A copula model for repeated measurements with non-ignorable non-monotone missing outcome

A normal copula-based selection model is proposed for continuous longitudinal data with a non-ignorable non-monotone missing-data process. The normal copula is used to combine the distribution of the outcome of interest and that of the missing-data indicators given the covariates. Parameters in the model are estimated by a pseudo-likelihood method. We first use the GEE with a logistic link to estimate the parameters associated with the marginal distribution of the missing-data indicator given the covariates, assuming that covariates are always observed. Then we estimate other parameters by inserting the estimates from the first step into the full likelihood function. A simulation study is conducted to assess the robustness of the assumed model under different missing-data processes. The proposed method is then applied to one example from a community cohort study to demonstrate its capability to reduce bias.     

Estimation methods for marginal and association parameters for longitudinal binary data with nonignorable missing observations

In longitudinal studies, missing observations occur commonly. It has been well known that biased results could be produced if missingness is not properly handled in the analysis. Authors have developed many methods with the focus on either incomplete response or missing covariate observations, but rarely on both. The complexity of modeling and computational difficulty would be the major challenges in handling missingness in both response and covariate variables. In this paper, we develop methods using the pairwise likelihood formulation to handle longitudinal binary data with missing observations present in both response and covariate variables. We propose a unified framework to accommodate various types of missing data patterns. We evaluate the performance of the methods empirically under a variety of circumstances. In particular, we investigate issues on efficiency and robustness. We analyze longitudinal data from the National Population Health Study with the use of our methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Explicating the Conditions Under Which Multilevel Multiple Imputation Mitigates Bias Resulting from Random Coefficient-Dependent Missing Longitudinal Data

Random coefficient-dependent (RCD) missingness is a non-ignorable mechanism through which missing data can arise in longitudinal designs. RCD, for which we cannot test, is a problematic form of missingness that occurs if subject-specific random effects correlate with propensity for missingness or dropout. Particularly when covariate missingness is a problem, investigators typically handle missing longitudinal data by using single-level multiple imputation procedures implemented with long-format data, which ignores within-person dependency entirely, or implemented with wide-format (i.e., multivariate) data, which ignores some aspects of within-person dependency. When either of these standard approaches to handling missing longitudinal data is used, RCD missingness leads to parameter bias and incorrect inference. We explain why multilevel multiple imputation (MMI) should alleviate bias induced by a RCD missing data mechanism under conditions that contribute to stronger determinacy of random coefficients. We evaluate our hypothesis with a simulation study. Three design factors are considered: intraclass correlation (ICC; ranging from .25 to .75), number of waves (ranging from 4 to 8), and percent of missing data (ranging from 20 to 50%). We find that MMI greatly outperforms the single-level wide-format (multivariate) method for imputation under a RCD mechanism. For the MMI analyses, bias was most alleviated when the ICC is high, there were more waves of data, and when there was less missing data. Practical recommendations for handling longitudinal missing data are suggested.     

A multistate Markov chain model for longitudinal, categorical quality-of-life data subject to non-ignorable missingness

Quality-of-life (QOL) is an important outcome in clinical research, particularly in cancer clinical trials. Typically, data are collected longitudinally from patients during treatment and subsequent follow-up. Missing data are a common problem, and missingness may arise in a non-ignorable fashion. In particular, the probability that a patient misses an assessment may depend on the patient's QOL at the time of the scheduled assessment. We propose a Markov chain model for the analysis of categorical outcomes derived from QOL measures. Our model assumes that transitions between QOL states depend on covariates through generalized logit models or proportional odds models. To account for non-ignorable missingness, we incorporate logistic regression models for the conditional probabilities of observing measurements, given their actual values. The model can accommodate time-dependent covariates. Estimation is by maximum likelihood, summing over all possible values of the missing measurements. We describe options for selecting parsimonious models, and we study the finite-sample properties of the estimators by simulation. We apply the techniques to data from a breast cancer clinical trial in which QOL assessments were made longitudinally, and in which missing data frequently arose.     

