We outline a framework for multiple imputation of nonignorable item nonresponse when the marginal distributions of some of the variables with missing values are known. In particular, our framework ensures that (i) the completed datasets result in design-based estimates of totals that are plausible, given the margins, and (ii) the completed datasets maintain associations across variables as posited in the imputation models. To do so, we propose an additive nonignorable model for nonresponse, coupled with a rejection sampling step. The rejection sampling step favors completed datasets that result in design-based estimates that are plausible given the known margins. We illustrate the framework using simulations with stratified sampling.
How to deal with nonignorable response is often a challenging problem encountered in statistical analysis with missing data. Parametric model assumption for the response mechanism is often made and there is no way to validate the model assumption with missing data. We consider a semiparametric response model that relaxes the parametric model assumption in the response mechanism. Two types of efficient estimators, profile maximum likelihood estimator and profile calibration estimator, are proposed and their asymptotic properties are investigated. Two extensive simulation studies are used to com…
Statistical inference with nonresponse is quite challenging, especially when the response mechanism is nonignorable. The existing methods often require correct model specifications for both outcome and response models. However, due to nonresponse, both models cannot be verified from data directly and model misspecification can lead to a seriously biased inference. To overcome this limitation, we develop a robust and efficient semiparametric method based on the profile likelihood. The proposed method uses the robust semiparametric response model, in which fully unspecified function of study var…
When the response mechanism is believed to be not missing at random (NMAR), a valid analysis requires stronger assumptions on the response mechanism than standard statistical methods would otherwise require. Semiparametric estimators have been developed under the model assumptions on the response mechanism. In this paper, a new statistical test is proposed to guarantee model identifiability without using any instrumental variable. Furthermore, we develop optimal semiparametric estimation for parameters such as the population mean. Specifically, we propose two semiparametric optimal estimators …
We present an approach to inform decisions about nonresponse follow-up sampling. The basic idea is (i) to create completed samples by imputing nonrespondents' data under various assumptions about the nonresponse mechanisms, (ii) take hypothetical samples of varying sizes from the completed samples, and (iii) compute and compare measures of accuracy and cost for different proposed sample sizes. As part of the methodology, we present a new approach for generating imputations for multivariate continuous data with nonignorable unit nonresponse. We fit mixtures of multivariate normal distributions …
Nonmonotone missing data arise routinely in empirical studies of social and health sciences, and when ignored, can induce selection bias and loss of efficiency. In practice, it is common to account for nonresponse under a missing-at-random assumption which although convenient, is rarely appropriate when nonresponse is nonmonotone. Likelihood and Bayesian missing data methodologies often require specification of a parametric model for the full data law, thus a priori ruling out any prospect for semiparametric inference. In this paper, we propose an all-purpose approach which delivers semiparame…
Often, government agencies and survey organizations know the population counts or percentages for some of the variables in a survey. These may be available from auxiliary sources, for example, administrative databases or other high quality surveys. We present and illustrate a model-based framework for leveraging such auxiliary marginal information when handling unit and item nonresponse. We show how one can use the margins to specify different missingness mechanisms for each type of nonresponse. We use the framework to impute missing values in voter turnout in a subset of data from the U.S.\ C…
Statistical inference with nonresponse is quite challenging, especially when the response mechanism is nonignorable. In this case, the validity of statistical inference depends on untestable correct specification of the response model. To avoid the misspecification, we propose semiparametric Bayesian estimation in which an outcome model is parametric, but the response model is semiparametric in that we do not assume any parametric form for the nonresponse variable. We adopt penalized spline methods to estimate the unknown function. We also consider a fully nonparametric approach to modeling th…
Many imputation methods are based on statistical models that assume that the variable of interest is a noisy observation of a function of the auxiliary variables or covariates. Misspecification of this model may lead to severe errors in estimates and to misleading conclusions. A new imputation method for item nonresponse in surveys is proposed based on a nonparametric estimation of the functional dependence between the variable of interest and the auxiliary variables. We consider the use of smoothing spline estimation within an additive model framework to flexibly build an imputation model in …
We study a class of missingness mechanisms, called sequentially additive nonignorable, for modeling multivariate data with item nonresponse. These mechanisms explicitly allow the probability of nonresponse for each variable to depend on the value of that variable, thereby representing nonignorable missingness mechanisms. These missing data models are identified by making use of auxiliary information on marginal distributions, such as marginal probabilities for multivariate categorical variables or moments for numeric variables. We present theory proving identification results, and illustrate t…
