Fractional Imputation Research Articles

Bayesian jackknife empirical likelihood‐based inference for missing data and causal inference

AbstractMissing data reduce the representativeness of the sample and can lead to inference problems. In this article, we apply the Bayesian jackknife empirical likelihood (BJEL) method for inference on data that are missing at random, as well as for causal inference. The semiparametric fractional imputation estimator, propensity score‐weighted estimator, and doubly robust estimator are used for constructing the jackknife pseudo values, which are needed for conducting BJEL‐based inference with missing data. Existing methods, such as normal approximation and JEL, are compared with the BJEL approach in a simulation study. The proposed approach shows better performance in many scenarios in terms of credible intervals. Furthermore, we demonstrate the application of the proposed approach for causal inference problems in a study of risk factors for impaired kidney function.

Sampling Importance Resampling Algorithm with Nonignorable Missing Response Variable Based on Smoothed Quantile Regression

The presence of nonignorable missing response variables often leads to complex conditional distribution patterns that cannot be effectively captured through mean regression. In contrast, quantile regression offers valuable insights into the conditional distribution. Consequently, this article places emphasis on the quantile regression approach to address nonrandom missing data. Taking inspiration from fractional imputation, this paper proposes a novel smoothed quantile regression estimation equation based on a sampling importance resampling (SIR) algorithm instead of nonparametric kernel regression methods. Additionally, we present an augmented inverse probability weighting (AIPW) smoothed quantile regression estimation equation to reduce the influence of potential misspecification in a working model. The consistency and asymptotic normality of the empirical likelihood estimators corresponding to the above estimating equations are proven under the assumption of a correctly specified parameter working model. Furthermore, we demonstrate that the AIPW estimation equation converges to an IPW estimation equation when a parameter working model is misspecified, thus illustrating the robustness of the AIPW estimation approach. Through numerical simulations, we examine the finite sample properties of the proposed method when the working models are both correctly specified and misspecified. Furthermore, we apply the proposed method to analyze HIV—CD4 data, thereby exploring variations in treatment effects and the influence of other covariates across different quantiles.

IPLSL and IPLSQ: Two types of imputation PLS algorithms for hierarchical latent variable model

In many applications of hierarchical latent variable model, manifest variables could be missing due to various reasons. Improper missing data techniques or data imputation algorithms may lead to incredible data sets and inaccurate estimates. In our paper, we propose two types of imputation partial least square (PLS) algorithms to deal with missing data problems in hierarchical latent variable model. One is the imputation PLS algorithms based on linear regression (IPLSL), the other is the imputation PLS algorithms based on quantile regression (IPLSQ). In IPLSL, we apply the idea of complete case analysis (CC), inverse probability weighting (IPW) and fractional imputation (FI) to one kind of PLS algorithm which is different from the existing repeated indicators approach, two-step approach and hybrid approach. In IPLSQ, we also apply CC, IPW and FI to a modified PLS algorithm based on quantile regression. Compared with IPLSL, IPLSQ has the advantages in capturing overall view of structural relationships at different quantiles and highlighting the changing relations according to the explored quantile of interest. Together with mean and median imputation, we investigate the performances of our IPLSL and IPLSQ algorithms through simulation studies and then apply them to national science and technology innovation capability study based on part of World Bank database and Global Innovation Index report.

Parametric Fractional Imputation for Longitudinal Data with Intermittent Missing Values

Longitudinal data analysis had been widely developed in the past three decades. Longitudinal data are common in many fields such as public health, medicine, biological and social sciences. Longitudinal data have special nature as the individual may be observed during a long period of time. Hence, missing values are common in longitudinal data. The presence of missing values leads to biased results and complicates the analysis. The missing values have two patterns: intermittent and dropout. The missing data mechanisms are missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). The appropriate analysis relies heavily on the assumed mechanism and pattern. The parametric fractional imputation is developed to handle longitudinal data with intermittent missing pattern. The maximum likelihood estimates are obtained and the Jackkife method is used to obtain the standard errors of the parameters estimates. Finally a simulation study is conducted to validate the proposed approach. Also, the proposed approach is applied to a real data.

