our methods do not require parametric assumptions, allow for flexible data-adaptive estimation of effect modification, and are doubly robust. We derive asymptotic properties under weak conditions, and use the methods to study infant mortality effects of neonatal intensive care units with high versus low technical capacity, using travel time as an instrument.
Author: Edward H. Kennedy
Publisher:
ISBN:
Category :
Languages : en
Pages : 248
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Book Description
Semiparametric doubly robust methods for causal inference help protect against bias due to model misspecification, while also reducing sensitivity to the curse of dimensionality (e.g., when high-dimensional covariate adjustment is necessary). However, doubly robust methods have not yet been developed in numerous important settings. In particular, standard semiparametric theory mostly only considers independent and identically distributed samples and smooth parameters that can be estimated at classical root-n rates. In this dissertation we extend this theory and develop novel methodology for three settings outside these bounds: (1) matched cohort studies, (2) nonparametric dose-response estimation, and (3) complex high-dimensional effects with continuous instrumental variables. After giving an introduction in Chapter 1, we show in Chapter 2 that, for matched cohort studies, efficient and doubly robust estimators of effects on the treated are computationally equivalent to standard estimators that ignore the non-standard sampling. We also show that matched cohort studies are often more efficient than random sampling for estimating effects on the treated, and derive the optimal number of matches for given matching variables. We apply our methods in a study of the effect of hysterectomy on the risk of cardiovascular disease. In Chapter 3 we develop a novel approach for causal dose-response curve estimation that is doubly robust without requiring any parametric assumptions, and which naturally incorporates general off-the-shelf machine learning. We derive asymptotic properties for a kernel-based version of our approach and propose a data-driven method for bandwidth selection. The methods are used to study the effect of hospital nurse staffing on excess readmissions penalties. In Chapter 4 we develop novel estimators of the local instrumental variable curve, which represents the treatment effect among compliers who would take treatment when the instrument passes some threshold. Our methods do not require parametric assumptions, allow for flexible data-adaptive estimation of effect modification, and are doubly robust. We derive asymptotic properties under weak conditions, and use the methods to study infant mortality effects of neonatal intensive care units with high versus low technical capacity, using travel time as an instrument.
Author: Wenjing Zheng
Publisher:
ISBN:
Category :
Languages : en
Pages : 169
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Book Description
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.
Author: Di Shu
Publisher:
ISBN:
Category : Econometrics
Languages : en
Pages : 239
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Book Description
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.
Author: Heng Shu
Publisher:
ISBN:
Category :
Languages : en
Pages : 118
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In this dissertation, we develop improved estimation of average treatment effect on the treatment (ATT) which achieves double robustness, local efficiency, intrinsic efficiency and sample boundedness, using a calibrated likelihood approach. Moreover, we consider an extension of two-group causal inference problem to a general data combination problem, and develop estimators achieving desirable properties beyond double robustness and local efficiency. The proposed methods are shown, both theoretically and numerically, to be superior in robustness, efficiency or both to various existing estimators. In the first part, we review existing estimators on average treatment effect (ATE), mainly based on Tan (2006, 2010). This review provides a useful basis for improved estimation of average treatment effect on the treated (ATT). In the second part, we propose new methods to estimate the average treatment effect on the treated (ATT), which is of extensive interest in Econometrics, Biostatistics and other research fields. This problem seems to be often treated as a simple modification or extension of that of estimating overall average treatment effects (ATE). But the propensity score is no longer ancillary for estimation of ATT, in contrast with estimation of ATE. We study the efficient influence function and the corresponding semiparametric variance bound for the estimation of ATT under three different assumptions: a nonparametric model, a correct propensity score model and known propensity score. Then we construct Augmented Inverse Probability Weighted (AIPW) estimators which are locally efficient and doubly robust. Furthermore, we develop calibrated regression and likelihood estimators that are not only doubly robust and locally efficient, but also intrinsically e cient and sample bounded. Two simulations and real data analysis on a job training program are provided to demonstrate the advantage of our estimators compared with existing estimators. In the third part, we extend our methods to a general data combination problem for moment restriction models (Chen et al. 2008). Similarly, we derive augmented inverse probability weighted (AIPW) estimators that are locally efficient and doubly robust. Moreover, we develop calibrated regression and likelihood estimators which achieve double robustness, local efficiency and intrinsic efficiency. For illustration, we take the linear two-sample instrumental variable problem as an example, and derive all the relevant estimators by applying the general estimators in this specific example. Finally, a simulation study and an Econometric application on a public housing project are provided to demonstrate the superior performance of our improved estimators.
