Nonignorable Missing Data Research Articles

Weighted composite quantile regression analysis for nonignorable missing data using nonresponse instrument

ABSTRACTEfficient statistical inference on nonignorable missing data is a challenging problem. This paper proposes a new estimation procedure based on composite quantile regression (CQR) for linear regression models with nonignorable missing data, that is applicable even with high-dimensional covariates. A parametric model is assumed for modelling response probability, which is estimated by the empirical likelihood approach. Local identifiability of the proposed strategy is guaranteed on the basis of an instrumental variable approach. A set of data-based adaptive weights constructed via an empirical likelihood method is used to weight CQR functions. The proposed method is resistant to heavy-tailed errors or outliers in the response. An adaptive penalisation method for variable selection is proposed to achieve sparsity with high-dimensional covariates. Limiting distributions of the proposed estimators are derived. Simulation studies are conducted to investigate the finite sample performance of the proposed methodologies. An application to the ACTG 175 data is analysed.

Approximate conditional likelihood for generalized linear models with general missing data mechanism

The generalized linear model is an indispensable tool for analyzing non-Gaussian response data, with both canonical and non-canonical link functions comprehensively used. When missing values are present, many existing methods in the literature heavily depend on an unverifiable assumption of the missing data mechanism, and they fail when the assumption is violated. This paper proposes a missing data mechanism that is as generally applicable as possible, which includes both ignorable and nonignorable missing data cases, as well as both scenarios of missing values in response and covariate. Under this general missing data mechanism, the authors adopt an approximate conditional likelihood method to estimate unknown parameters. The authors rigorously establish the regularity conditions under which the unknown parameters are identifiable under the approximate conditional likelihood approach. For parameters that are identifiable, the authors prove the asymptotic normality of the estimators obtained by maximizing the approximate conditional likelihood. Some simulation studies are conducted to evaluate finite sample performance of the proposed estimators as well as estimators from some existing methods. Finally, the authors present a biomarker analysis in prostate cancer study to illustrate the proposed method.

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

ABSTRACTQuantitle 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.

Treatment of nonignorable missing data when modeling unobserved heterogeneity with finite mixture models.

Multiple imputation has become a widely accepted technique to deal with the problem of incomplete data. Typically, imputation of missing values and the statistical analysis are performed separately. Therefore, the imputation model has to be consistent with the analysis model. If the data are analyzed with a mixture model, the parameter estimates are usually obtained iteratively. Thus, if the data are missing not at random, parameter estimation and treatment of missingness should be combined. We solve both problems by simultaneously imputing values using the data augmentation method and estimating parameters using the EM algorithm. This iterative procedure ensures that the missing values are properly imputed given the current parameter estimates. Properties of the parameter estimates were investigated in a simulation study. The results are illustrated using data from the National Health and Nutrition Examination Survey.

Latent Variable Model for Weight Gain Prevention Data with Informative Intermittent Missingness

Missing data is a common problem in longitudinal studies because of the characteristics of repeated measurements. Herein is proposed a latent variable model for nonignorable intermittent missing data in which the latent variables are used as random effects in modeling and link longitudinal responses and missingness process. In this methodology, the latent variables are assumed to be normally distributed with zero-mean, and the values of variance-covariance are calculated through maximum likelihood estimations. Parameter estimates and standard errors of the proposed method are compared with the mixed model and the complete-case analysis in the simulations and the application to the weight gain prevention among women (WGPW) data set. In the simulation results with respect to bias, mean squared error, and coverage of confidence interval, the proposed model performs better than the other two methods in different scenarios. Relatively, the proposed latent variable model and the mixed model do a better job for between-subject effects compared to within-subject effects. The converse is true for the complete case analysis. The simulation results also provide support for application of this proposed latent variable model to the WGPW data set.