Covariate adjustment in randomized trials with binary outcomes: targeted maximum likelihood estimation

Covariate adjustment using linear models for continuous outcomes in randomized trials has been shown to increase efficiency and power over the unadjusted method in estimating the marginal effect of treatment. However, for binary outcomes, investigators generally rely on the unadjusted estimate as the literature indicates that covariate-adjusted estimates based on the logistic regression models are less efficient. The crucial step that has been missing when adjusting for covariates is that one must integrate/average the adjusted estimate over those covariates in order to obtain the marginal effect. We apply the method of targeted maximum likelihood estimation (tMLE) to obtain estimators for the marginal effect using covariate adjustment for binary outcomes. We show that the covariate adjustment in randomized trials using the logistic regression models can be mapped, by averaging over the covariate(s), to obtain a fully robust and efficient estimator of the marginal effect, which equals a targeted maximum likelihood estimator. This tMLE is obtained by simply adding a clever covariate to a fixed initial regression. We present simulation studies that demonstrate that this tMLE increases efficiency and power over the unadjusted method, particularly for smaller sample sizes, even when the regression model is mis-specified.     

Analysis of non-ignorable missing and left-censored longitudinal data using a weighted random effects tobit model

In a longitudinal study with response data collected during a hospital stay, observations may be missing because of the subject's discharge from the hospital prior to completion of the study or the death of the subject, resulting in non-ignorable missing data. In addition to non-ignorable missingness, there is left-censoring in the response measurements because of the inherent limit of detection. For analyzing non-ignorable missing and left-censored longitudinal data, we have proposed to extend the theory of random effects tobit regression model to weighted random effects tobit regression model. The weights are computed on the basis of inverse probability weighted augmented methodology. An extensive simulation study was performed to compare the performance of the proposed model with a number of competitive models. The simulation study shows that the estimates are consistent and that the root mean square errors of the estimates are minimal for the use of augmented inverse probability weights in the random effects tobit model. The proposed method is also applied to the non-ignorable missing and left-censored interleukin-6 biomarker data obtained from the Genetic and Inflammatory Markers of Sepsis study.     

Recovery of information from multiple imputation: a simulation study

ABSTRACT: BACKGROUND: Multiple imputation is becoming increasingly popular for handling missing data. However, it is often implemented without adequate consideration of whether it offers any advantage over complete case analysis for the research question of interest, or whether potential gains may be offset by bias from a poorly fitting imputation model, particularly as the amount of missing data increases. METHODS: Simulated datasets (n = 1000) drawn from a synthetic population were used to explore information recovery from multiple imputation in estimating the coefficient of a binary exposure variable when various proportions of data (10-90%) were set missing at random in a highly-skewed continuous covariate or in the binary exposure. Imputation was performed using multivariate normal imputation (MVNI), with a simple or zero-skewness log transformation to manage non-normality. Bias, precision, mean-squared error and coverage for a set of regression parameter estimates were compared between multiple imputation and complete case analyses. RESULTS: For missingness in the continuous covariate, multiple imputation produced less bias and greater precision for the effect of the binary exposure variable, compared with complete case analysis, with larger gains in precision with more missing data. However, even with only moderate missingness, large bias and substantial under-coverage were apparent in estimating the continuous covariate's effect when skewness was not adequately addressed. For missingness in the binary covariate, all estimates had negligible bias but gains in precision from multiple imputation were minimal, particularly for the coefficient of the binary exposure. CONCLUSIONS: Although multiple imputation can be useful if covariates required for confounding adjustment are missing, benefits are likely to be minimal when data are missing in the exposure variable of interest. Furthermore, when there are large amounts of missingness, multiple imputation can become unreliable and introduce bias not present in a complete case analysis if the imputation model is not appropriate. Epidemiologists dealing with missing data should keep in mind the potential limitations as well as the potential benefits of multiple imputation. Further work is needed to provide clearer guidelines on effective application of this method.     

