Using validation sets for outcomes can greatly improve the estimation of vaccine efficacy (VE) in the field (Halloran and Longini, 2001; Halloran and others, 2003). Most statistical methods for using validation sets rely on the assumption that outcomes on those with no cultures are missing at random (MAR). However, often the validation sets will not be chosen at random. For example, confirmational cultures are often done on people with influenza-like illness as part of routine influenza surveillance. VE estimates based on such non-MAR validation sets could be biased. Here we propose frequentist and Bayesian approaches for estimating VE in the presence of validation bias. Our work builds on the ideas of Rotnitzky and others (1998, 2001), Scharfstein and others (1999, 2003), and Robins and others (2000). Our methods require expert opinion about the nature of the validation selection bias. In a re-analysis of an influenza vaccine study, we found, using the beliefs of a flu expert, that within any plausible range of selection bias the VE estimate based on the validation sets is much higher than the point estimate using just the non-specific case definition. Our approach is generally applicable to studies with missing binary outcomes with categorical covariates.
Bayesian; Expert opinion; Identifiability; Influenza; Missing data; Selection model; Vaccine efficacy
In this paper, we develop a Bayesian method for joint analysis of longitudinal measurements and competing risks failure time data. The model allows one to analyze the longitudinal outcome with nonignorable missing data induced by multiple types of events, to analyze survival data with dependent censoring for the key event, and to draw inferences on multiple endpoints simultaneously. Compared with the likelihood approach, the Bayesian method has several advantages. It is computationally more tractable for high-dimensional random effects. It is also convenient to draw inference. Moreover, it provides a means to incorporate prior information that may help to improve estimation accuracy. An illustration is given using a clinical trial data of scleroderma lung disease. The performance of our method is evaluated by simulation studies.
joint modeling; competing risks; longitudinal data; Bayesian approach
For decades, statisticians, philosophers, medical investigators and others interested in data analysis have argued that the Bayesian paradigm is the proper approach for reporting the results of scientific analyses for use by clients and readers. To date, the methods have been too complicated for non-statisticians to use. In this paper we argue that the World-Wide Web provides the perfect environment to put the Bayesian paradigm into practice: the likelihood function of the data is parsimoniously represented on the server side, the reader uses the client to represent her prior belief, and a downloaded program (a Java applet) performs the combination. In our approach, a different applet can be used for each likelihood function, prior belief can be assessed graphically, and calculation results can be reported in a variety of ways. We present a prototype implementation, BayesApplet, for two-arm clinical trials with normally-distributed outcomes, a prominent model for clinical trials. The primary implication of this work is that publishing medical research results on the Web can take a form beyond or different from that currently used on paper, and can have a profound impact on the publication and use of research results.
While Berkson’s bias is widely recognized in the epidemiologic literature, it remains underappreciated as a model of both selection bias and bias due to missing data. Simple causal diagrams and 2×2 tables illustrate how Berkson’s bias connects to collider bias and selection bias more generally, and show the strong analogies between Berksonian selection bias and bias due to missing data. In some situations, considerations of whether data are missing at random or missing not at random is less important than the causal structure of the missing-data process. While dealing with missing data always relies on strong assumptions about unobserved variables, the intuitions built with simple examples can provide a better understanding of approaches to missing data in real-world situations.
In the analysis of missing data, sensitivity analyses are commonly used to check the sensitivity of the parameters of interest with respect to the missing data mechanism and other distributional and modeling assumptions. In this article, we formally develop a general local influence method to carry out sensitivity analyses of minor perturbations to generalized linear models in the presence of missing covariate data. We examine two types of perturbation schemes (the single-case and global perturbation schemes) for perturbing various assumptions in this setting. We show that the metric tensor of a perturbation manifold provides useful information for selecting an appropriate perturbation. We also develop several local influence measures to identify influential points and test model misspecification. Simulation studies are conducted to evaluate our methods, and real datasets are analyzed to illustrate the use of our local influence measures.
