Nonignorable Dropouts Research Articles

Quantile partially linear additive model for data with dropouts and an application to modeling cognitive decline.

The National Alzheimer's Coordinating Center Uniform Data Set includes test results from a battery of cognitive exams. Motivated by the need to model the cognitive ability of low-performing patients we create a composite score from ten tests and propose to model this score using a partially linear quantile regression model for longitudinal studies with non-ignorable dropouts. Quantile regression allows for modeling non-central tendencies. The partially linear model accommodates nonlinear relationships between some of the covariates and cognitive ability. The data set includes patients that leave the study prior to the conclusion. Ignoring such dropouts will result in biased estimates if the probability of dropout depends on the response. To handle this challenge, we propose a weighted quantile regression estimator where the weights are inversely proportional to the estimated probability a subject remains in the study. We prove that this weighted estimator is a consistent and efficient estimator of both linear and nonlinear effects.

Improved multiple quantile regression estimation with nonignorable dropouts

Improved multiple quantile regression estimation with nonignorable dropouts

Robust statistical inference for longitudinal data with nonignorable dropouts

In this paper, we propose robust statistical inference and variable selection method for generalized linear models that accommodate the outliers, nonignorable dropouts and within-subject correlations. The purpose of our study is threefold. First, we construct the robust and bias-corrected generalized estimating equations (GEEs) by combining the Mallows-type weights, Huber's score function and inverse probability weighting approaches to against the influence of outliers and account for nonignorable dropouts. Subsequently, the generalized method of moments is utilized to estimate the parameters in the nonignorable dropout propensity based on sufficient instrumental estimating equations. Second, in order to incorporate the within-subject correlations under an informative working correlation structure, we borrow the idea of quadratic inference function and hybrid-GEE to obtain the improved empirical likelihood procedures. The asymptotic properties of the proposed estimators and their confidence regions are derived. Third, the robust variable selection and algorithm are investigated. We evaluate the performance of proposed estimators through simulation and illustrate our method in an application to HIV-CD4 data.

Smoothed quantile regression with nonignorable dropouts

In this paper, we adopt a three-stage estimation procedure and statistical inference methods for quantile regression (QR) based on empirical likelihood (EL) approach with nonignorable dropouts. In the first stage, we consider a parametric model on the dropout propensity of response and handle the parameter identifiability issue by using nonresponse instrument. With the estimated dropout propensity, in the second stage the inverse probability weighting and kernel smoothing methods are applied to construct the bias-corrected and smoothed generalized estimating equations for nonignorable dropouts. In the third stage, borrowing the matrix expansion idea of quadratic inference function, we obtain the proposed estimators that can accommodate the within-subject correlations and improve the estimation efficiency simultaneously. A class of improved estimators and their confidence regions for QR coefficient are derived. Further, the penalized EL method and algorithm for variable selection are investigated. Simulation studies and a real example on HIV-CD4 data set are also provided to show the performance of the proposed estimators.

Generalized partial linear models with nonignorable dropouts

Generalized partial linear models with nonignorable dropouts

Improved composite quantile regression and variable selection with nonignorable dropouts

With nonignorable dropouts and outliers, we propose robust statistical inference and variable selection methods for linear quantile regression models based on composite quantile regression and empirical likelihood (EL) that accommodate both the within-subject correlations and nonignorable dropouts. The purpose of our study is threefold. First, we apply the generalized method of moments to estimate the parameters in the nonignorable dropout propensity based on an instrument. Subsequently, the inverse probability weighting and kernel smoothing approaches are applied to obtain the smoothed and bias-corrected generalized estimating equations. Second, we borrow the idea of quadratic inference function to construct the improved EL procedure for nonignorable dropouts. The asymptotic properties of the proposed estimators and their confidence regions are derived. Third, the penalized EL method and algorithm for variable selection are investigated. The finite-sample performance of the proposed estimators is studied through simulation, and an application to HIV-CD4 data set is also presented.

