PARAMETRIC FRACTIONAL IMPUTATION FOR NON‐IGNORABLE CATEGORICAL MISSING DATA WITH FOLLOW‐UP

Incomplete data subject to non‐ignorable non‐response are often encountered in practice and have a non‐identifiability problem. A follow‐up sample is randomly selected from the set of non‐respondents to avoid the non‐identifiability problem and get complete responses. Glynn, Laird, & Rubin analyzed non‐ignorable missing data with a follow‐up sample under a pattern mixture model. In this article, maximum likelihood estimation of parameters of the categorical missing data is considered with a follow‐up sample under a selection model. To estimate the parameters with non‐ignorable missing data, the EM algorithm with weighting, proposed by Ibrahim, is used. That is, in the E‐step, the weighted mean is calculated using the fractional weights for imputed data. Variances are estimated using the approximated jacknife method. Simulation results are presented to compare the proposed method with previously presented methods.     

Fractional Regression Hot Deck Imputation Weight Adjustment

Fractional regression hot deck imputation (FRHDI) imputes multiple values for each instance of a missing dependent variable. The imputed values are equal to the predicted value plus multiple random residuals. Fractional weights enable variance estimation and preserve correlations. In some circumstances with some starting weight values, existing procedures for computing FRHDI weights can produce negative values. We discuss procedures for constructing non-negative adjusted fractional weights for FRHDI and study performance of the algorithm using simulation. The algorithm can be used effectively with FRDHI procedures for handling missing data in the context of a complex sample survey.     

Nonparametric curve estimation with missing data: A general empirical process approach

A general nonparametric imputation procedure, based on kernel regression, is proposed to estimate points as well as set- and function-indexed parameters when the data are missing at random (MAR). The proposed method works by imputing a specific function of a missing value (and not the missing value itself), where the form of this specific function is dictated by the parameter of interest. Both single and multiple imputations are considered. The associated empirical processes provide the right tool to study the uniform convergence properties of the resulting estimators. Our estimators include, as special cases, the imputation estimator of the mean, the estimator of the distribution function proposed by Cheng and Chu [1996. Kernel estimation of distribution functions and quantiles with missing data. Statist. Sinica 6, 63–78], imputation estimators of a marginal density, and imputation estimators of regression functions.     

Augmented inverse probability weighted fractional imputation in quantile regression

By employing all the observed information and the optimal augmentation term, we propose an augmented inverse probability weighted fractional imputation method (AFI) to handle covariates missing at random in quantile regression. Compared with the existing completely case analysis, inverse probability weighting, multiple imputation and fractional imputation based on quantile regression model with missing covarites, we carry out simulation study to investigate its performance in estimation accuracy and efficiency, computational efficiency and estimation robustness. We also talk about the influence of imputation replicates in our AFI. Finally, we apply our methodology to part of the National Health and Nutrition Examination Survey data.     

缺失偏态数据下线性回归模型的统计推断

研究缺失偏态数据下线性回归模型的参数估计问题,针对缺失偏态数据,为克服样本分布扭曲缺点和提高模型的回归系数、尺度参数和偏度参数的估计效果,提出了一种适合偏态数据下线性回归模型中缺失数据的修正回归插补方法.通过随机模拟和实例研究,并与均值插补、回归插补、随机回归插补方法比较,结果表明所提出的修正回归插补方法是有效可行的.     

基于分层模型的缺失数据插补方法研究

大规模抽样调查多采用复杂抽样设计,得到具有分层嵌套结构的调查数据集,其中不可避免会遇到数据缺失问题,针对分层结构含缺失数据集的插补策略目前鲜有研究。本文将Gibbs算法应用到分层含缺失数据集的多重插补过程中,分别研究了固定效应模型插补法和随机效应模型插补法,进而通过理论推导和数值模拟,在不同组内相关系数、群组规模、数据缺失比例等情形下,从参数估计结果的无偏性和有效性两方面,比较不同方法的插补效果,给出插补模型的选择建议。研究结果表明,采用随机效应模型作为插补模型时,得到的参数估计结果更准确,而固定效应模型作为插补模型操作相对简便,在数据缺失比例较小、组内相关系数较大、群组规模较大等情形下,可以采用固定效应插补模型,否则建议采用随机效应插补模型。     