We introduce a nonresponse mechanism for multivariate missing data in which each study variable and its nonresponse indicator are conditionally independent given the remaining variables and their nonresponse indicators. This is a nonignorable missingness mechanism, in that nonresponse for any item can depend on values of other items that are themselves missing. We show that, under this itemwise conditionally independent nonresponse assumption, one can define and identify nonparametric saturated classes of joint multivariate models for the study variables and their missingness indicators. We al…
Imputation is a popular technique for handling item nonresponse in survey sampling. Parametric imputation is based on a parametric model for imputation and is less robust against the failure of the imputation model. Nonparametric imputation is fully robust but is not applicable when the dimension of covariates is large due to the curse of dimensionality. Semiparametric imputation is another robust imputation based on a flexible model where the number of model parameters can increase with the sample size. In this paper, we propose another semiparametric imputation based on a more flexible model…
We consider identification and inference about mean functionals of observed covariates and an outcome variable subject to nonignorable missingness. By leveraging a shadow variable, we establish a necessary and sufficient condition for identification of the mean functional even if the full data distribution is not identified. We further characterize a necessary condition for $\sqrt{n}$-estimability of the mean functional. This condition naturally strengthens the identifying condition, and it requires the existence of a function as a solution to a representer equation that connects the shadow va…
Item nonresponse is a common issue in surveys. Because unadjusted estimators may be biased in the presence of nonresponse, it is common practice to impute the missing values with the objective of reducing the nonresponse bias as much as possible. However, commonly used imputation procedures may lead to unstable estimators of population totals/means when influential units are present in the set of respondents. In this article, we consider the class of multiply robust imputation procedures that provide some protection against the failure of underlying model assumptions. We develop an efficient v…
Imputation is a popular technique for handling missing data. We consider a nonparametric approach to imputation using the kernel ridge regression technique and propose consistent variance estimation. The proposed variance estimator is based on a linearization approach which employs the entropy method to estimate the density ratio. The root-n consistency of the imputation estimator is established when a Sobolev space is utilized in the kernel ridge regression imputation, which enables us to develop the proposed variance estimator. Synthetic data experiments are presented to confirm our theory.
The Current Population Survey is the gold-standard data source for studying who turns out to vote in elections. However, it suffers from potentially nonignorable unit and item nonresponse. Fortunately, after elections, the total number of voters is known from administrative sources and can be used to adjust for potential nonresponse bias. We present a model-based approach to utilize this known voter turnout rate, as well as other population marginal distributions of demographic variables, in multiple imputation for unit and item nonresponse. In doing so, we ensure that the imputations produce …
Nonresponse is common in surveys. When the response probability of a survey variable $Y$ depends on $Y$ through an observed auxiliary categorical variable $Z$ (i.e., the response probability of $Y$ is conditionally independent of $Y$ given $Z$), a simple method often used in practice is to use $Z$ categories as imputation cells and construct estimators by imputing nonrespondents or reweighting respondents within each imputation cell. This simple method, however, is inefficient when some $Z$ categories have small sizes and ad hoc methods are often applied to collapse small imputation cells. Ass…
Item nonresponse is frequently encountered in practice. Ignoring missing data can lose efficiency and lead to misleading inference. Fractional imputation is a frequentist approach of imputation for handling missing data. However, the parametric fractional imputation of \cite{kim2011parametric} may be subject to bias under model misspecification. In this paper, we propose a novel semiparametric fractional imputation method using Gaussian mixture models. The proposed method is computationally efficient and leads to robust estimation. The proposed method is further extended to incorporate the cat…
Predictive mean matching imputation is popular for handling item nonresponse in survey sampling. In this article, we study the asymptotic properties of the predictive mean matching estimator of the population mean. For variance estimation, the conventional bootstrap inference for matching estimators with fixed matches has been shown to be invalid due to the nonsmoothness nature of the matching estimator. We propose asymptotically valid replication variance estimation. The key strategy is to construct replicates of the estimator directly based on linear terms, instead of individual records of v…
