Estimation of Regression Coefficients When Some Regressors are not Always Observed
01 Sep 1994-Journal of the American Statistical Association (Taylor & Francis Group)-Vol. 89, Iss: 427, pp 846-866
TL;DR: In this paper, a new class of semiparametric estimators, based on inverse probability weighted estimating equations, were proposed for parameter vector α 0 of the conditional mean model when the data are missing at random in the sense of Rubin and the missingness probabilities are either known or can be parametrically modeled.
Abstract: In applied problems it is common to specify a model for the conditional mean of a response given a set of regressors. A subset of the regressors may be missing for some study subjects either by design or happenstance. In this article we propose a new class of semiparametric estimators, based on inverse probability weighted estimating equations, that are consistent for parameter vector α0 of the conditional mean model when the data are missing at random in the sense of Rubin and the missingness probabilities are either known or can be parametrically modeled. We show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0, and then deriving a representation for the efficient score...
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TL;DR: 2 general approaches that come highly recommended: maximum likelihood (ML) and Bayesian multiple imputation (MI) are presented and may eventually extend the ML and MI methods that currently represent the state of the art.
Abstract: Statistical procedures for missing data have vastly improved, yet misconception and unsound practice still abound. The authors frame the missing-data problem, review methods, offer advice, and raise issues that remain unresolved. They clear up common misunderstandings regarding the missing at random (MAR) concept. They summarize the evidence against older procedures and, with few exceptions, discourage their use. They present, in both technical and practical language, 2 general approaches that come highly recommended: maximum likelihood (ML) and Bayesian multiple imputation (MI). Newer developments are discussed, including some for dealing with missing data that are not MAR. Although not yet in the mainstream, these procedures may eventually extend the ML and MI methods that currently represent the state of the art.
10,568 citations
Cites background from "Estimation of Regression Coefficien..."
...1 In Rubin’s (1976) definition, Equation 1 is not required to hold for all possible values of R, but only for the R that actually appeared in the sample....
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...1 In Rubin’s (1976) definition, Equation 1 is not required to hold for all possible values of R, but only for the R that actually appeared in the sample. This technical point clarifies certain issues. For example, suppose that an experiment produced no missing values even though it could have. In that case, Equation 1 would hold because Ymis is empty, and Rubin’s (1976) results indicate that one should simply analyze the complete data without worrying about the fact that missing values could have arisen in hypothetical repetitions of the experiment....
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TL;DR: Essential features of multiple imputation are reviewed, with answers to frequently asked questions about using the method in practice.
Abstract: In recent years, multiple imputation has emerged as a convenient and flexible paradigm for analysing data with missing values. Essential features of multiple imputation are reviewed, with answers to frequently asked questions about using the method in practice.
3,387 citations
TL;DR: A non-response problem in survival analysis where the occurrence of missing data in the risk factor is related to mortality is studied, and multiple imputation is used to impute missing blood pressure and then analyse the data under a variety of non- response models.
Abstract: This paper studies a non-response problem in survival analysis where the occurrence of missing data in the risk factor is related to mortality. In a study to determine the influence of blood pressure on survival in the very old (85+ years), blood pressure measurements are missing in about 12.5 per cent of the sample. The available data suggest that the process that created the missing data depends jointly on survival and the unknown blood pressure, thereby distorting the relation of interest. Multiple imputation is used to impute missing blood pressure and then analyse the data under a variety of non-response models. One special modelling problem is treated in detail; the construction of a predictive model for drawing imputations if the number of variables is large. Risk estimates for these data appear robust to even large departures from the simplest non-response model, and are similar to those derived under deletion of the incomplete records.
2,147 citations
TL;DR: The results of simulation studies are presented which demonstrate that the finite sample performance of DR estimators is as impressive as theory would predict and the proposed method is applied to a cardiovascular clinical trial.
Abstract: The goal of this article is to construct doubly robust (DR) estimators in ignorable missing data and causal inference models. In a missing data model, an estimator is DR if it remains consistent when either (but not necessarily both) a model for the missingness mechanism or a model for the distribution of the complete data is correctly specified. Because with observational data one can never be sure that either a missingness model or a complete data model is correct, perhaps the best that can be hoped for is to find a DR estimator. DR estimators, in contrast to standard likelihood-based or (nonaugmented) inverse probability-weighted estimators, give the analyst two chances, instead of only one, to make a valid inference. In a causal inference model, an estimator is DR if it remains consistent when either a model for the treatment assignment mechanism or a model for the distribution of the counterfactual data is correctly specified. Because with observational data one can never be sure that a model for the treatment assignment mechanism or a model for the counterfactual data is correct, inference based on DR estimators should improve upon previous approaches. Indeed, we present the results of simulation studies which demonstrate that the finite sample performance of DR estimators is as impressive as theory would predict. The proposed method is applied to a cardiovascular clinical trial.
1,769 citations
Book•
30 Jul 2007TL;DR: In this article, the authors proposed a method to estimate causal effects by conditioning on observed variables to block backdoor paths in observational social science research, but the method is limited to the case of causal exposure and identification criteria for conditioning estimators.