On the identifiability and estimation of generalized linear models with parametric nonignorable missing data mechanism

We address the problem of identifying and estimating generalized linear models when the response variable is nonignorably missing. Three types of monotone missing data mechanism are assumed, including Logit model, Probit model and complementary Log–log model. In this situation, likelihood based on observed data may not be identifiable. In this article, we prove the model parameters are identifiable under very mild conditions and then construct estimators based on a likelihood-based approach. The proposed estimators are shown to be consistent and asymptotically normal. Simulation studies demonstrate that the proposed inference procedure performs well in many settings. We apply the proposed method to a data set from research in Chinese Household Income Project study.

A likelihood-based approach for multivariate one-sided tests with missing data

ABSTRACTInequality-restricted hypotheses testing methods containing multivariate one-sided testing methods are useful in practice, especially in multiple comparison problems. In practice, multivariate and longitudinal data often contain missing values since it may be difficult to observe all values for each variable. However, although missing values are common for multivariate data, statistical methods for multivariate one-sided tests with missing values are quite limited. In this article, motivated by a dataset in a recent collaborative project, we develop two likelihood-based methods for multivariate one-sided tests with missing values, where the missing data patterns can be arbitrary and the missing data mechanisms may be non-ignorable. Although non-ignorable missing data are not testable based on observed data, statistical methods addressing this issue can be used for sensitivity analysis and might lead to more reliable results, since ignoring informative missingness may lead to biased analysis. We analyse the real dataset in details under various possible missing data mechanisms and report interesting findings which are previously unavailable. We also derive some asymptotic results and evaluate our new tests using simulations.

Empirical likelihood method for non-ignorable missing data problems.

Missing response problem is ubiquitous in survey sampling, medical, social science and epidemiology studies. It is well known that non-ignorable missing is the most difficult missing data problem where the missing of a response depends on its own value. In statistical literature, unlike the ignorable missing data problem, not many papers on non-ignorable missing data are available except for the full parametric model based approach. In this paper we study a semiparametric model for non-ignorable missing data in which the missing probability is known up to some parameters, but the underlying distributions are not specified. By employing Owen (1988)'s empirical likelihood method we can obtain the constrained maximum empirical likelihood estimators of the parameters in the missing probability and the mean response which are shown to be asymptotically normal. Moreover the likelihood ratio statistic can be used to test whether the missing of the responses is non-ignorable or completely at random. The theoretical results are confirmed by a simulation study. As an illustration, the analysis of a real AIDS trial data shows that the missing of CD4 counts around two years are non-ignorable and the sample mean based on observed data only is biased.

Bayesian local influence analysis of general estimating equations with nonignorable missing data

Bayesian empirical likelihood (BEL) method with missing data depends heavily on the prior specification and missing data mechanism assumptions. It is well known that the resulting Bayesian estimations and tests may be sensitive to these assumptions and observations. To this end, a Bayesian local influence procedure is proposed to assess the effect of various perturbations to the individual observations, priors, estimating equations (EEs) and missing data mechanism in general EEs with nonignorable missing data. A perturbation model is introduced to simultaneously characterize various perturbations, and a Bayesian perturbation manifold is constructed to characterize the intrinsic structure of these perturbations. The first- and second-order adjusted local influence measures are developed to quantify the effect of various perturbations. The proposed methods are adopted to systematically investigate the tenability of nonignorable missing mechanism assumption, the sensitivity of the choice of the nonresponse instrumental variable and the sensitivity of EEs assumption, and goodness-of-fit statistics are presented to assess the plausibility of the posited EEs. Simulation studies are conducted to investigate the performance of the proposed methodologies. An example is analyzed.

Unfolding IRT Models for Likert-Type Items With a Don't Know Option.

Attitude surveys are widely used in the social sciences. It has been argued that the underlying response process to attitude items may be more aligned with the ideal-point (unfolding) process than with the cumulative (dominance) process, and therefore, unfolding item response theory (IRT) models are more appropriate than dominance IRT models for these surveys. Missing data and don't know (DK) responses are common in attitude surveys, and they may not be ignorable in the likelihood for parameter estimation. Existing unfolding IRT models often treat missing data or DK as missing at random. In this study, a new class of unfolding IRT models for nonignorable missing data and DK were developed, in which the missingness and DK were assumed to measure a hierarchy of latent traits, which may be correlated with the latent attitude that a test intended to measure. The Bayesian approach with Markov chain Monte Carlo methods was used to estimate the parameters of the new models. Simulation studies demonstrated that the parameters were recovered fairly well, and ignoring nonignorable missingness or DK resulted in poor parameter estimates. An empirical example of a religious belief scale about health was given.