Author: Linbo Wang
Publisher:
ISBN:
Category :
Languages : en
Pages : 108
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Most complex observational and randomized studies are motivated by the potential of drawing causal statements. However, a usual statistical analysis may yield estimates that do not have causal interpretations. In fact, unlike most other parameters, identification of causal parameters usually relies on untestable assumptions. Moreover, even under these identification assumptions, estimation of causal parameters often relies on nuisance models. The parameter estimation in the nuisance models is crucial to obtain robust causal effect estimates. My research attempts to address these methodological hallenges. In Chapter 2 we study robust estimation of propensity score weights. The propensity score plays a central role in inferring causal effects from observational studies. In particular, weighting and subclassification are two principal approaches to estimate the average causal effect based on estimated propensity scores. Unlike the conventional version of the propensity score subclassification estimator, if the propensity score model is correctly specified, the weighting methods offer consistent and possibly efficient estimation of the average causal effect. However, this theoretical appeal may be diminished in practice by sensitivity to misspecification of the propensity score model. In contrast, subclassification methods are usually more robust to model misspecification. We hence propose to use subclassification for robust estimation of propensity score weights. Our approach is based on the intuition that the inverse probability weighting estimator can be seen as the limit of subclassification estimators as the number of subclasses goes to infinity. By formalizing this intuition, we propose novel propensity score weighting estimators that are both consistent and robust to model misspecification. Empirical studies show that the proposed estimators perform favorably compared to existing methods. In Chapter 3 we study identification and estimation of causal effects with outcomes truncated by death. It is common that in medical studies, the outcome of interest is truncated by death, meaning that a subject had died before the outcome could be measured. In this case, restricted analysis among survivors may be subject to selection bias. It is hence of interest to estimate the survivor average causal effect (SACE), defined as the average causal effect among subjects who would survive under either exposure. In this chapter, we consider the identification and estimation problems of the SACE. We propose to identify a substitution variable for the latent membership of the always-survivor group. The identifiability conditions required for a substitution variable are similar in idea to conditions for an instrumental variable. We show that the SACE is identifiable with use of a substitution variable, and propose novel model parameterizations for estimation of the SACE under our identification assumptions. Our approaches are illustrated via simulation studies and two data analyses. In Chapter 4, we study causal analysis of ordinal treatments and binary outcomes under truncation by death. It is common that in multi-arm randomized trials, the outcome of interest is “truncated by death,” meaning that it is only observed or well-defined conditioning on an intermediate outcome. In this case, in addition to pairwise contrasts, the joint inference for all treatment arms is also of interest. Under a monotonicity assumption we present methods for both pairwise and joint causal analyses of ordinal treatments and binary outcomes in presence of truncation by death. We illustrate via examples the appropriateness of our assumptions in different scientific contexts.
Author: Mark J. van der Laan
Publisher: Springer
ISBN: 3319653040
Category : Mathematics
Languages : en
Pages : 655
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Book Description
This textbook for graduate students in statistics, data science, and public health deals with the practical challenges that come with big, complex, and dynamic data. It presents a scientific roadmap to translate real-world data science applications into formal statistical estimation problems by using the general template of targeted maximum likelihood estimators. These targeted machine learning algorithms estimate quantities of interest while still providing valid inference. Targeted learning methods within data science area critical component for solving scientific problems in the modern age. The techniques can answer complex questions including optimal rules for assigning treatment based on longitudinal data with time-dependent confounding, as well as other estimands in dependent data structures, such as networks. Included in Targeted Learning in Data Science are demonstrations with soft ware packages and real data sets that present a case that targeted learning is crucial for the next generation of statisticians and data scientists. Th is book is a sequel to the first textbook on machine learning for causal inference, Targeted Learning, published in 2011. Mark van der Laan, PhD, is Jiann-Ping Hsu/Karl E. Peace Professor of Biostatistics and Statistics at UC Berkeley. His research interests include statistical methods in genomics, survival analysis, censored data, machine learning, semiparametric models, causal inference, and targeted learning. Dr. van der Laan received the 2004 Mortimer Spiegelman Award, the 2005 Van Dantzig Award, the 2005 COPSS Snedecor Award, the 2005 COPSS Presidential Award, and has graduated over 40 PhD students in biostatistics and statistics. Sherri Rose, PhD, is Associate Professor of Health Care Policy (Biostatistics) at Harvard Medical School. Her work is centered on developing and integrating innovative statistical approaches to advance human health. Dr. Rose’s methodological research focuses on nonparametric machine learning for causal inference and prediction. She co-leads the Health Policy Data Science Lab and currently serves as an associate editor for the Journal of the American Statistical Association and Biostatistics.