Abstract: Part I. Causality and Empirical Research in the Social Sciences: 1. Introduction Part II. Counterfactuals, Potential Outcomes, and Causal Graphs: 2. Counterfactuals and the potential-outcome model 3. Causal graphs Part III. Estimating Causal Effects by Conditioning on Observed Variables to Block Backdoor Paths: 4. Models of causal exposure and identification criteria for conditioning estimators 5. Matching estimators of causal effects 6. Regression estimators of causal effects 7. Weighted regression estimators of causal effects Part IV. Estimating Causal Effects When Backdoor Conditioning Is Ineffective: 8. Self-selection, heterogeneity, and causal graphs 9. Instrumental-variable estimators of causal effects 10. Mechanisms and causal explanation 11. Repeated observations and the estimation of causal effects Part V. Estimation When Causal Effects Are Not Point Identified by Observables: 12. Distributional assumptions, set identification, and sensitivity analysis Part VI. Conclusions: 13. Counterfactuals and the future of empirical research in observational social science.
1,701 citations
References
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Book•
01 Jan 1987
TL;DR: This work states that maximum Likelihood for General Patterns of Missing Data: Introduction and Theory with Ignorable Nonresponse and large-Sample Inference Based on Maximum Likelihood Estimates is likely to be high.
Abstract: Preface.PART I: OVERVIEW AND BASIC APPROACHES.Introduction.Missing Data in Experiments.Complete-Case and Available-Case Analysis, Including Weighting Methods.Single Imputation Methods.Estimation of Imputation Uncertainty.PART II: LIKELIHOOD-BASED APPROACHES TO THE ANALYSIS OF MISSING DATA.Theory of Inference Based on the Likelihood Function.Methods Based on Factoring the Likelihood, Ignoring the Missing-Data Mechanism.Maximum Likelihood for General Patterns of Missing Data: Introduction and Theory with Ignorable Nonresponse.Large-Sample Inference Based on Maximum Likelihood Estimates.Bayes and Multiple Imputation.PART III: LIKELIHOOD-BASED APPROACHES TO THE ANALYSIS OF MISSING DATA: APPLICATIONS TO SOME COMMON MODELS.Multivariate Normal Examples, Ignoring the Missing-Data Mechanism.Models for Robust Estimation.Models for Partially Classified Contingency Tables, Ignoring the Missing-Data Mechanism.Mixed Normal and Nonnormal Data with Missing Values, Ignoring the Missing-Data Mechanism.Nonignorable Missing-Data Models.References.Author Index.Subject Index.
18,201 citations
TL;DR: In this article, it was shown that ignoring the process that causes missing data when making sampling distribution inferences about the parameter of the data, θ, is generally appropriate if and only if the missing data are missing at random and the observed data are observed at random, and then such inferences are generally conditional on the observed pattern of missing data.
Abstract: Two results are presented concerning inference when data may be missing. First, ignoring the process that causes missing data when making sampling distribution inferences about the parameter of the data, θ, is generally appropriate if and only if the missing data are “missing at random” and the observed data are “observed at random,” and then such inferences are generally conditional on the observed pattern of missing data. Second, ignoring the process that causes missing data when making Bayesian inferences about θ is generally appropriate if and only if the missing data are missing at random and the parameter of the missing data is “independent” of θ. Examples and discussion indicating the implications of these results are included.
8,197 citations
7,643 citations
TL;DR: In this paper, two sampling schemes are discussed in connection with the problem of determining optimum selection probabilities according to the information available in a supplementary variable, which is a general technique for the treatment of samples drawn without replacement from finite universes when unequal selection probabilities are used.
Abstract: This paper presents a general technique for the treatment of samples drawn without replacement from finite universes when unequal selection probabilities are used. Two sampling schemes are discussed in connection with the problem of determining optimum selection probabilities according to the information available in a supplementary variable. Admittedly, these two schemes have limited application. They should prove useful, however, for the first stage of sampling with multi-stage designs, since both permit unbiased estimation of the sampling variance without resorting to additional assumptions. * Journal Paper No. J2139 of the Iowa Agricultural Experiment Station, Ames, Iowa, Project 1005. Presented to the Institute of Mathematical Statistics, March 17, 1951.
3,990 citations
"Estimation of Regression Coefficien..." refers background in this paper
...In this setting, Manski and Lerman (1977) and Kalbfleisch and Lawless (1988) generalized an idea of Horvitz and Thompson (1952) and proposed the complete case es-...
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Book•
01 Jan 1989
TL;DR: Inverse Boundary Value Problems (IBV) as discussed by the authors, the heat equation is replaced by the Tikhonov regularization and regularization by Discretization (TBD) method.
Abstract: Normed Spaces.- Bounded and Compact Operators.- Riesz Theory.- Dual Systems and Fredholm Alternative.- Regularization in Dual Systems.- Potential Theory.- Singular Integral Equations.- Sobolev Spaces.- The Heat Equation.- Operator Approximations .-Degenerate Kernel Approximation.- Quadrature Methods.- Projection Methods.- Iterative Solution and Stability.- Equations of the First Kind.- Tikhonov Regularization.- Regularization by Discretization.- Inverse Boundary Value Problems.- References.- Index.
2,323 citations