Influence measure; Local influence; Missing covariates; Perturbation manifold; Perturbation scheme
Randomized clinical trials are the gold standard for evaluating interventions as
randomized assignment equalizes known and unknown characteristics between
intervention groups. However, when participants miss visits, the ability to
conduct an intent-to-treat analysis and draw conclusions about a causal link is
compromised. As guidance to those performing clinical trials, this review is a
non-technical overview of the consequences of missing data and a prescription
for its treatment beyond the typical analytic approaches to the entire research
process. Examples of bias from incorrect analysis with missing data and
discussion of the advantages/disadvantages of analytic methods are given. As no
single analysis is definitive when missing data occurs, strategies for its
prevention throughout the course of a trial are presented. We aim to convey an
appreciation for how missing data influences results and an understanding of the
need for careful consideration of missing data during the design, planning,
conduct, and analytic stages.
missing data; clinical trial; intent to treat; MCAR; MAR; MNAR; study design
In this article, we first study parameter identifiability in randomized clinical trials with noncompliance and missing outcomes. We show that under certain conditions the parameters of interest are identifiable even under different types of completely nonignorable missing data: that is, the missing mechanism depends on the outcome. We then derive their maximum likelihood and moment estimators and evaluate their finite-sample properties in simulation studies in terms of bias, efficiency, and robustness. Our sensitivity analysis shows that the assumed nonignorable missing-data model has an important impact on the estimated complier average causal effect (CACE) parameter. Our new method provides some new and useful alternative nonignorable missing-data models over the existing latent ignorable model, which guarantees parameter identifiability, for estimating the CACE in a randomized clinical trial with noncompliance and missing data.
Causal inference; Identifiability; Maximum likelihood estimates; Missing data; Noncompliance; Nonignorable
Case-control studies are particularly susceptible to differential exposure misclassification when exposure status is determined following incident case status. Probabilistic bias analysis methods have been developed as ways to adjust standard effect estimates based on the sensitivity and specificity of exposure misclassification. The iterative sampling method advocated in probabilistic bias analysis bears a distinct resemblance to a Bayesian adjustment; however, it is not identical. Furthermore, without a formal theoretical framework (Bayesian or frequentist), the results of a probabilistic bias analysis remain somewhat difficult to interpret. We describe, both theoretically and empirically, the extent to which probabilistic bias analysis can be viewed as approximately Bayesian. While the differences between probabilistic bias analysis and Bayesian approaches to misclassification can be substantial, these situations often involve unrealistic prior specifications and are relatively easy to detect. Outside of these special cases, probabilistic bias analysis and Bayesian approaches to exposure misclassification in case-control studies appear to perform equally well.
Randomized trials with dropouts or censored data and discrete time-to-event type outcomes are frequently analyzed using the Kaplan–Meier or product limit (PL) estimation method. However, the PL method assumes that the censoring mechanism is noninformative and when this assumption is violated, the inferences may not be valid. We propose an expanded PL method using a Bayesian framework to incorporate informative censoring mechanism and perform sensitivity analysis on estimates of the cumulative incidence curves. The expanded method uses a model, which can be viewed as a pattern mixture model, where odds for having an event during the follow-up interval (tk−1,tk], conditional on being at risk at tk−1, differ across the patterns of missing data. The sensitivity parameters relate the odds of an event, between subjects from a missing-data pattern with the observed subjects for each interval. The large number of the sensitivity parameters is reduced by considering them as random and assumed to follow a log-normal distribution with prespecified mean and variance. Then we vary the mean and variance to explore sensitivity of inferences. The missing at random (MAR) mechanism is a special case of the expanded model, thus allowing exploration of the sensitivity to inferences as departures from the inferences under the MAR assumption. The proposed approach is applied to data from the TRial Of Preventing HYpertension.
Clinical trials; Hypertension; Ignorability index; Missing data; Pattern-mixture model; TROPHY trial
We explore a Bayesian approach to selection of variables that represent fixed and random effects in modeling of longitudinal binary outcomes with missing data caused by dropouts. We show via analytic results for a simple example that nonignorable missing data lead to biased parameter estimates. This bias results in selection of wrong effects asymptotically, which we can confirm via simulations for more complex settings. By jointly modeling the longitudinal binary data with the dropout process that possibly leads to nonignorable missing data, we are able to correct the bias in estimation and selection. Mixture priors with a point mass at zero are used to facilitate variable selection. We illustrate the proposed approach using a clinical trial for acute ischemic stroke.