Nearest neighbor imputation is popular for handling item nonresponse in survey sampling. In this article, we study the asymptotic properties of the nearest neighbor imputation estimator for general population parameters, including population means, proportions and quantiles. For variance estimation, the conventional bootstrap inference for matching estimators with fixed number of matches has been shown to be invalid due to the nonsmoothness nature of the matching estimator. We propose asymptotically valid replication variance estimation. The key strategy is to construct replicates of the estim…
Multiple imputation provides an effective way to handle missing data. When several possible models are under consideration for the data, the multiple imputation is typically performed under a single-best model selected from the candidate models. This single model selection approach ignores the uncertainty associated with the model selection and so leads to underestimation of the variance of multiple imputation estimator. In this paper, we propose a new multiple imputation procedure incorporating model uncertainty in the final inference. The proposed method incorporates possible candidate model…
Nonresponse is present in almost all surveys and can severely bias estimates. It is usually distinguished between unit and item nonresponse: in the former, we completely fail to have information from a unit selected in the sample, while in the latter, we observe only part of the information on the selected unit. Unit nonresponse is usually dealt with by reweighting: each unit selected in the sample has associated a sampling weight and an unknown response probability; the initial sampling weight is multiplied by the inverse of estimated response probability. Item nonresponse is usually dealt wi…
In epidemiological surveys, data missing not at random (MNAR) due to survey nonresponse may potentially lead to a bias in the risk factor estimates. We propose an approach based on Bayesian data augmentation and survival modelling to reduce the nonresponse bias. The approach requires additional information based on follow-up data. We present a case study of smoking prevalence using FINRISK data collected between 1972 and 2007 with a follow-up to the end of 2012 and compare it to other commonly applied missing at random (MAR) imputation approaches. A simulation experiment is carried out to stud…
Fractional imputation (FI) is a relatively new method of imputation for handling item nonresponse in survey sampling. In FI, several imputed values with their fractional weights are created for each missing item. Each fractional weight represents the conditional probability of the imputed value given the observed data, and the parameters in the conditional probabilities are often computed by an iterative method such as EM algorithm. The underlying model for FI can be fully parametric, semiparametric, or nonparametric, depending on plausibility of assumptions and the data structure. In this pap…
In order to overcome the problem of item nonresponse, random imputation methods are often used because they tend to preserve the distribution of the imputed variable. Among the random imputation methods, the random hot-deck has the interesting property of imputing observed values. A new random hot-deck imputation method is proposed. The key innovation of this method is that the selection of donors is viewed as a sampling problem and uses calibration and balanced sampling. This approach makes it possible to select donors such that if the auxiliary variables were imputed, their estimated totals …
Nonparametric and machine learning methods are flexible methods for obtaining accurate predictions. Nowadays, data sets with a large number of predictors and complex structures are fairly common. In the presence of item nonresponse, nonparametric and machine learning procedures may thus provide a useful alternative to traditional imputation procedures for deriving a set of imputed values. In this paper, we conduct an extensive empirical investigation that compares a number of imputation procedures in terms of bias and efficiency in a wide variety of settings, including high-dimensional data se…
Nonresponse is a common problem in survey sampling. Appropriate treatment can be challenging, especially when dealing with detailed breakdowns of totals. Often, the nearest neighbor imputation method is used to handle such incomplete multinomial data. In this article, we investigate the nearest neighbor ratio imputation estimator, in which auxiliary variables are used to identify the closest donor and the vector of proportions from the donor is applied to the total of the recipient to implement ratio imputation. To estimate the asymptotic variance, we first treat the nearest neighbor ratio imp…
Nonresponse frequently arises in practice, and simply ignoring it may lead to erroneous inference. Besides, the number of collected covariates may increase as the sample size in modern statistics, so parametric imputation or propensity score weighting usually leads to inefficiency without consideration of sparsity. In this paper, we propose a nonparametric imputation method with sparse learning by employing an efficient kernel-based learning gradient algorithm to identify truly informative covariates. Moreover, an augmented probability weighting framework is adopted to improve the estimation e…
We present a nonparametric Bayesian joint model for multivariate continuous and categorical variables, with the intention of developing a flexible engine for multiple imputation of missing values. The model fuses Dirichlet process mixtures of multinomial distributions for categorical variables with Dirichlet process mixtures of multivariate normal distributions for continuous variables. We incorporate dependence between the continuous and categorical variables by (i) modeling the means of the normal distributions as component-specific functions of the categorical variables and (ii) forming dis…
