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

In biomedical research with longitudinal designs, missing values due to intermittent non-response or premature withdrawal are usually 'non-ignorable' in the sense that unobserved values are related to the patterns of missingness. By drawing the framework of a shared-parameter mechanism, the process yielding the repeated count measures and the process yielding missing values can be modelled separately, conditionally on a group of shared parameters. For chronic diseases, Markov transition models can be used to study the transitional features of the pathologic processes. In this paper, Markov Chain Monte Carlo algorithms are developed to fit a random-effects Markov transition model for incomplete count repeated measures, within which random effects are shared by the counting process and the missing-data mechanism. Assuming a Poisson distribution for the count measures, the transition probabilities are estimated using a Poisson regression model. The missingness mechanism is modelled with a multinomial-logit regression to calculate the transition probabilities of the missingness indicators. The method is demonstrated using both simulated data sets and a practical data set from a smoking cessation clinical trial.

Longitudinal studies collect information on a sample of individuals which is followed over time to analyze the effects of individual and time-dependent characteristics on the observed response. These studies often suffer from attrition: individuals drop out of the study before its completion time and thus present incomplete data records. When the missing mechanism, once conditioned on other (observed) variables, does not depend on current (eventually unobserved) values of the response variable, the dropout mechanism is known to be ignorable. We propose a selection model extending semiparametric variance component models for longitudinal binary responses to allow for dependence between the missing data mechanism and the primary response process. The model is applied to a data set from a methadone maintenance treatment programme held in Sidney, 1986.

Rasch mixture models can be a useful tool when checking the assumption of measurement invariance for a single Rasch model. They provide advantages compared to manifest differential item functioning (DIF) tests when the DIF groups are only weakly correlated with the manifest covariates available. Unlike in single Rasch models, estimation of Rasch mixture models is sensitive to the specification of the ability distribution even when the conditional maximum likelihood approach is used. It is demonstrated in a simulation study how differences in ability can influence the latent classes of a Rasch mixture model. If the aim is only DIF detection, it is not of interest to uncover such ability differences as one is only interested in a latent group structure regarding the item difficulties. To avoid any confounding effect of ability differences (or impact), a new score distribution for the Rasch mixture model is introduced here. It ensures the estimation of the Rasch mixture model to be independent of the ability distribution and thus restricts the mixture to be sensitive to latent structure in the item difficulties only. Its usefulness is demonstrated in a simulation study, and its application is illustrated in a study of verbal aggression.

The analysis of longitudinal data with non-ignorable missingness remains an active area in biostatistics research. This article discusses various random effects and latent process models which have been proposed for analyzing longitudinal binary data subject to both non-ignorable intermittent missing data and dropout. These models account for non-ignorable missingness by introducing random effects or a latent process which is shared between the response model and the model for the missing-data mechanism. We discuss various random effects and latent processes approaches and compare these approaches with analyses from an opiate clinical trial data set, which had high proportion of intermittent missingness and dropout. We also compare these random effect and latent process approaches with other methods for accounting for non-ignorable missingness using this data set.

Background:Composite measures, like the Disease Activity Score for 28 joints (DAS28),are key primary outcomes in rheumatoid arthritis (RA) trials. DAS28 combinesfour different components in a continuous measure. When one or more of thesecomponents are missing the overall composite score is also missing atintermediate or trial endpoint assessments.Objectives:This study examined missing data patterns and mechanisms in a longitudinal RAtrial to evaluate how best to handle missingness when analysing compositeoutcomes.Design:The Tumour-Necrosis-Factor Inhibitors against Combination Intensive Therapy(TACIT) trial was an open label, pragmatic randomized multicentre two armnon-inferiority study. Patients were followed up for 12 months, with monthlymeasurement of the composite outcome and its components. Active RA patientswere randomized to conventional disease modifying drugs (cDMARDs) or TumourNecrosis Factor-α inhibitors (TNFis).Methods:The TACIT trial was used to explore the extent of missing data in thecomposite outcome, DAS28. Patterns of missing data in components and thecomposite outcome were examined graphically. Longitudinal multivariablelogistic regression analysis assessed missing data mechanisms duringfollow-up.Results:Two hundred and five patients were randomized: at 12 months 59/205 (29%) hadunobserved composite outcome and 146/205 (71%) had an observed DAS28outcome; however, 34/146 had one or more intermediate assessments missing.We observed mixed missing data patterns, especially for the missingcomposite outcome due to one component missing rather than patient notattending thier visit. Age and gender predicted missingness components,providing strong evidence the missing observations were unlikely to beMissing Completely at Random (MCAR).Conclusion:Researchers should undertake detailed evaluations of missing data patternsand mechanisms at the final and intermediate time points, whether or not theoutcome variable is a composite outcome. In addition, the impact ontreatment estimates in patients who only provide data at milestoneassessments need to be assessed.Trial Registration ISRCTN Number:37438295