A tractable method to account for high-dimensional nonignorable missing data in intensive longitudinal data

Despite the need for sensitivity analysis to nonignorable missingness in intensive longitudinal data (ILD), such analysis is greatly hindered by novel ILD features, such as large data volume and complex nonmonotonic missing-data patterns. Likelihood of alternative models permitting nonignorable missingness often involves very high-dimensional integrals, causing curse of dimensionality and rendering solutions computationally prohibitive to obtain. We aim to overcome this challenge by developing a computationally feasible method, nonlinear indexes of local sensitivity to nonignorability (NISNI). We use linear mixed effects models for the incomplete outcome and covariates. We use Markov multinomial models to describe complex missing-data patterns and mechanisms in ILD, thereby permitting missingness probabilities to depend directly on missing data. Using a second-order Taylor series to approximate likelihood under nonignorability, we develop formulas and closed-form expressions for NISNI. Our approach permits the outcome and covariates to be missing simultaneously, as is often the case in ILD, and can capture U-shaped impact of nonignorability in the neighborhood of the missing at random model without fitting alternative models or evaluating integrals. We evaluate performance of this method using simulated data and real ILD collected by the ecological momentary assessment method.     

Discrete-time survival data with longitudinal covariates

Survival analysis has been conventionally performed on a continuous time scale. In practice, the survival time is often recorded or handled on a discrete scale; when this is the case, the discrete-time survival analysis would provide analysis results more relevant to the actual data scale. Besides, data on time-dependent covariates in the survival analysis are usually collected through intermittent follow-ups, resulting in the missing and mismeasured covariate data. In this work, we propose the sufficient discrete hazard (SDH) approach to discrete-time survival analysis with longitudinal covariates that are subject to missingness and mismeasurement. The SDH method employs the conditional score idea available for dealing with mismeasured covariates, and the penalized least squares for estimating the missing covariate value using the regression spline basis. The SDH method is developed for the single event analysis with the logistic discrete hazard model, and for the competing risks analysis with the multinomial logit model. Simulation results revel good finite-sample performances of the proposed estimator and the associated asymptotic theory. The proposed SDH method is applied to the scleroderma lung study data, where the time to medication withdrawal and time to death were recorded discretely in months, for illustration.     

Bayesian semiparametric regression for longitudinal binary processes with missing data

Longitudinal studies with binary repeated measures are widespread in biomedical research. Marginal regression approaches for balanced binary data are well developed, whereas for binary process data, where measurement times are irregular and may differ by individuals, likelihood-based methods for marginal regression analysis are less well developed. In this article, we develop a Bayesian regression model for analyzing longitudinal binary process data, with emphasis on dealing with missingness. We focus on the settings where data are missing at random (MAR), which require a correctly specified joint distribution for the repeated measures in order to draw valid likelihood-based inference about the marginal mean. To provide maximum flexibility, the proposed model specifies both the marginal mean and serial dependence structures using nonparametric smooth functions. Serial dependence is allowed to depend on the time lag between adjacent outcomes as well as other relevant covariates. Inference is fully Bayesian. Using simulations, we show that adequate modeling of the serial dependence structure is necessary for valid inference of the marginal mean when the binary process data are MAR. Longitudinal viral load data from the HIV Epidemiology Research Study are analyzed for illustration.     

Semiparametric regression models for repeated measures of mortal cohorts with non‐monotone missing outcomes and time‐dependent covariates

We propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non‐response to estimate regression parameters interpreted as conditioned on being alive. Our proposed method accommodates outcomes and time‐dependent covariates that are missing not at random with non‐monotone missingness patterns via inverse‐probability weighting. Missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse‐probability‐weighted estimating equation. Thus, we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates. The approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition, inflammation, and age. Copyright © 2010 John Wiley & Sons, Ltd.     