Bayesian variable selection; Bias; Dropout; Missing data; Model selection
This paper considers agency in the setting of embodied or active inference. In brief, we associate a sense of agency with prior beliefs about action and ask what sorts of beliefs underlie optimal behavior. In particular, we consider prior beliefs that action minimizes the Kullback–Leibler (KL) divergence between desired states and attainable states in the future. This allows one to formulate bounded rationality as approximate Bayesian inference that optimizes a free energy bound on model evidence. We show that constructs like expected utility, exploration bonuses, softmax choice rules and optimism bias emerge as natural consequences of this formulation. Previous accounts of active inference have focused on predictive coding and Bayesian filtering schemes for minimizing free energy. Here, we consider variational Bayes as an alternative scheme that provides formal constraints on the computational anatomy of inference and action—constraints that are remarkably consistent with neuroanatomy. Furthermore, this scheme contextualizes optimal decision theory and economic (utilitarian) formulations as pure inference problems. For example, expected utility theory emerges as a special case of free energy minimization, where the sensitivity or inverse temperature (of softmax functions and quantal response equilibria) has a unique and Bayes-optimal solution—that minimizes free energy. This sensitivity corresponds to the precision of beliefs about behavior, such that attainable goals are afforded a higher precision or confidence. In turn, this means that optimal behavior entails a representation of confidence about outcomes that are under an agent's control.
active inference; agency; Bayesian; bounded rationality; embodied cognition; free energy; inference; utility theory
Systematic reviewer authors intending to include all randomized participants in their meta-analyses need to make assumptions about the outcomes of participants with missing data.
The objective of this paper is to provide systematic reviewer authors with a relatively simple guidance for addressing dichotomous data for participants excluded from analyses of randomized trials.
This guide is based on a review of the Cochrane handbook and published methodological research. The guide deals with participants excluded from the analysis who were considered ‘non-adherent to the protocol’ but for whom data are available, and participants with missing data.
Systematic reviewer authors should include data from ‘non-adherent’ participants excluded from the primary study authors' analysis but for whom data are available. For missing, unavailable participant data, authors may conduct a complete case analysis (excluding those with missing data) as the primary analysis. Alternatively, they may conduct a primary analysis that makes plausible assumptions about the outcomes of participants with missing data. When the primary analysis suggests important benefit, sensitivity meta-analyses using relatively extreme assumptions that may vary in plausibility can inform the extent to which risk of bias impacts the confidence in the results of the primary analysis. The more plausible assumptions draw on the outcome event rates within the trial or in all trials included in the meta-analysis. The proposed guide does not take into account the uncertainty associated with assumed events.
This guide proposes methods for handling participants excluded from analyses of randomized trials. These methods can help in establishing the extent to which risk of bias impacts meta-analysis results.
Longitudinal studies with binary repeated measures are widespread in biomedical research. Marginal regression approaches for balanced binary data are well developed, while 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, 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 missing at random. Longitudinal viral load data from the HIV Epidemiology Research Study (HERS) are analyzed for illustration.
Repeated measures; Marginal model; Nonparametric regression; Penalized splines; HIV/AIDS; Antiviral treatment
Missing outcome data are very common in smoking cessation trials. It is often assumed that all such missing data are from participants who have been unsuccessful in giving up smoking (“missing=smoking”). Here we use data from a recent Internet based smoking cessation trial in order to investigate which of a set of a priori chosen baseline variables are predictive of missingness, and the evidence for and against the “missing=smoking” assumption.
We use a selection model, which models the probability that the outcome is observed given the outcome and other variables. The selection model includes a parameter for which zero indicates that the data are Missing at Random (MAR) and large values indicate “missing=smoking”. We examine the evidence for the predictive power of baseline variables in the context of a sensitivity analysis. We use data on the number and type of attempts made to obtain outcome data in order to estimate the association between smoking status and the missing data indicator.