Mycophenolate mofetil versus oral cyclophosphamide in scleroderma-related interstitial lung disease (SLS II): a randomised controlled, double-blind, parallel group trial

SummaryBACKGROUNDTwelve months of oral cyclophosphamide (CYC) has been shown to alter the progression of scleroderma-related interstitial lung disease (SSc-ILD) when compared to placebo. However, toxicity was a concern and without continued treatment the efficacy disappeared by 24 months. We hypothesized that a two-year course of mycophenolate mofetil (MMF) would be safer, better tolerated and produce longer lasting improvements than CYC.METHODSPatients with SSc-ILD meeting defined dyspnea, pulmonary function and high-resolution computed tomography (HRCT) criteria were randomized in a double-blind, two-arm trial at 14 medical centers. MMF (target dose 1500 mg twice daily) was administered for 24 months in one arm and oral CYC (target dose 2·0 mg/kg/day) administered for 12 months followed by placebo for 12 months in the other arm. The primary endpoint, change in forced vital capacity as a percent of the predicted normal value (FVC %) over the course of 24 months, was assessed in a modified intention-to-treat analysis using an inferential joint model combining a mixed effects model for longitudinal outcomes and a survival model to handle non-ignorable missing data. The study was registered with ClinicalTrials.gov, number NCT00883129, and is closed.RESULTSBetween November, 2009, and January, 2013, 142 patients were randomized. 126 patients (63 MMF; 63 CYC) with acceptable baseline HRCT studies and at least one outcome measure were included in the analysis. The adjusted FVC % (primary endpoint) improved from baseline to 24 months by 2.17 in the MMF arm (95% CI, 0.53–3.84) and 2·86 in the CYC arm (95% confidence interval 1·19–4·58) with no significant between-treatment difference (p=0·24), indicating that the trial was negative for the primary endpoint. However, in a post-hoc analysis of the primary endpoint, within-treatment improvements from baseline to 24 months were noted in both the CYC and MMF arms. A greater number of patients on CYC than on MMF prematurely withdrew from study drug (32 vs 20) and failed treatment (2 vs 0), and the time to stopping treatment was significantly shorter in the CYC arm (p=0·019). Sixteen deaths occurred (11 CYC; 5 MMF) with most due to progressive ILD. Leukopenia (30 vs 4 patients) and thrombocytopenia (4 vs 0 patients) occurred more often in patients treated with CYC. In post-hoc analyses, within- (but not between-) treatment improvements were also noted in defined secondary outcomes including skin score, dyspnea and whole-lung HRCT scores.INTERPRETATIONTreatment of SSc-ILD with MMF for two years or CYC for one year both resulted in significant improvements in pre-specified measures of lung function, dyspnea, lung imaging, and skin disease over the 2-year course of the study. While MMF was better tolerated and associated with less toxicity, the hypothesis that it would have greater efficacy at 24 months than CYC was not confirmed. These findings support the potential clinical impact of both CYC and MMF for progressive SSc-ILD, as well as the current preference for MMF due to its better tolerability and toxicity profile.FUNDINGNational Heart, Lung and Blood Institute/National Institutes of Health with drug supply provided by Hoffmann-La Roche/Genentech.

Estimation of response from longitudinal binary data with nonignorable missing values in migraine trials.