Covariate adjustment in clinical trials with non-ignorable missing data and non-compliance

Estimating causal effects in psychiatric clinical trials is often complicated by treatment non-compliance and missing outcomes. While new estimators have recently been proposed to address these problems, they do not allow for inclusion of continuous covariates. We propose estimators that adjust for continuous covariates in addition to non-compliance and missing data. Using simulations, we compare mean squared errors for the new estimators with those of previously established estimators. We then illustrate our findings in a study examining the efficacy of clozapine versus haloperidol in the treatment of refractory schizophrenia. For data with continuous or binary outcomes in the presence of non-compliance, non-ignorable missing data, and a covariate effect, the new estimators generally performed better than the previously established estimators. In the clozapine trial, the new estimators gave point and interval estimates similar to established estimators. We recommend the new estimators as they are unbiased even when outcomes are not missing at random and they are more efficient than established estimators in the presence of covariate effects under the widest variety of circumstances.     

Missing data in longitudinal studies: cross-sectional multiple imputation provides similar estimates to full-information maximum likelihood

PurposeThe aim of this research was to examine, in an exploratory manner, whether cross-sectional multiple imputation generates valid parameter estimates for a latent growth curve model in a longitudinal data set with nonmonotone missingness.MethodsA simulated longitudinal data set of N = 5000 was generated and consisted of a continuous dependent variable, assessed at three measurement occasions and a categorical time-invariant independent variable. Missing data had a nonmonotone pattern and the proportion of missingness increased from the initial to the final measurement occasion (5%–20%). Three methods were considered to deal with missing data: listwise deletion, full-information maximum likelihood, and multiple imputation. A latent growth curve model was specified and analysis of variance was used to compare parameter estimates between the full data set and missing data approaches.ResultsMultiple imputation resulted in significantly lower slope variance compared with the full data set. There were no differences in any parameter estimates between the multiple imputation and full-information maximum likelihood approaches.ConclusionsThis study suggested that in longitudinal studies with nonmonotone missingness, cross-sectional imputation at each time point may be viable and produces estimates comparable with those obtained with full-information maximum likelihood. Future research pursuing the validity of this method is warranted.     

A transition model for quality‐of‐life data with non‐ignorable non‐monotone missing data

In this paper, we consider a full likelihood method to analyze continuous longitudinal responses with non‐ignorable non‐monotone missing data. We consider a transition probability model for the missingness mechanism. A first‐order Markov dependence structure is assumed for both the missingness mechanism and observed data. This process fits the natural data structure in the longitudinal framework. Our main interest is in estimating the parameters of the marginal model and evaluating the missing‐at‐random assumption in the Effects of Public Information Study, a cancer‐related study recently conducted at the University of Pennsylvania. We also present a simulation study to assess the performance of the model. Copyright © 2012 John Wiley & Sons, Ltd.     

THE RELATIONSHIP BETWEEN HOT-DECK MULTIPLE IMPUTATION AND WEIGHTED LIKELIHOOD

Hot-deck imputation is an intuitively simple and popular method of accommodating incomplete data. Users of the method will often use the usual multiple imputation variance estimator which is not appropriate in this case. However, no variance expression has yet been derived for this easily implemented method applied to missing covariates in regression models. The simple hot-deck method is in fact asymptotically equivalent to the mean-score method for the estimation of a regression model parameter, so that hot-deck can be understood in the context of likelihood methods. Both of these methods accommodate data where missingness may depend on the observed variables but not on the unobserved value of the incomplete covariate, that is, missing at random (MAR). The asymptotic properties of hot-deck are derived here for the case where the fully observed variables are categorical, though the incomplete covariate(s) may be continuous. Simulation studies indicate that the two methods compare well in small samples and for small numbers of imputations. Current users of hot-deck may now conduct their analysis using mean-score, which is a weighted likelihood method and can thus be implemented by a single pass through the data using any standard package which accommodates weighted regression models. Valid inference is now straightforward using the variance expression provided here. The equivalence of mean-score and hot-deck is illustrated using three clinical data sets where an important covariate is missing for a large number of study subjects. © 1997 by John Wiley & Sons, Ltd.     

