Missing data mechanisms and their implications on the analysis of categorical data

We review some issues related to the implications of different missing data mechanisms on statistical inference for contingency tables and consider simulation studies to compare the results obtained under such models to those where the units with missing data are disregarded. We confirm that although, in general, analyses under the correct missing at random and missing completely at random models are more efficient even for small sample sizes, there are exceptions where they may not improve the results obtained by ignoring the partially classified data. We show that under the missing not at random (MNAR) model, estimates on the boundary of the parameter space as well as lack of identifiability of the parameters of saturated models may be associated with undesirable asymptotic properties of maximum likelihood estimators and likelihood ratio tests; even in standard cases the bias of the estimators may be low only for very large samples. We also show that the probability of a boundary solution obtained under the correct MNAR model may be large even for large samples and that, consequently, we may not always conclude that a MNAR model is misspecified because the estimate is on the boundary of the parameter space.     

CBPS-based estimation for linear models with responses missing at random

In this article, based on the covariate balancing propensity score (CBPS), estimators for the regression coefficients and the population mean are obtained, when the responses of linear models are missing at random. It is proved that the proposed estimators are asymptotically normal. In simulation studies and real example, the proposed estimators show improved performance relative to usual augmented inverse probability weighted estimators.     

Inference and Missing Data: Asymptotic Results

In Rubin (1976) the missing at random (MAR) and missing completely at random (MCAR) conditions are discussed. It is concluded that the MAR condition allows one to ignore the missing data mechanism when doing likelihood or Bayesian inference but also that the stronger MCAR condition is in some sense the weakest generally sufficient condition allowing (conditional) frequentist inference while ignoring the missing data mechanism. In this paper it is shown that (a slightly strengthened version of) the MAR condition is sufficient to yield ordinary large sample results for estimators and test statistics and thus may be used for (asymptotic) frequentist inference.     

A resampling method by perturbing the estimating functions for quantile regression with missing data

In this article, we propose a resampling method based on perturbing the estimating functions to compute the asymptotic variances of quantile regression estimators under missing at random condition. We prove that the conditional distributions of the resampling estimators are asymptotically equivalent to the distributions of quantile regression estimators. Our method can deal with complex situations, where the response and part of covariates are missing. Numerical results based on simulated and real data are provided under several designs.     

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.     

Rate/Mean Regression for Multiple-Sequence Recurrent Event Data with Missing Event Category

Abstract.  Censored recurrent event data frequently arise in biomedical studies. Often, the events are not homogenous, and may be categorized. We propose semiparametric regression methods for analysing multiple-category recurrent event data and consider the setting where event times are always known, but the information used to categorize events may be missing. Application of existing methods after censoring events of unknown category (i.e. 'complete-case' methods) produces consistent estimators only when event types are missing completely at random, an assumption which will frequently fail in practice. We propose methods, based on weighted estimating equations, which are applicable when event category missingness is missing at random. Parameter estimators are shown to be consistent and asymptotically normal. Finite sample properties are examined through simulations and the proposed methods are applied to an end-stage renal disease data set obtained from a national organ failure registry.     

A robust pairwise likelihood method for incomplete longitudinal binary data arising in clusters

Clustered longitudinal data feature cross‐sectional associations within clusters, serial dependence within subjects, and associations between responses at different time points from different subjects within the same cluster. Generalized estimating equations are often used for inference with data of this sort since they do not require full specification of the response model. When data are incomplete, however, they require data to be missing completely at random unless inverse probability weights are introduced based on a model for the missing data process. The authors propose a robust approach for incomplete clustered longitudinal data using composite likelihood. Specifically, pairwise likelihood methods are described for conducting robust estimation with minimal model assumptions made. The authors also show that the resulting estimates remain valid for a wide variety of missing data problems including missing at random mechanisms and so in such cases there is no need to model the missing data process. In addition to describing the asymptotic properties of the resulting estimators, it is shown that the method performs well empirically through simulation studies for complete and incomplete data. Pairwise likelihood estimators are also compared with estimators obtained from inverse probability weighted alternating logistic regression. An application to data from the Waterloo Smoking Prevention Project is provided for illustration. The Canadian Journal of Statistics 39: 34–51; 2011 © 2010 Statistical Society of Canada     