Variable selection for high-dimensional generalized linear model with block-missing data

In modern scientific research, multiblock missing data emerges with synthesizing information across multiple studies. However, existing imputation methods for handling block-wise missing data either focus on the single-block missing pattern or heavily rely on the model structure. In this study, we propose a single regression-based imputation algorithm for multiblock missing data. First, we conduct a sparse precision matrix estimation based on the structure of block-wise missing data. Second, we impute the missing blocks with their means conditional on the observed blocks. Theoretical results about variable selection and estimation consistency are established in the context of a generalized linear model. Moreover, simulation studies show that compared with existing methods, the proposed imputation procedure is robust to various missing mechanisms because of the good properties of regression imputation. An application to Alzheimer's Disease Neuroimaging Initiative data also confirms the superiority of our proposed method.     

Nonparametric conditional mean imputation

Imputation is a much used method for handling missing data. It is appealing as it separates the missing data part of the analysis, which is handled by imputation, and the estimation part, which is handled by complete data methods. Most imputation methods, however, either rely on strict parametric assumptions or are rather ad hoc in which case they often only work approximately under even stricter assumptions. In this paper a non-parametric imputation method is proposed. Since it is non-parametric it works under quite general assumptions. In particular, a model for the complete data is not required in the imputation step, and the complete data method used after the imputation may be a general estimating equation for estimating a finite-dimensional parameter. Large sample results for the resulting estimator are given.     

Validity and efficiency in analyzing ordinal responses with missing observations

This article addresses issues in creating public-use data files in the presence of missing ordinal responses and subsequent statistical analyses of the dataset by users. The authors propose a fully efficient fractional imputation (FI) procedure for ordinal responses with missing observations. The proposed imputation strategy retrieves the missing values through the full conditional distribution of the response given the covariates and results in a single imputed data file that can be analyzed by different data users with different scientific objectives. Two most critical aspects of statistical analyses based on the imputed data set,  validity  and  efficiency, are examined through regression analysis involving the ordinal response and a selected set of covariates. It is shown through both theoretical development and simulation studies that, when the ordinal responses are missing at random, the proposed FI procedure leads to valid and highly efficient inferences as compared to existing methods. Variance estimation using the fractionally imputed data set is also discussed. The Canadian Journal of Statistics 48: 138–151; 2020 © 2019 Statistical Society of Canada     

Empirical likelihood confidence intervals under imputation for missing survey data from stratified simple random sampling

Missing observations due to non‐response are commonly encountered in data collected from sample surveys. The focus of this article is on item non‐response which is often handled by filling in (or imputing) missing values using the observed responses (donors). Random imputation (single or fractional) is used within homogeneous imputation classes that are formed on the basis of categorical auxiliary variables observed on all the sampled units. A uniform response rate within classes is assumed, but that rate is allowed to vary across classes. We construct confidence intervals (CIs) for a population parameter that is defined as the solution to a smooth estimating equation with data collected using stratified simple random sampling. The imputation classes are assumed to be formed across strata. Fractional imputation with a fixed number of random draws is used to obtain an imputed estimating function. An empirical likelihood inference method under the fractional imputation is proposed and its asymptotic properties are derived. Two asymptotically correct bootstrap methods are developed for constructing the desired CIs. In a simulation study, the proposed bootstrap methods are shown to outperform traditional bootstrap methods and some non‐bootstrap competitors under various simulation settings. The Canadian Journal of Statistics 47: 281–301; 2019 © 2019 Statistical Society of Canada     

Imputation for statistical inference with coarse data

Coarse data is a general type of incomplete data that includes grouped data, censored data, and missing data. The likelihood‐based estimation approach with coarse data is challenging because the likelihood function is in integral form. The Monte Carlo EM algorithm of Wei & Tanner [Wei & Tanner (1990). Journal of the American Statistical Association, 85, 699–704] is adapted to compute the maximum likelihood estimator in the presence of coarse data. Stochastic coarse data is also covered and the computation can be implemented using the parametric fractional imputation method proposed by Kim [Kim (2011). Biometrika, 98, 119–132]. Results from a limited simulation study are presented. The proposed method is also applied to the Korean Longitudinal Study of Aging (KLoSA). The Canadian Journal of Statistics 40: 604–618; 2012 © 2012 Statistical Society of Canada     