NON-RESPONSE MODELS FOR THE ANALYSIS OF NON-MONOTONE IGNORABLE MISSING DATA

We discuss a new class of ignorable non-monotone missing data models – the randomized monotone missingness (RMM) models. We argue that the RMM models represent the most general plausible physical mechanism for generating non-monotone ignorable data. We show that there exists ignorable missing data processes that are not RMM. We argue that it may therefore be inappropriate to analyse non-monotone missing data under the assumption that the missingness mechanism is ignorable, if a statistical test has rejected the hypothesis that the missing data process is RMM representable. We use RMM models to analyse data from a case-control study of the effects of radiation on breast cancer. © 1997 by John Wiley & Sons, Ltd.     

Logistic regression models with missing covariate values for complex survey data

Maximum likelihood methods are used to incorporate partially observed covariate values in fitting logistic regression models. We extend these methods to data collected through complex surveys using the pseudo-likelihood approach. One can obtain parameter estimates of the logistic regression model using standard statistical software and their standard errors by Taylor series expansion or the jackknife method. We apply the approach to data from a two-phase survey screening for dementia in a community sample of African Americans age 65 and older living in Indianapolis. The binary response variable is dementia and the covariate with missing values is a daily functioning score collected from interviews with a relative of the study subject. © 1997 John Wiley & Sons, Ltd.     

Longitudinal latent variable models given incompletely observed biomarkers and covariates

In this paper, we analyze a two‐level latent variable model for longitudinal data from the National Growth and Health Study where surrogate outcomes or biomarkers and covariates are subject to missingness at any of the levels. A conventional method for efficient handling of missing data is to re‐express the desired model as a joint distribution of variables, including the biomarkers, that are subject to missingness conditional on all of the covariates that are completely observed, and estimate the joint model by maximum likelihood, which is then transformed to the desired model. The joint model, however, identifies more parameters than desired, in general. We show that the over‐identified joint model produces biased estimation of the latent variable model and describe how to impose constraints on the joint model so that it has a one‐to‐one correspondence with the desired model for unbiased estimation. The constrained joint model handles missing data efficiently under the assumption of ignorable missing data and is estimated by a modified application of the expectation‐maximization algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

Identification of significant host factors for HIV dynamics modelled by non-linear mixed-effects models

Non-linear mixed-effects models are powerful tools for modelling HIV viral dynamics. In AIDS clinical trials, the viral load measurements for each subject are often sparse. In such cases, linearization procedures are usually used for inferences. Under such linearization procedures, however, standard covariate selection methods based on the approximate likelihood, such as the likelihood ratio test, may not be reliable. In order to identify significant host factors for HIV dynamics, in this paper we consider two alternative approaches for covariate selection: one is based on individual non-linear least square estimates and the other is based on individual empirical Bayes estimates. Our simulation study shows that, if the within-individual data are sparse and the between-individual variation is large, the two alternative covariate selection methods are more reliable than the likelihood ratio test, and the more powerful method based on individual empirical Bayes estimates is especially preferable. We also consider the missing data in covariates. The commonly used missing data methods may lead to misleading results. We recommend a multiple imputation method to handle missing covariates. A real data set from an AIDS clinical trial is analysed based on various covariate selection methods and missing data methods.     

Robust estimation in mixed linear models with non-monotone missingness

We introduce a model to account for abrupt changes among repeated measures with non-monotone missingness. Development of likelihood inferences for such models is hard because it involves intractable integration to obtain the marginal likelihood. We use hierarchical likelihood to overcome such difficulty. Abrupt changes among repeated measures can be well described by introducing random effects in the dispersion. A simulation study shows that the resulting estimator is efficient, robust against misspecification of fatness of tails. For illustration we use a schizophrenic behaviour data presented by Rubin and Wu.