BackgroundMissing data due to attrition are rampant in substance abuse clinical trials. However, missing data are often ignored in the presentation of substance abuse clinical trials. This paper demonstrates missing data methods which may be used for hypothesis testing.MethodsMethods involving stratifying and weighting individuals based on missing data pattern are shown to produce tests that are robust to missing data mechanisms in terms of Type I error and power. In this article, we describe several methods of combining data that may be used for testing hypotheses of the treatment effect. Furthermore, illustrations of each test's Type I error and power under different missing data percentages and mechanisms are quantified using a Monte-Carlo simulation study.ResultsType I error rates were similar for each method, while powers depended on missing data assumptions. Specifically, power was greatest for the weighted, compared to un-weighted methods, especially for greater missing data percentages.ConclusionResults of this study as well as extant literature demonstrate the need for standards of design and analysis specific to substance abuse clinical trials. Given the known substantial attrition rates and concern for the missing data mechanism in substance abuse clinical trials, investigators need to incorporate missing data methods a priori. That is, missing data methods should be specified at the outset of the study and not after the data have been collected.

Summary We propose a method for estimating parameters in generalized linear models with missing covariates and a non-ignorable missing data mechanism. We use a multinomial model for the missing data indicators and propose a joint distribution for them which can be written as a sequence of one-dimensional conditional distributions, with each one-dimensional conditional distribution consisting of a logistic regression. We allow the covariates to be either categorical or continuous. The joint covariate distribution is also modelled via a sequence of one-dimensional conditional distributions, and the response variable is assumed to be completely observed. We derive the E- and M-steps of the EM algorithm with non-ignorable missing covariate data. For categorical covariates, we derive a closed form expression for the E- and M-steps of the EM algorithm for obtaining the maximum likelihood estimates (MLEs). For continuous covariates, we use a Monte Carlo version of the EM algorithm to obtain the MLEs via the Gibbs sampler. Computational techniques for Gibbs sampling are proposed and implemented. The parametric form of the assumed missing data mechanism itself is not ‘testable' from the data, and thus the non-ignorable modelling considered here can be viewed as a sensitivity analysis concerning a more complicated model. Therefore, although a model may have ‘passed' the tests for a certain missing data mechanism, this does not mean that we have captured, even approximately, the correct missing data mechanism. Hence, model checking for the missing data mechanism and sensitivity analyses play an important role in this problem and are discussed in detail. Several simulations are given to demonstrate the methodology. In addition, a real data set from a melanoma cancer clinical trial is presented to illustrate the methods proposed.

BackgroundTraditional approaches to identify missing mechanisms are usually based on the hypothesis test and confronted with both theoretical and practical challenges. It has been proved that the Bayesian network is powerful in integrating, analyzing and visualizing information, and some previous researches have verified the promising features of Bayesian network to deal with the aforementioned challenges in missing mechanism identification. Based on the above reasons, this paper explores the application of Bayesian network to the identification of missing mechanisms for the first time, and proposes a new method, the Bayesian network-based missing mechanism identification (BN-MMI) method, to identify missing mechanism in medical research.MethodsThe procedure of BN-MMI method consists three easy-to-implement steps: estimating the missing data structure by the Bayesian network; assessing the credibility of the estimated missing data structure; and identifying the missing mechanism from the estimated missing data structure. The BN-MMI method is verified by simulation research and empirical research.ResultsThe simulation study verified the validity, consistency and robustness of BN-MMI method, and indicated its outperformance in contrast to the traditional logistic regression method. In addition, the empirical study illustrated the applicability of BN-MMI method in the real world by an example of medical record data.ConclusionsIt was confirmed that the BN-MMI method itself, together with human knowledge and expertise, could identify the missing mechanisms according to the probabilistic dependence/independence relations among variables of interest. At the same time, our research shed light upon the potential application of BN-MMI method to a broader range of missing data issues in medical studies.

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.