Estimating the population mean in the case of missing data using simple random sampling

In this paper, we suggest three new ratio estimators of the population mean using quartiles of the auxiliary variable when there are missing data from the sample units. The suggested estimators are investigated under the simple random sampling method. We obtain the mean square errors equations for these estimators. The suggested estimators are compared with the sample mean and ratio estimators in the case of missing data. Also, they are compared with estimators in Singh and Horn [Compromised imputation in survey sampling, Metrika 51 (2000), pp. 267–276], Singh and Deo [Imputation by power transformation, Statist. Papers 45 (2003), pp. 555–579], and Kadilar and Cingi [Estimators for the population mean in the case of missing data, Commun. Stat.-Theory Methods, 37 (2008), pp. 2226–2236] and present under which conditions the proposed estimators are more efficient than other estimators. In terms of accuracy and of the coverage of the bootstrap confidence intervals, the suggested estimators performed better than other estimators.     

Asymptotic bias in the linear mixed effects model under non-ignorable missing data mechanisms

Summary.  In longitudinal studies, missingness of data is often an unavoidable problem. Estimators from the linear mixed effects model assume that missing data are missing at random. However, estimators are biased when this assumption is not met. In the paper, theoretical results for the asymptotic bias are established under non-ignorable drop-out, drop-in and other missing data patterns. The asymptotic bias is large when the drop-out subjects have only one or no observation, especially for slope-related parameters of the linear mixed effects model. In the drop-in case, intercept-related parameter estimators show substantial asymptotic bias when subjects enter late in the study. Eight other missing data patterns are considered and these produce asymptotic biases of a variety of magnitudes.     

Semiparametric estimation for weighted average derivatives with responses missing at random

When responses are missing at random, we propose a semiparametric direct estimator for the missing probability and density-weighted average derivatives of a general nonparametric multiple regression function. An estimator for the normalized version of the weighted average derivatives is constructed as well using instrumental variables regression. The proposed estimators are computationally simple and asymptotically normal, and provide a solution to the problem of estimating index coefficients of single-index models with responses missing at random. The developed theory generalizes the method of the density-weighted average derivatives estimation of Powell et al. (1989) for the non-missing data case. Monte Carlo simulation studies are conducted to study the performance of the methods.     

Missing data estimators in the general linear model: an evaluation of simulated data as an experimental design

Previous simulations have reported second order missing data estimators to be superior to the more straightforward first order procedures such as mean value replacement. These simulations however were based on deterministic comparisonsbetween regression criteria even though simulated sampling is a random procedure. In this paper a simulation structured asan experimental design allows statistical testing of the various missing data estimators for the various regression criteria as well as different regression specifications. Our results indicate that although no missing data estimator is globally best many of the computationally simpler first order methods perform as well as the more expensive higher order estimators, contrary to some previous findings.     

Inverse Probability Weighting with Missing Predictors of Treatment Assignment or Missingness

Inverse probability weighting (IPW) can deal with confounding in non randomized studies. The inverse weights are probabilities of treatment assignment (propensity scores), estimated by regressing assignment on predictors. Problems arise if predictors can be missing. Solutions previously proposed include assuming assignment depends only on observed predictors and multiple imputation (MI) of missing predictors. For the MI approach, it was recommended that missingness indicators be used with the other predictors. We determine when the two MI approaches, (with/without missingness indicators) yield consistent estimators and compare their efficiencies.We find that, although including indicators can reduce bias when predictors are missing not at random, it can induce bias when they are missing at random. We propose a consistent variance estimator and investigate performance of the simpler Rubin’s Rules variance estimator. In simulations we find both estimators perform well. IPW is also used to correct bias when an analysis model is fitted to incomplete data by restricting to complete cases. Here, weights are inverse probabilities of being a complete case. We explain how the same MI methods can be used in this situation to deal with missing predictors in the weight model, and illustrate this approach using data from the National Child Development Survey.     