Efficient Robust Estimation for Linear Models with Missing Response at Random

Coefficient estimation in linear regression models with missing data is routinely carried out in the mean regression framework. However, the mean regression theory breaks down if the error variance is infinite. In addition, correct specification of the likelihood function for existing imputation approach is often challenging in practice, especially for skewed data. In this paper, we develop a novel composite quantile regression and a weighted quantile average estimation procedure for parameter estimation in linear regression models when some responses are missing at random. Instead of imputing the missing response by randomly drawing from its conditional distribution, we propose to impute both missing and observed responses by their estimated conditional quantiles given the observed data and to use the parametrically estimated propensity scores to weigh check functions that define a regression parameter. Both estimation procedures are resistant to heavy‐tailed errors or outliers in the response and can achieve nice robustness and efficiency. Moreover, we propose adaptive penalization methods to simultaneously select significant variables and estimate unknown parameters. Asymptotic properties of the proposed estimators are carefully investigated. An efficient algorithm is developed for fast implementation of the proposed methodologies. We also discuss a model selection criterion, which is based on an IC_Q ‐type statistic, to select the penalty parameters. The performance of the proposed methods is illustrated via simulated and real data sets.     

A comparison study of nonparametric imputation methods

Consider estimation of a population mean of a response variable when the observations are missing at random with respect to the covariate. Two common approaches to imputing the missing values are the nonparametric regression weighting method and the Horvitz-Thompson (HT) inverse weighting approach. The regression approach includes the kernel regression imputation and the nearest neighbor imputation. The HT approach, employing inverse kernel-estimated weights, includes the basic estimator, the ratio estimator and the estimator using inverse kernel-weighted residuals. Asymptotic normality of the nearest neighbor imputation estimators is derived and compared to kernel regression imputation estimator under standard regularity conditions of the regression function and the missing pattern function. A comprehensive simulation study shows that the basic HT estimator is most sensitive to discontinuity in the missing data patterns, and the nearest neighbors estimators can be insensitive to missing data patterns unbalanced with respect to the distribution of the covariate. Empirical studies show that the nearest neighbor imputation method is most effective among these imputation methods for estimating a finite population mean and for classifying the species of the iris flower data.     

Likelihood‐based Inference with Missing Data Under Missing‐at‐Random

Likelihood‐based inference with missing data is challenging because the observed log likelihood is often an (intractable) integration over the missing data distribution, which also depends on the unknown parameter. Approximating the integral by Monte Carlo sampling does not necessarily lead to a valid likelihood over the entire parameter space because the Monte Carlo samples are generated from a distribution with a fixed parameter value. We consider approximating the observed log likelihood based on importance sampling. In the proposed method, the dependency of the integral on the parameter is properly reflected through fractional weights. We discuss constructing a confidence interval using the profile likelihood ratio test. A Newton–Raphson algorithm is employed to find the interval end points. Two limited simulation studies show the advantage of the Wilks inference over the Wald inference in terms of power, parameter space conformity and computational efficiency. A real data example on salamander mating shows that our method also works well with high‐dimensional missing data.     

Efficient estimation in a regression model with missing responses

This article examines methods to efficiently estimate the mean response in a linear model with an unknown error distribution under the assumption that the responses are missing at random. We show how the asymptotic variance is affected by the estimator of the regression parameter, and by the imputation method. To estimate the regression parameter, the ordinary least squares is efficient only if the error distribution happens to be normal. If the errors are not normal, then we propose a one step improvement estimator or a maximum empirical likelihood estimator to efficiently estimate the parameter.To investigate the imputation’s impact on the estimation of the mean response, we compare the listwise deletion method and the propensity score method (which do not use imputation at all), and two imputation methods. We demonstrate that listwise deletion and the propensity score method are inefficient. Partial imputation, where only the missing responses are imputed, is compared to full imputation, where both missing and non-missing responses are imputed. Our results reveal that, in general, full imputation is better than partial imputation. However, when the regression parameter is estimated very poorly, the partial imputation will outperform full imputation. The efficient estimator for the mean response is the full imputation estimator that utilizes an efficient estimator of the parameter.     