Improved kth power expectile regression with nonignorable dropouts

The kth () power expectile regression (ER) can balance robustness and effectiveness between the ordinary quantile regression and ER simultaneously. Motivated by a longitudinal ACTG 193A data with nonignorable dropouts, we propose a two-stage estimation procedure and statistical inference methods based on the kth power ER and empirical likelihood to accommodate both the within-subject correlations and nonignorable dropouts. Firstly, we construct the bias-corrected generalized estimating equations by combining the kth power ER and inverse probability weighting approaches. Subsequently, the generalized method of moments is utilized to estimate the parameters in the nonignorable dropout propensity based on sufficient instrumental estimating equations. Secondly, in order to incorporate the within-subject correlations under an informative working correlation structure, we borrow the idea of quadratic inference function to obtain the improved empirical likelihood procedures. The asymptotic properties of the corresponding estimators and their confidence regions are derived. The finite-sample performance of the proposed estimators is studied through simulation and an application to the ACTG 193A data is also presented.

Improved empirical likelihood inference and variable selection for generalized linear models with longitudinal nonignorable dropouts

In this paper, we propose improved statistical inference and variable selection methods for generalized linear models based on empirical likelihood approach that accommodates both the within-subject correlations and nonignorable dropouts. We first apply the generalized method of moments to estimate the parameters in the nonignorable dropout propensity based on an instrument. The inverse probability weighting is applied to obtain the bias-corrected generalized estimating equations (GEEs), and then we borrow the idea of quadratic inference function and hybrid GEE to construct the empirical likelihood procedures for longitudinal data with nonignorable dropouts, respectively. Two different classes of estimators and their confidence regions are derived. Further, the penalized EL method and algorithm for variable selection are investigated. The finite-sample performance of the proposed estimators is studied through simulation, and an application to HIV-CD4 data set is also presented.

A monotone data augmentation algorithm for longitudinal data analysis via multivariate skew-t, skew-normal or t distributions.

The mixed effects model for repeated measures has been widely used for the analysis of longitudinal clinical data collected at a number of fixed time points. We propose a robust extension of the mixed effects model for repeated measures for skewed and heavy-tailed data on basis of the multivariate skew-t distribution, and it includes the multivariate normal, t, and skew-normal distributions as special cases. An efficient Markov chain Monte Carlo algorithm is developed using the monotone data augmentation and parameter expansion techniques. We employ the algorithm to perform controlled pattern imputations for sensitivity analyses of longitudinal clinical trials with nonignorable dropouts. The proposed methods are illustrated by real data analyses. Sample SAS programs for the analyses are provided in the online supplementary material.

A Bayesian dose-finding design for phase I/II clinical trials with nonignorable dropouts.

Phase I/II trials utilize both toxicity and efficacy data to achieve efficient dose finding. However, due to the requirement of assessing efficacy outcome, which often takes a long period of time to be evaluated, the duration of phase I/II trials is often longer than that of the conventional dose-finding trials. As a result, phase I/II trials are susceptible to the missing data problem caused by patient dropout, and the missing efficacy outcomes are often nonignorable in the sense that patients who do not experience treatment efficacy are more likely to drop out of the trial. We propose a Bayesian phase I/II trial design to accommodate nonignorable dropouts. We treat toxicity as a binary outcome and efficacy as a time-to-event outcome. We model the marginal distribution of toxicity using a logistic regression and jointly model the times to efficacy and dropout using proportional hazard models to adjust for nonignorable dropouts. The correlation between times to efficacy and dropout is modeled using a shared frailty. We propose a two-stage dose-finding algorithm to adaptively assign patients to desirable doses. Simulation studies show that the proposed design has desirable operating characteristics. Our design selects the target dose with a high probability and assigns most patients to the target dose.

Handling non-ignorable dropouts in longitudinal data: a conditional model based on a latent Markov heterogeneity structure

We illustrate a class of conditional models for the analysis of longitudinal data suffering attrition in random effects models framework, where the subject-specific random effects are assumed to be discrete and to follow a time-dependent latent process. The latent process accounts for unobserved heterogeneity and correlation between individuals in a dynamic fashion, and for dependence between the observed process and the missing data mechanism. Of particular interest is the case where the missing mechanism is non-ignorable. To deal with the topic we introduce a conditional to dropout model. A shape change in the random effects distribution is considered by directly modeling the effect of the missing data process on the evolution of the latent structure. To estimate the resulting model, we rely on the conditional maximum likelihood approach and for this aim we outline an EM algorithm. The proposal is illustrated via simulations and then applied on a dataset concerning skin cancers. Comparisons with other well-established methods are provided as well.