Estimating Average Treatment Effects Utilizing Fractional Imputation when Confounders are Subject to Missingness

Abstract The problem of missingness in observational data is ubiquitous. When the confounders are missing at random, multiple imputation is commonly used; however, the method requires congeniality conditions for valid inferences, which may not be satisfied when estimating average causal treatment effects. Alternatively, fractional imputation, proposed by Kim 2011, has been implemented to handling missing values in regression context. In this article, we develop fractional imputation methods for estimating the average treatment effects with confounders missing at random. We show that the fractional imputation estimator of the average treatment effect is asymptotically normal, which permits a consistent variance estimate. Via simulation study, we compare fractional imputation’s accuracy and precision with that of multiple imputation.

Semiparametric Fractional Imputation Using Gaussian Mixture Models for Handling Multivariate Missing Data

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 may be subject to bias under model misspecification. In this article, we propose a novel semiparametric fractional imputation (SFI) 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 categorical auxiliary information. The asymptotic model consistency and -consistency of the SFI estimator are also established. Some simulation studies are presented to check the finite sample performance of the proposed method. Supplementary materials for this article are available online.

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.

A robust joint modeling approach for longitudinal data with informative dropouts

This article proposes a robust method for analysing longitudinal continuous responses with informative dropouts and potential outliers by using the multivariate t-distribution. We specify a dropout mechanism and a missing covariate distribution and incorporate them into the complete data log-likelihood. Unlike the existing approaches which mainly focus on the inference of regression mean and dropouts process, our approach aims to reveal the dynamics in the location function, marginal scale function and association by joint parsimonious modeling the location and dependence structure. A parametric fractional imputation algorithm is developed to speed up the computation associated with the EM algorithm for maximum likelihood estimation with missing data. The resulting estimators are shown to be consistent and asymptotically normally distributed. Data examples and simulations demonstrate the effectiveness of the proposed approach.

Validity and efficiency in analyzing ordinal responses with missing observations

AbstractThis 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

A stochastic variant of the EM algorithm to fit mixed (discrete and continuous) longitudinal data with nonignorable missingness

In longitudinal studies data are collected on the same set of units for more than one occasion. In medical studies it is very common to have mixed Poisson and continuous longitudinal data. In such studies, for different reasons, some intended measurements might not be available resulting in a missing data setting. When the probability of missingness is related to the missing values, the missingness mechanism is termed nonrandom. The stochastic expectation-maximization (SEM) algorithm and the parametric fractional imputation (PFI) method are developed to handle nonrandom missingness in mixed discrete and continuous longitudinal data assuming different covariance structures for the continuous outcome. The proposed techniques are evaluated using simulation studies. Also, the proposed techniques are applied to the interstitial cystitis data base (ICDB) data.

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

AbstractMissing 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

A fast imputation algorithm in quantile regression.

In many applications, some covariates could be missing for various reasons. Regression quantiles could be either biased or under-powered when ignoring the missing data. Multiple imputation and EM-based augment approach have been proposed to fully utilize the data with missing covariates for quantile regression. Both methods however are computationally expensive. We propose a fast imputation algorithm (FI) to handle the missing covariates in quantile regression, which is an extension of the fractional imputation in likelihood based regressions. FI and modified imputation algorithms (FIIPW and MIIPW) are compared to existing MI and IPW approaches in the simulation studies, and applied to part of of the National Collaborative Perinatal Project study.

Fully Efficient Joint Fractional Imputation for Incomplete Bivariate Ordinal Responses

Fully Efficient Joint Fractional Imputation for Incomplete Bivariate Ordinal Responses

Analysis of inaccurate data with mixture measurement error models

Measurement error, the difference between a measured (observed) value of quantity and its true value, is perceived as a possible source of estimation bias in many surveys. To correct for such bias, a validation sample can be used in addition to the original sample for adjustment of measurement error. Depending on the type of validation sample, we can either use the internal calibration approach or the external calibration approach. Motivated by Korean Longitudinal Study of Aging (KLoSA), we propose a novel application of fractional imputation to correct for measurement error in the analysis of survey data. The proposed method is to create imputed values of the unobserved true variables, which are mis-measured in the main study, by using validation subsample. Furthermore, the proposed method can be directly applicable when the measurement error model is a mixture distribution. Variance estimation using Taylor linearization is developed. Results from a limited simulation study are also presented.