We apply our methods to the iQuit smoking cessation trial data. From the sensitivity analysis, we obtain strong evidence that older participants are more likely to provide outcome data. The model for the number and type of attempts to obtain outcome data confirms that age is a good predictor of missing data. There is weak evidence from this model that participants who have successfully given up smoking are more likely to provide outcome data but this evidence does not support the “missing=smoking” assumption. The probability that participants with missing outcome data are not smoking at the end of the trial is estimated to be between 0.14 and 0.19.
Those conducting smoking cessation trials, and wishing to perform an analysis that assumes the data are MAR, should collect and incorporate baseline variables into their models that are thought to be good predictors of missing data in order to make this assumption more plausible. However they should also consider the possibility of Missing Not at Random (MNAR) models that make or allow for less extreme assumptions than “missing=smoking”.
There is accumulating evidence that prior knowledge about expectations plays an important role in perception. The Bayesian framework is the standard computational approach to explain how prior knowledge about the distribution of expected stimuli is incorporated with noisy observations in order to improve performance. However, it is unclear what information about the prior distribution is acquired by the perceptual system over short periods of time and how this information is utilized in the process of perceptual decision making. Here we address this question using a simple two-tone discrimination task. We find that the “contraction bias”, in which small magnitudes are overestimated and large magnitudes are underestimated, dominates the pattern of responses of human participants. This contraction bias is consistent with the Bayesian hypothesis in which the true prior information is available to the decision-maker. However, a trial-by-trial analysis of the pattern of responses reveals that the contribution of most recent trials to performance is overweighted compared with the predictions of a standard Bayesian model. Moreover, we study participants' performance in a-typical distributions of stimuli and demonstrate substantial deviations from the ideal Bayesian detector, suggesting that the brain utilizes a heuristic approximation of the Bayesian inference. We propose a biologically plausible model, in which decision in the two-tone discrimination task is based on a comparison between the second tone and an exponentially-decaying average of the first tone and past tones. We show that this model accounts for both the contraction bias and the deviations from the ideal Bayesian detector hypothesis. These findings demonstrate the power of Bayesian-like heuristics in the brain, as well as their limitations in their failure to fully adapt to novel environments.
In this paper we study how history affects perception using an auditory delayed comparison task, in which human participants repeatedly compare the frequencies of two, temporally-separated pure tones. We demonstrate that the history of the experiment has a substantial effect on participants' performance: when both tones are high relative to past stimuli, people tend to report that the 2nd tone was higher, and when they are relatively low, they tend to report that the 1st tone was higher. Interestingly, only the most recent trials bias performance, which can be interpreted as if the participants assume that the statistics of stimuli in the experiment is highly volatile. Moreover, this bias persists even in settings, in which it is detrimental to performance. These results demonstrate the abilities, as well as limitations, of the cognitive system when incorporating expectations in perception.
This article studies a general joint model for longitudinal measurements and competing risks survival data. The model consists of a linear mixed effects sub-model for the longitudinal outcome, a proportional cause-specific hazards frailty sub-model for the competing risks survival data, and a regression sub-model for the variance–covariance matrix of the multivariate latent random effects based on a modified Cholesky decomposition. The model provides a useful approach to adjust for non-ignorable missing data due to dropout for the longitudinal outcome, enables analysis of the survival outcome with informative censoring and intermittently measured time-dependent covariates, as well as joint analysis of the longitudinal and survival outcomes. Unlike previously studied joint models, our model allows for heterogeneous random covariance matrices. It also offers a framework to assess the homogeneous covariance assumption of existing joint models. A Bayesian MCMC procedure is developed for parameter estimation and inference. Its performances and frequentist properties are investigated using simulations. A real data example is used to illustrate the usefulness of the approach.
Cause-specific hazard; Bayesian analysis; Cholesky decomposition; Mixed effects model; MCMC; Modeling covariance matrices
Having substantial missing data is a common problem in administrative and cancer registry data. We propose a sensitivity analysis to evaluate the impact of a covariate that is potentially missing not at random in survival analyses using Weibull proportional hazards regressions. We apply the method to an investigation of the impact of missing grade on post-surgical mortality outcomes in individuals with metastatic kidney cancer. Data came from the Surveillance Epidemiology and End Results (SEER) registry which provides population based information on those undergoing cytoreductive nephrectomy. Tumor grade is an important component of risk stratification for patients with both localized and metastatic kidney cancer. Many individuals in SEER with metastatic kidney cancer are missing tumor grade information. We found that surgery was protective, but that the magnitude of the effect depended on assumptions about the relationship of grade with missingness.
