When data is partially missing at random, imputation and importance weighting are often used to estimate moments of the unobserved population. In this paper, we study 1-nearest neighbor (1NN) importance weighting, which estimates moments by replacing missing data with the complete data that is the nearest neighbor in the non-missing covariate space. We define an empirical measure, the 1NN measure, and show that it is weakly consistent for the measure of the missing data. The main idea behind this result is that the 1NN measure is performing inverse probability weighting in the limit. We study applications to missing data and mitigating the impact of covariate shift in prediction tasks.
@cite_21 outlines many of the common methods for dealing with missing data, including imputation, importance weighting, and likelihood based methods (such as the EM algorithm). Importance weighting, or propensity scoring, is commonly used for correcting sampling selection bias @cite_13 , which arises in survey sampling and causal inference. Tree-based methods and SVMs have been used to obtain estimates of the ideal importance weights in @cite_0 @cite_15 . Hot-deck imputation has been studied empirically in @cite_2 @cite_26 @cite_17 but is not a commonly used as model-based approaches.
@cite_26 @cite_21 @cite_0 @cite_2 @cite_15 @cite_13 @cite_17
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Hot deck imputation is a procedure in which missing items are replaced with values from respondents. A model supporting such procedures is the model in which response probabilities are assumed equal within imputation cells. An efficient version of hot deck imputation is described for the cell response model and a computationally efficient variance estimator is given. An approximation to the fully efficient procedure in which a small number of values are imputed for each nonrespondent is described. Variance estimation procedures are illustrated in a Monte Carlo study. 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. Abstract Causal effect modeling with naturalistic rather than experimental data is challenging. In observational studies participants in different treatment conditions may also differ on pretreatment characteristics that influence outcomes. Propensity score methods can theoretically eliminate these confounds for all observed covariates, but accurate estimation of propensity scores is impeded by large numbers of covariates, uncertain functional forms for their associations with treatment selection, and other problems. This article demonstrates that boosting, a modern statistical technique, can overcome many of these obstacles. The authors illustrate this approach with a study of adolescent probationers in substance abuse treatment programs. Propensity score weights estimated using boosting eliminate most pretreatment group differences and substantially alter the apparent relative effects of adolescent substance abuse treatment. Covariate data which are missing or measured with error form the subject of a growing body of statistical literature. Parametric methods have not been widely adopted, quite possibly due to the necessity of specifying the form of a 'nuisance function' not required for complete data analysis, and the non-robustness of the methods to mis-specification. A non-parametric counterpart of multiple imputation, known as 'hot deck', was proposed by Rubin (1987) and has been used by the Census Bureau to complete public-use databases. However, inference using this method has not been possible due to the distribution theory not being available. Recently, it has been shown that the hot deck estimator has the same asymptotic distribution as the 'mean score' estimator, so that inference using hot deck is now possible. The method is intuitively appealing and easily implemented. Furthermore, it accommodates missingness which depends on outcome, which is an important generalization of many currently available methods. In this paper, the hot deck multiple imputation method is explained, its asymptotic distribution presented and its application to data analysis demonstrated by an example. Machine learning techniques such as classification and regression trees (CART) have been suggested as promising alternatives to logistic regression for the estimation of propensity scores. The authors examined the performance of various CART-based propensity score models using simulated data. Hypothetical studies of varying sample sizes (n=500, 1000, 2000) with a binary exposure, continuous outcome, and ten covariates were simulated under seven scenarios differing by degree of non-linear and non-additive associations between covariates and the exposure. Propensity score weights were estimated using logistic regression (all main effects), CART, pruned CART, and the ensemble methods of bagged CART, random forests, and boosted CART. Performance metrics included covariate balance, standard error, percent absolute bias, and 95 confidence interval coverage. All methods displayed generally acceptable performance under conditions of either non-linearity or non-additivity alone. However, under conditions of both moderate non-additivity and moderate non-linearity, logistic regression had subpar performance, while ensemble methods provided substantially better bias reduction and more consistent 95 CI coverage. The results suggest that ensemble methods, especially boosted CART, may be useful for propensity score weighting. Abstract : The results of observational studies are often disputed because of nonrandom treatment assignment. For example, patients at greater risk may be overrepresented in some treatment group. This paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies. The propensity score is the (estimated) conditional probability of assignment to a particular treatment given a vector of observed covariates. Both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates. Applications include: matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching, multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations, and visual representation of multivariate adjustment by a two-dimensional plot. (Author) Hot deck imputation is a method for handling missing data in which each missing value is replaced with an observed response from a "similar" unit. Despite being used extensively in practice, the theory is not as well developed as that of other imputation methods. We have found that no consensus exists as to the best way to apply the hot deck and obtain inferences from the completed data set. Here we review different forms of the hot deck and existing research on its statistical properties. We describe applications of the hot deck currently in use, including the U.S. Census Bureau's hot deck for the Current Population Survey (CPS). We also provide an extended example of variations of the hot deck applied to the third National Health and Nutrition Examination Survey (NHANES III). Some potential areas for future research are highlighted. Copyright (c) 2010 The Authors. Journal compilation (c) 2010 International Statistical Institute.