Semiparametric Analysis of Isotonic Errors-in-Variables Regression Models with Missing Response

This article is concerned with the estimation problem in the semiparametric isotonic regression model when the covariates are measured with additive errors and the response is missing at random. An inverse marginal probability weighted imputation approach is developed to estimate the regression parameters and a least-square approach under monotone constraint is employed to estimate the functional component. We show that the proposed estimator of the regression parameter is root-n consistent and asymptotically normal and the isotonic estimator of the functional component, at a fixed point, is cubic root-n consistent. A simulation study is conducted to examine the finite-sample properties of the proposed estimators. A data set is used to demonstrate the proposed approach.     

Missing data techniques for multilevel data: implications of model misspecification

When modeling multilevel data, it is important to accurately represent the interdependence of observations within clusters. Ignoring data clustering may result in parameter misestimation. However, it is not well established to what degree parameter estimates are affected by model misspecification when applying missing data techniques (MDTs) to incomplete multilevel data. We compare the performance of three MDTs with incomplete hierarchical data. We consider the impact of imputation model misspecification on the quality of parameter estimates by employing multiple imputation under assumptions of a normal model (MI/NM) with two-level cross-sectional data when values are missing at random on the dependent variable at rates of 10%, 30%, and 50%. Five criteria are used to compare estimates from MI/NM to estimates from MI assuming a linear mixed model (MI/LMM) and maximum likelihood estimation to the same incomplete data sets. With 10% missing data (MD), techniques performed similarly for fixed-effects estimates, but variance components were biased with MI/NM. Effects of model misspecification worsened at higher rates of MD, with the hierarchical structure of the data markedly underrepresented by biased variance component estimates. MI/LMM and maximum likelihood provided generally accurate and unbiased parameter estimates but performance was negatively affected by increased rates of MD.     

Using data augmentation to correct for non-ignorable non-response when surrogate data are available: an application to the distribution of hourly pay

Summary.  The paper develops a data augmentation method to estimate the distribution function of a variable, which is partially observed, under a non-ignorable missing data mechanism, and where surrogate data are available. An application to the estimation of hourly pay distributions using UK Labour Force Survey data provides the main motivation. In addition to considering a standard parametric data augmentation method, we consider the use of hot deck imputation methods as part of the data augmentation procedure to improve the robustness of the method. The method proposed is compared with standard methods that are based on an ignorable missing data mechanism, both in a simulation study and in the Labour Force Survey application. The focus is on reducing bias in point estimation, but variance estimation using multiple imputation is also considered briefly.     

基于随机森林模型的分类数据缺失值插补

缺失数据是影响调查问卷数据质量的重要因素,对调查问卷中的缺失值进行插补可以显著提高调查数据的质量。调查问卷的数据类型多以分类型数据为主,数据挖掘技术中的分类算法是处理属性分类问题的常用方法,随机森林模型是众多分类算法中精度较高的方法之一。将随机森林模型引入调查问卷缺失数据的插补研究中,提出了基于随机森林模型的分类数据缺失值插补方法,并根据不同的缺失模式探讨了相应的插补步骤。通过与其它方法的实证模拟比较,表明随机森林插补法得到的插补值准确度更优、可信度更高。     

Skew-normal factor analysis models with incomplete data

Traditional factor analysis (FA) rests on the assumption of multivariate normality. However, in some practical situations, the data do not meet this assumption; thus, the statistical inference made from such data may be misleading. This paper aims at providing some new tools for the skew-normal (SN) FA model when missing values occur in the data. In such a model, the latent factors are assumed to follow a restricted version of multivariate SN distribution with additional shape parameters for accommodating skewness. We develop an analytically feasible expectation conditional maximization algorithm for carrying out parameter estimation and imputation of missing values under missing at random mechanisms. The practical utility of the proposed methodology is illustrated with two real data examples and the results are compared with those obtained from the traditional FA counterparts.