A MIXTURE DROPOUT MECHANISM IN A LONGITUDINAL STUDY WITH TWO TIME POINTS: A METHADONE STUDY

One of the most important issues that confront statisticians in longitudinal studies is dropouts. A variety of reasons may lead to withdrawal from a study and produce two different missingness mechanisms, namely, missing at random and non-ignorable dropouts. Nevertheless, none of these mechanisms is tenable in most studies. In addition, it may be that not all of dropouts are nonignorable. Many dropout handling methods have been employed by assuming only one of these dropout mechanisms. In this study, the dropout indicator is improved to take into account both dropout mechanisms. In this two-stage approach, a selection model is combined with an imputation method for dropout process in a longitudinal study with two time points. Simulation studies in a variety of situations are conducted to evaluate this approach in estimating the mean of the response variable at the second time point. This parameter is estimated by using maximum likelihood method. The results of the simulation studies indicate the superiority of the proposed method to the existing ones in estimating the mean of the variable with dropouts. In addition, this method is performed on a methadone dataset of 161 patients admitted to an Iranian clinic to estimate the final methadone dose.

Bayesian Latent-Class Mixed-Effect Hybrid Models for Dyadic Longitudinal Data with Non-Ignorable Dropouts

The analysis of longitudinal dyadic data is challenging due to the complicated correlations within and between dyads, as well as possibly non-ignorable dropouts. Based on a mixed-effects hybrid model, we propose an approach to analyze longitudinal dyadic data with non-ignorable dropouts. We factorize the joint distribution of the measurement and dropout processes into three components: the marginal distribution of random effects, the conditional distribution of the dropout process given the random effects, and the conditional distribution of the measurement process given the random effects and missing data patterns. We model the conditional dropout process using a discrete survival model, and the conditional measurement process using a latent-class pattern-mixture model. These models account for the dyadic interdependence using the "actor" and "partner" effects and dyad-specific random effects. We use the latent-dropout-class approach to address the problem of a large number of missing data patterns caused by the dyadic data structure. We evaluate the performance of the proposed method using a simulation study, and apply our method to a longitudinal dyadic data set that arose from a prostate cancer trial.

BAYESIAN MODELING LONGITUDINAL DYADIC DATA WITH NONIGNORABLE DROPOUT, WITH APPLICATION TO A BREAST CANCER STUDY.

Dyadic data are common in the social and behavioral sciences, in which members of dyads are correlated due to the interdependence structure within dyads. The analysis of longitudinal dyadic data becomes complex when nonignorable dropouts occur. We propose a fully Bayesian selection-model-based approach to analyze longitudinal dyadic data with nonignorable dropouts. We model repeated measures on subjects by a transition model and account for within-dyad correlations by random effects. In the model, we allow subject's outcome to depend on his/her own characteristics and measure history, as well as those of the other member in the dyad. We further account for the nonignorable missing data mechanism using a selection model in which the probability of dropout depends on the missing outcome. We propose a Gibbs sampler algorithm to fit the model. Simulation studies show that the proposed method effectively addresses the problem of nonignorable dropouts. We illustrate our methodology using a longitudinal breast cancer study.

A two‐part mixed‐effects pattern‐mixture model to handle zero‐inflation and incompleteness in a longitudinal setting

Two-part regression models are frequently used to analyze longitudinal count data with excess zeros, where the same set of subjects is repeatedly observed over time. In this context, several sources of heterogeneity may arise at individual level that affect the observed process. Further, longitudinal studies often suffer from missing values: individuals dropout of the study before its completion, and thus present incomplete data records. In this paper, we propose a finite mixture of hurdle models to face the heterogeneity problem, which is handled by introducing random effects with a discrete distribution; a pattern-mixture approach is specified to deal with non-ignorable missing values. This approach helps us to consider overdispersed counts, while allowing for association between the two parts of the model, and for non-ignorable dropouts. The effectiveness of the proposal is tested through a simulation study. Finally, an application to real data on skin cancer is provided.

Application of Multiple Imputation in Analysis of Data from Clinical Trials with Treatment Related Dropouts

Many longitudinal clinical studies suffer from nonignorable dropouts. Ramakrishnan and Wang (2005) proposed a mixed effects analysis treating missing data as missing due to truncation (MDT), estimated the parameters under a multivariate truncated normal model, and suggested an adjustment to the degrees of freedom to account for the implicit estimation of the missing data. We propose a multiple imputation (MI) in conjunction with the MDT method as an alternative to accurately accommodate the uncertainty introduced by imputation. The data used in Ramakrishnan and Wang (2005) is considered for illustration. A comparison of various methods using a simulation study is presented.