Wavelet estimation of density for censored data with censoring indicator missing at random

In this paper, we define the nonlinear wavelet estimator of density for the right censoring model with the censoring indicator missing at random (MAR), and develop its asymptotic expression for mean integrated squared error (MISE). Unlike for kernel estimator, the MISE expression of the estimator is not affected by the presence of discontinuities in the curve. Meanwhile, asymptotic normality of the estimator is established. The proposed estimator can reduce to the estimator defined by Li [Non-linear wavelet-based density estimators under random censorship. J Statist Plann Inference. 2003;117(1):35–58] when the censoring indicator MAR does not occur and a bandwidth in non-parametric estimation is close to zero. Also, we define another two nonlinear wavelet estimators of the density. A simulation is done to show the performance of the three proposed estimators.     

The weighted least square based estimators with censoring indicators missing at random

In this paper, we study linear regression analysis when some of the censoring indicators are missing at random. We define regression calibration estimate, imputation estimate and inverse probability weighted estimate for the regression coefficient vector based on the weighted least squared approach due to Stute (1993), and prove all the estimators are asymptotically normal. A simulation study was conducted to evaluate the finite properties of the proposed estimators, and a real data example is provided to illustrate our methods.     

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.     

Efficient inverse probability weighting method for quantile regression with nonignorable missing data

Quantitle regression (QR) is a popular approach to estimate functional relations between variables for all portions of a probability distribution. Parameter estimation in QR with missing data is one of the most challenging issues in statistics. Regression quantiles can be substantially biased when observations are subject to missingness. We study several inverse probability weighting (IPW) estimators for parameters in QR when covariates or responses are subject to missing not at random. Maximum likelihood and semiparametric likelihood methods are employed to estimate the respondent probability function. To achieve nice efficiency properties, we develop an empirical likelihood (EL) approach to QR with the auxiliary information from the calibration constraints. The proposed methods are less sensitive to misspecified missing mechanisms. Asymptotic properties of the proposed IPW estimators are shown under general settings. The efficiency gain of EL-based IPW estimator is quantified theoretically. Simulation studies and a data set on the work limitation of injured workers from Canada are used to illustrated our proposed methodologies.     

Information attainable in some randomly incomplete data models

The Fisher information is intricately linked to the asymptotic (first-order) optimality of maximum likelihood estimators for parametric complete-data models. When data are missing completely at random in a multivariate setup, it is shown that information in a single observation is well-defined and it plays the same role as in the complete-data model in characterizing the first-order asymptotic optimality properties of associated maximum likelihood estimators; computational aspects are also thoroughly appraised. As an illustration, the logistic regression model with incomplete binary responses and an incomplete categorical covariate is worked out.     

Product-limit Estimators and Cox Regression with Missing Censoring Information

The Kaplan–Meier estimator of a survival function requires that the censoring indicator is always observed. A method of survival function estimation is developed when the censoring indicators are missing completely at random (MCAR). The resulting estimator is a smooth functional of the Nelson–Aalen estimators of certain cumulative transition intensities. The asymptotic properties of this estimator are derived. A simulation study shows that the proposed estimator has greater efficiency than competing MCAR-based estimators. The approach is extended to the Cox model setting for the estimation of a conditional survival function given a covariate.     

Partially linear single-index model with missing responses at random

This paper considers semiparametric partially linear single-index model with missing responses at random. Imputation approach is developed to estimate the regression coefficients, single-index coefficients and the nonparametric function, respectively. The imputation estimators for the regression coefficients and single-index coefficients are obtained by a stepwise approach. These estimators are shown to be asymptotically normal, and the estimator for the nonparametric function is proved to be asymptotically normal at any fixed point. The bandwidth problem is also considered in this paper, a delete-one cross validation method is used to select the optimal bandwidth. A simulation study is conducted to evaluate the proposed methods.     

Probability density estimation with data missing at random when covariables are present

This paper addresses the problem of the probability density estimation in the presence of covariates when data are missing at random (MAR). The inverse probability weighted method is used to define a nonparametric and a semiparametric weighted probability density estimators. A regression calibration technique is also used to define an imputed estimator. It is shown that all the estimators are asymptotically normal with the same asymptotic variance as that of the inverse probability weighted estimator with known selection probability function and weights. Also, we establish the mean squared error (MSE) bounds and obtain the MSE convergence rates. A simulation is carried out to assess the proposed estimators in terms of the bias and standard error.