Author: Shandong Zhao
Publisher:
ISBN: 9781321301366
Category :
Languages : en
Pages : 145
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Propensity score methods have become a part of the standard toolkit for applied researchers who wish to ascertain causal effects from observational data. While they were originally developed for binary treatments, several researchers have proposed generalizations of the propensity score methodology for non-binary treatment regimes. In this article, we firstly review three main methods that generalize propensity scores in this direction, namely, inverse propensity weighting (IPW), the propensity function (P-FUNCTION), and the generalized propensity score (GPS), along with recent extensions of the GPS that aim to improve its robustness. We compare the assumptions, theoretical properties, and empirical performance of these methods. We propose three new methods that provide robust causal estimation based on the P-FUNCTION and GPS. While our proposed P-FUNCTION-based estimator preforms well, we generally advise caution in that all available methods can be biased by model misspecification and extrapolation. In a related line of research, we consider adjustment for posttreatment covariates in causal inference. Even in a randomized experiment, observations might have different compliance performance under treatment and control assignment. This posttreatment covariate cannot be adjusted using standard statistical methods. We review the principal stratification framework which allows for modeling this effect as part of its Bayesian hierarchical models. We generalize the current model to add the possibility of adjusting for pretreatment covariates. We also propose a new estimator of the average treatment effect over the entire population. In a third line of research, we discuss the spectral line detection problem in high energy astrophysics. We carefully review how this problem can be statistically formulated as a precise hypothesis test with point null hypothesis, why a usual likelihood ratio test does not apply for problem of this nature, and a doable fix to correctly quantify the p-value using the likelihood ratio test statistic via posterior predictive p-values. However, as p-values (including posterior predictive p-values) tend to overstate the evidence for the alternative hypothesis for precise hypothesis testing, we review a Bayesian alternative method to do the line detection problem using the Bayes factor. Although Bayes factors are generally criticized to be sensitive to the choice of prior distributions, we show that such prior dependence can reflect different scientific questions and thus be sensible. In fact, p-values have similar "subjective influence'' in that testing for the existance of a line at a fixed location or in an area with broad range can lead to very different conclusions. This is usually known as the look elsewhere effect in astrophysics.
Author: Mark J. van der Laan
Publisher: Springer Science & Business Media
ISBN: 1441997822
Category : Mathematics
Languages : en
Pages : 628
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Book Description
The statistics profession is at a unique point in history. The need for valid statistical tools is greater than ever; data sets are massive, often measuring hundreds of thousands of measurements for a single subject. The field is ready to move towards clear objective benchmarks under which tools can be evaluated. Targeted learning allows (1) the full generalization and utilization of cross-validation as an estimator selection tool so that the subjective choices made by humans are now made by the machine, and (2) targeting the fitting of the probability distribution of the data toward the target parameter representing the scientific question of interest. This book is aimed at both statisticians and applied researchers interested in causal inference and general effect estimation for observational and experimental data. Part I is an accessible introduction to super learning and the targeted maximum likelihood estimator, including related concepts necessary to understand and apply these methods. Parts II-IX handle complex data structures and topics applied researchers will immediately recognize from their own research, including time-to-event outcomes, direct and indirect effects, positivity violations, case-control studies, censored data, longitudinal data, and genomic studies.
Author: Romain Sébastien Neugebauer
Publisher:
ISBN:
Category :
Languages : en
Pages : 324
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Author: Yue Wang
Publisher:
ISBN:
Category :
Languages : en
Pages : 298
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