In migraine trials pain relief responses from a headache at specific time points and sustained pain relief response over a period of time are important efficacy measures. When there are missing records of individual time point pain scores and/or headache recurrences during a migraine trial, the common approach used in practice to estimate the sustained response is statistically inconsistent even if the data are missing completely at random. Methods dealing with nonignorable longitudinal missing data usually assume certain models for the missing mechanism which can not be checked as they involve unobserved data. Taking advantage of the specific definition of the ‘sustained pain relief’ response, we propose two estimating methods based on intuitive imputation, which do not require model assumptions on the missing probability or specification of the correlation structure among the longitudinal observations. The consistency of the proposed methods is discussed in theory and their empirical performances are assessed through intensive simulation studies. The simulation results show that the proposed methods perform well in terms of reducing bias and mean square error except in several extreme cases which are unlikely to happen in real trials. The application of the proposed methods is illustrated in a real data analysis.

Bayesian joint modeling for assessing the progression of chronic kidney disease in children.

Joint models are rich and flexible models for analyzing longitudinal data with nonignorable missing data mechanisms. This article proposes a Bayesian random-effects joint model to assess the evolution of a longitudinal process in terms of a linear mixed-effects model that accounts for heterogeneity between the subjects, serial correlation, and measurement error. Dropout is modeled in terms of a survival model with competing risks and left truncation. The model is applied to data coming from ReVaPIR, a project involving children with chronic kidney disease whose evolution is mainly assessed through longitudinal measurements of glomerular filtration rate.

Semiparametric Inference of the Complier Average Causal Effect with Nonignorable Missing Outcomes

Noncompliance and missing data often occur in randomized trials, which complicate the inference of causal effects. When both noncompliance and missing data are present, previous papers proposed moment and maximum likelihood estimators for binary and normally distributed continuous outcomes under the latent ignorable missing data mechanism. However, the latent ignorable missing data mechanism may be violated in practice, because the missing data mechanism may depend directly on the missing outcome itself. Under noncompliance and an outcome-dependent nonignorable missing data mechanism, previous studies showed the identifiability of complier average causal effect for discrete outcomes. In this article, we study the semiparametric identifiability and estimation of complier average causal effect in randomized clinical trials with both all-or-none noncompliance and outcome-dependent nonignorable missing continuous outcomes, and propose a two-step maximum likelihood estimator in order to eliminate the infinite dimensional nuisance parameter. Our method does not need to specify a parametric form for the missing data mechanism. We also evaluate the finite sample property of our method via extensive simulation studies and sensitivity analysis, with an application to a double-blinded psychiatric clinical trial.

A Hierarchical Model for Accuracy and Choice on Standardized Tests.

This paper assesses the psychometric value of allowing test-takers choice in standardized testing. New theoretical results examine the conditions where allowing choice improves score precision. A hierarchical framework is presented for jointly modeling the accuracy of cognitive responses and item choices. The statistical methodology is disseminated in the 'cIRT' R package. An 'answer two, choose one' (A2C1) test administration design is introduced to avoid challenges associated with nonignorable missing data. Experimental results suggest that the A2C1 design and payout structure encouraged subjects to choose items consistent with their cognitive trait levels. Substantively, the experimental data suggest that item choices yielded comparable information and discrimination ability as cognitive items. Given there are no clear guidelines for writing more or less discriminating items, one practical implication is that choice can serve as a mechanism to improve score precision.

Semiparametric Pseudo-Likelihoods in Generalized Linear Models With Nonignorable Missing Data

We consider identifiability and estimation in a generalized linear model in which the response variable and some covariates have missing values and the missing data mechanism is nonignorable and unspecified. We adopt a pseudo-likelihood approach that makes use of an instrumental variable to help identifying unknown parameters in the presence of nonignorable missing data. Explicit conditions for the identifiability of parameters are given. Some asymptotic properties of the parameter estimators based on maximizing the pseudo-likelihood are established. Explicit asymptotic covariance matrix and its estimator are also derived in some cases. For the numerical maximization of the pseudo-likelihood, we develop a two-step iteration algorithm that decomposes a nonconcave maximization problem into two problems of maximizing concave functions. Some simulation results and an application to a dataset from cotton factory workers are also presented.