A Bayesian Model for Longitudinal Count Data with Non-Ignorable Dropout

Asthma is an important chronic disease of childhood. An intervention programme for managing asthma was designed on principles of self-regulation and was evaluated by a randomized longitudinal study.The study focused on several outcomes, and, typically, missing data remained a pervasive problem. We develop a pattern-mixture model to evaluate the outcome of intervention on the number of hospitalizations with non-ignorable dropouts. Pattern-mixture models are not generally identifiable as no data may be available to estimate a number of model parameters. Sensitivity analyses are performed by imposing structures on the unidentified parameters.We propose a parameterization which permits sensitivity analyses on clustered longitudinal count data that have missing values due to non-ignorable missing data mechanisms. This parameterization is expressed as ratios between event rates across missing data patterns and the observed data pattern and thus measures departures from an ignorable missing data mechanism. Sensitivity analyses are performed within a Bayesian framework by averaging over different prior distributions on the event ratios. This model has the advantage of providing an intuitive and flexible framework for incorporating the uncertainty of the missing data mechanism in the final analysis.

An autoregressive linear mixed effects model for the analysis of longitudinal data which include dropouts and show profiles approaching asymptotes

We are interested in longitudinal data of a continuous response that show profiles with an initial sharp change and approaching asymptotes for each patient, and many patients drop out with a reason related to the response. In this paper, we focus on a model that assumes a dropout process is missing at random (MAR). In this dropout process, we can obtain consistent maximum likelihood estimators as long as both the mean and covariance structures are correctly specified. However, parsimonious covariance structures for the profiles approaching asymptotes are unclear. An autoregressive linear mixed effects model can express the profile with random individual asymptotes. We show that this model provides a new parsimonious covariance structure. The covariance structure at steady state is compound symmetry and the other elements of the covariance depend on the measurement points. In simulation studies, the estimate of the asymptote is unbiased in MAR dropouts, but biased in non-ignorable dropouts. We also applied this model to actual schizophrenia trial data.

Analysis of incomplete quality of life data in advanced stage cancer: A practical application of multiple imputation

This paper presents a practical approach to analyzing incomplete quality of life (QOL) data that contains non-ignorable dropouts in patients with advanced non-small-cell lung cancer (NSCLC). QOL scores for the physical domain at baseline and at the end of the first and second courses of chemotherapy were compared between two treatment groups in a phase III trial. One hundred and 103 eligible patients were randomized to receive cisplatin and irinotecan (CPT-P) or cisplatin and vindesine, respectively; of those two groups, 83 and 85, respectively, completed a QOL questionnaire at least at baseline. A multiple imputation incorporating auxiliary QOL variables was implemented as one of alternatives of sensitivity analyses; these were complete case, available case, and pattern mixture analyses. Although larger sensitivity to missing data was found for CPT-P treatment, none of the alternative analyses demonstrated a significant difference in estimated slopes over time between the groups. This study presents an analytical approach for dealing with the complex problem of missing QOL data. It must be noted, however, that the validity of the multiple imputation method we present is not certain unless we can specify sufficiently informative auxiliary variables to ensure the conversion of non-ignorable missingness to ignorable.

Modeling longitudinal data with nonignorable dropouts using a latent dropout class model.

In longitudinal studies with dropout, pattern-mixture models form an attractive modeling framework to account for nonignorable missing data. However, pattern-mixture models assume that the components of the mixture distribution are entirely determined by the dropout times. That is, two subjects with the same dropout time have the same distribution for their response with probability one. As that is unlikely to be the case, this assumption made lead to classification error. In addition, if there are certain dropout patterns with very few subjects, which often occurs when the number of observation times is relatively large, pattern-specific parameters may be weakly identified or require identifying restrictions. We propose an alternative approach, which is a latent-class model. The dropout time is assumed to be related to the unobserved (latent) class membership, where the number of classes is less than the number of observed patterns; a regression model for the response is specified conditional on the latent variable. This is a type of shared-parameter model, where the shared "parameter" is discrete. Parameter estimates are obtained using the method of maximum likelihood. Averaging the estimates of the conditional parameters over the distribution of the latent variable yields estimates of the marginal regression parameters. The methodology is illustrated using longitudinal data on depression from a study of HIV in women.