Missing data are frequently encountered in longitudinal clinical trials. To better monitor and understand the progress over time, one must handle the missing data appropriately and examine whether the missing data mechanism is ignorable or nonignorable. In this article, we develop a new probit model for longitudinal binary response data. It resolves a challenging issue for estimating the variance of the random effects, and substantially improves the convergence and mixing of the Gibbs sampling algorithm. We show that when improper uniform priors are specified for the regression coefficients of the joint multinomial model via a sequence of one-dimensional conditional distributions for the missing data indicators under nonignorable missingness, the joint posterior distribution is improper. A variation of Jeffreys prior is thus established as a remedy for the improper posterior distribution. In addition, an efficient Gibbs sampling algorithm is developed using a collapsing technique. Two model assessment criteria, the deviance information criterion (DIC) and the logarithm of the pseudomarginal likelihood (LPML), are used to guide the choices of prior specifications and to compare the models under different missing data mechanisms. We report on extensive simulations conducted to investigate the empirical performance of the proposed methods. The proposed methodology is further illustrated using data from an HIV prevention clinical trial.

We analytically derive the fixed-effects estimates in unconditional linear growth curve models by typical linear mixed-effects modelling (TLME) and by a pattern-mixture (PM) approach with random-slope-dependent two-missing-pattern missing not at random (MNAR) longitudinal data. Results showed that when the missingness mechanism is random-slope-dependent MNAR, TLME estimates of both the mean intercept and mean slope are biased because of incorrect weights used in the estimation. More specifically, the estimate of the mean slope is biased towards the mean slope for completers, whereas the estimate of the mean intercept is biased towards the opposite direction as compared to the estimate of the mean slope. We also discuss why the PM approach can provide unbiased fixed-effects estimates for random-coefficients-dependent MNAR data but does not work well for missing at random or outcome-dependent MNAR data. A small simulation study was conducted to illustrate the results and to compare results from TLME and PM. Results from an empirical data analysis showed that the conceptual finding can be generalized to other real conditions even when some assumptions for the analytical derivation cannot be met. Implications from the analytical and empirical results were discussed and sensitivity analysis was suggested for longitudinal data analysis with missing data.

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.

BackgroundProcedural skills are a key element in the training of future physiotherapists. Procedural skills relate to the acquisition of appropriate motor skills, which allow the safe application of clinical procedures to patients. In order to evaluate procedural skills in physiotherapy education validated assessment instruments are required. Recently the assessment of procedural skills in physiotherapy education (APSPT) tool was developed. The overall aim of this study was to establish the structural validity of the APSPT. In order to do this the following objectives were examined: i) the fit of the items of APSPT to the Rasch-model, ii) the fit of the overall score to the Rasch model, iii) the difficulty of each test item and iv) whether the difficulty levels of the individual test items cover the whole capacity spectrum of students in pre-registration physiotherapy education.MethodsFor this observational cross-sectional measurement properties study a convenience sample of 69 undergraduate pre-registration physiotherapy students of the HES-SO Valais-Wallis was recruited. Participants were instructed to perform a task procedure on a simulated patient. The performance was evaluated with the APSPT. A conditional maximum likelihood approach was used to estimate the parameters of a partial credit model for polytomous item responses. Item fit, ordering of thresholds, targeting and goodness of fit to the Rasch model was assessed.ResultsItem fit statistics showed that 25 items of the APSPT showed adequate fit to the Rasch model. Disordering of item thresholds did not occur and the targeting of the APSPT was adequate to measure the abilities of the included participants. Undimensionality and subgroup homogeneity were confirmed.ConclusionThis study presented evidence for the structural validity of the APSPT. Undimensionality of the APSPT was confirmed and therefore presents evidence that the latent dimension of procedural skills in physiotherapy education consists of several subcategories. However, the results should be interpreted with caution given the small sample size.

ABSTRACTSemiparametric mixed‐effects double regression models have demonstrated satisfactory efficacy across diverse applications of longitudinal studies. However, the estimation of these models often relies on the assumptions of normally distributed errors and complete data with no missing values. Therefore, such restrictions may limit the practical usage of these models when analyzing longitudinal data that exhibit heavy‐tailed behaviors and/or contain missing values. This paper introduces a Bayesian quantile regression‐based semiparametric mixed‐effects double regression model for examining longitudinal data with non‐ignorable missing responses. Here, the quantile regression is used to address non‐normality issues, and the missing mechanism is defined through a logistic regression model. Our proposed algorithm can concurrently model both the mean and variance of the mixed effects as functions of predictors while investigating the predictor effects at different quantiles of interest. Additionally, we utilize the Bayesian adaptive LASSO hierarchical model to devise an effective Metropolis‐Hastings‐within‐Gibbs computation algorithm for both estimation and variable selection purposes. Finally, we conduct different simulation studies and a real‐data example to demonstrate the successful implementation of our proposed Bayesian methodology.

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.