Semiparametric estimation for measurement error models with validation data

AbstractWe consider regression problems where error‐prone surrogates of true predictors are collected in a primary data set while accurate measurements of the predictors are available only in a relatively small validation data set. We propose a new class of semiparametric estimators for the regression coefficients based on expected estimating equations, where the relationship between the surrogates and the true predictors is modelled nonparametrically using a kernel smoother trained with the validation data. The new methods are developed under two different scenarios where the response variable is either observed or not observed in the validation data set. The proposed estimators have a natural connection with the fractional imputation method. They are consistent, asymptotically unbiased, and normal in both scenarios. Our simulation studies show that the proposed estimators are superior to competitors in terms of bias and mean squared error and are quite robust against the misspecification of the regression model and bandwidth selection. A real data application to the Korean Longitudinal Study of Aging is presented for illustration. The Canadian Journal of Statistics 45: 185–201; 2017 © 2017 Statistical Society of Canada

Semiparametric fractional imputation using empirical likelihood in survey sampling

ABSTRACTThe empirical likelihood method is a powerful tool for incorporating moment conditions in statistical inference. We propose a novel application of the empirical likelihood for handling item non-response in survey sampling. The proposed method takes the form of fractional imputation but it does not require parametric model assumptions. Instead, only the first moment condition based on a regression model is assumed and the empirical likelihood method is applied to the observed residuals to get the fractional weights. The resulting semiparametric fractional imputation provides -consistent estimates for various parameters. Variance estimation is implemented using a jackknife method. Two limited simulation studies are presented to compare several imputation estimators.

OUP accepted manuscript

SummaryItem nonresponse in surveys is often treated through some form of imputation. We introduce multiply robust imputation in finite population sampling. This is closely related to multiple robustness, which extends double robustness. In practice, multiple nonresponse models and multiple imputation models may be fitted, each involving different subsets of covariates and possibly different link functions. An imputation procedure is said to be multiply robust if the resulting estimator is consistent when all models but one are misspecified. A jackknife variance estimator is proposed and shown to be consistent. Random and fractional imputation procedures are discussed. A simulation study suggests that the proposed estimation procedures have low bias and high efficiency.

Nonparametric overdose control with late-onset toxicity in phase I clinical trials.

Under the framework of Bayesian model selection, we propose a nonparametric overdose control (NOC) design for dose finding in phase I clinical trials. Each dose assignment is guided via a feasibility bound, which thereby can control the number of patients allocated to excessively toxic dose levels. Several aspects of the NOC design are explored, including the coherence property in dose assignment, calibration of design parameters, and selection of the maximum tolerated dose (MTD). We further propose a fractional NOC (fNOC) design in conjunction with a so-called fractional imputation approach, to account for late-onset toxicity outcomes. Extensive simulation studies have been conducted to show that both the NOC and fNOC designs have robust and satisfactory finite-sample performance compared with the existing dose-finding designs. The proposed methods also possess several desirable properties: treating patients more safely and also neutralizing the aggressive escalation to overly toxic doses when the toxicity outcomes are late-onset. The fNOC design is exemplified with a real cancer phase I trial.

Fractional Imputation in Survey Sampling: A Comparative Review

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 paper, we give an overview of FI, introduce key ideas and methods to readers who are new to the FI literature, and highlight some new development. We also provide guidance on practical implementation of FI and valid inferential tools after imputation. We demonstrate the empirical performance of FI with respect to multiple imputation using a pseudo finite population generated from a sample in Monthly Retail Trade Survey in US Census Bureau.

An imputation approach for handling mixed-mode surveys

Mixed-mode surveys are becoming more popular recently because of their convenience for users, but different mode effects can complicate the comparability of the survey results. Motivated by the Private Education Expenditure Survey (PEES) of Korea, we propose a novel application of fractional imputation to handle mixed-mode survey data. The proposed method is applied to create imputed values of the unobserved counterfactual outcome variables in the mixed-mode surveys. The proposed method is directly applicable when the choice of survey mode is self-selected. Variance estimation using Taylor linearization is developed. Results from a limited simulation study are also presented.