Comparison Between Two Controlled Multiple Imputation Methods for Sensitivity Analyses of Time-to-Event Data With Possibly Informative Censoring

Controlled imputation methods provide general and flexible sensitivity analyses to address nonignorable missing data. For time-to-event data with possibly informative censoring, we compare two popular methods for imputing the censored event time conditional on the time of follow-up discontinuation. One is the delta-adjusted method that specifies that the hazard of having an event for subjects who discontinued before the time point is multiplicatively increased relative to the hazard for subjects who continued beyond the time point. The other is the reference-based method that specifies that the hazard for experimental subjects who discontinued lies between the hazard for experimental subjects who continued and the hazard for the reference control (e.g., placebo) subjects. We consider both piecewise constant and nonparametric baseline hazard functions, Bayesian and frequentist imputations, and Rubin’s and bootstrap variances for the multiple imputation estimator. We show that both the reference-based and delta-adjusted sensitivity analyses control the one-sided Type I error rate (in the direction of a difference favoring the experimental treatment). In addition, when the bootstrap variance is used for inference, the reference-based sensitivity analysis has better power than the delta-adjusted sensitivity analysis for the same underlying treatment effect.

Probabilistic Principal Component Analysis to Identify Profiles of Physical Activity Behaviours in the Presence of Non-Ignorable Missing Data

SummaryThe paper is motivated by an accelerometer-based study of physical activity (PA) behaviours in a large cohort of UK school-aged children. Advances in research on PA are accompanied by a growing number of results that are contributing to form a complex picture of PA behaviours in children. One source of such complexity is intimately related to the multiplicity of dimensions associated with PA. Currently a comprehensive individual accelerometer summary can include a large number of outcomes and this clearly poses challenges for the analysis. We explore the application of principal component analysis to accelerometer measurements that are aggregated daily over several days of the week and are affected by missingness. The probabilistic approach to principal component analysis with latent scores is extended to include non-ignorable missing data. The extended likelihood is maximized through a Monte Carlo EM algorithm via adaptive rejection Metropolis sampling. Our findings suggest that physical activity and inactivity are two dimensions over which children aggregate into distinct behavioural profiles, characterized by gender and season but not by anthropometric factors.

Managing nonignorable missing data with clustered multinomial responses

Clustered multinomial responses are common in public health studies. In this situation, the baseline logit random effects model is usually suggested as a general modelling approach. When nonignorable missing outcomes exist, naïve methods such as complete case analysis or likelihood methods ignoring missing information may distort the conclusions that are drawn. While methods to deal with binary and ordinal outcomes have been proposed, no easily implementable method is specifically available for missing clustered nominal responses. Joint modelling is usually one of the available choices but has high complexity in terms of likelihood. The numerical integration of both missing data and random effects is challenging. In this study, we have derived a closed form of likelihood. A simplified likelihood is also proposed, which is an extension of a previous study. One advantage is that both methods are easily implemented with commonly used software. We illustrate our proposed methods using the Global Youth Tobacco Survey and compare the results obtained by naïve methods that ignore missing data with the results obtained using the proposed methods. Our approaches restore the parameter estimates and predicted probability of each category to an acceptable extent. Analysis guidelines for the use of our methods are provided.

Power calculations for delta-adjusted pattern-mixture models.

Pattern-mixture models provide a general and flexible framework for sensitivity analyses of nonignorable missing data in longitudinal studies. The delta-adjusted pattern-mixture models handle missing data in a clinically interpretable manner and have been used as sensitivity analyses addressing the effectiveness hypothesis, while a likelihood-based approach that assumes data are missing at random is often used as the primary analysis addressing the efficacy hypothesis. We describe a method for power calculations for delta-adjusted pattern-mixture model sensitivity analyses in confirmatory clinical trials. To apply the method, we only need to specify the pattern probabilities at postbaseline time points, the expected treatment differences at postbaseline time points, the conditional covariance matrix of postbaseline measurements given the baseline measurement, and the delta-adjustment method for the pattern-mixture model. We use an example to illustrate and compare various delta-adjusted pattern-mixture models and use simulations to confirm the analytic results.