When a randomized controlled trial has missing outcome data, any analysis is based on untestable assumptions, e.g. that the data are missing at random, or less commonly on other assumptions about the missing data mechanism. Given such assumptions, there is an extensive literature on suitable methods of analysis. However, little is known about what assumptions are appropriate. We use two sources of ancillary data to explore the missing data mechanism in a trial of adherence therapy in patients with schizophrenia: carer-reported (proxy) outcomes and the number of contact attempts. This requires additional assumptions to be made whose plausibility we discuss. Proxy outcomes are found to be unhelpful in this trial because they are insufficiently associated with patient outcome and because the ancillary assumptions are implausible. The number of attempts required to achieve a follow-up interview is helpful and suggests that these data are unlikely to depart far from being missing at random. We also perform sensitivity analyses to departures from missingness at random, based on the investigators’ prior beliefs elicited at the start of the trial. Wider use of techniques such as these will help to inform the choice of suitable assumptions for the analysis of randomized controlled trials.
Informatively missing; Missing data; Missingness not at random; Prior elicitation; Proxy data; Repeated attempts; Sensitivity analysis
To perform a Bayesian analysis of the Mycotic Ulcer Treatment Trial I (MUTT I) using expert opinion as a prior belief.
MUTT I was a randomized clinical trial comparing topical natamycin or voriconazole for treating filamentous fungal keratitis. A questionnaire elicited expert opinion on the best treatment of fungal keratitis before MUTT I results were available. A Bayesian analysis was performed using the questionnaire data as a prior belief and the MUTT I primary outcome (3-month visual acuity) by frequentist analysis as a likelihood.
Corneal experts had a 41.1% prior belief that natamycin improved 3-month visual acuity compared with voriconazole. The Bayesian analysis found a 98.4% belief for natamycin treatment compared with voriconazole treatment for filamentous cases as a group (mean improvement 1.1 Snellen lines, 95% credible interval 0.1–2.1). The Bayesian analysis estimated a smaller treatment effect than the MUTT I frequentist analysis result of 1.8-line improvement with natamycin versus voriconazole (95% confidence interval 0.5–3.0, P = 0.006). For Fusarium cases, the posterior demonstrated a 99.7% belief for natamycin treatment, whereas non-Fusarium cases had a 57.3% belief.
The Bayesian analysis suggests that natamycin is superior to voriconazole when filamentous cases are analyzed as a group. Subgroup analysis of Fusarium cases found improvement with natamycin compared with voriconazole, whereas there was almost no difference between treatments for non-Fusarium cases. These results were consistent with, though smaller in effect size than, the MUTT I primary outcome by frequentist analysis. The accordance between analyses further validates the trial results. (ClinicalTrials.gov number, NCT00996736.)
We elicited the opinions of corneal specialists on treating filamentous fungal keratitis, to perform a Bayesian analysis of the Mycotic Ulcer Treatment Trial I. The Bayesian analysis result was consistent with, but suggested a smaller treatment effect than, the frequentist result.
fungal keratitis; corneal ulceration; clinical trial; Bayesian; statistics
The vast amount of biological knowledge accumulated over the years has allowed researchers to identify various biochemical interactions and define different families of pathways. There is an increased interest in identifying pathways and pathway elements involved in particular biological processes. Drug discovery efforts, for example, are focused on identifying biomarkers as well as pathways related to a disease. We propose a Bayesian model that addresses this question by incorporating information on pathways and gene networks in the analysis of DNA microarray data. Such information is used to define pathway summaries, specify prior distributions, and structure the MCMC moves to fit the model. We illustrate the method with an application to gene expression data with censored survival outcomes. In addition to identifying markers that would have been missed otherwise and improving prediction accuracy, the integration of existing biological knowledge into the analysis provides a better understanding of underlying molecular processes.
Bayesian variable selection; gene expression; Markov chain Monte Carlo; Markov random field prior; pathway selection
Treatment noncompliance and missing outcomes at posttreatment assessments are common problems in field experiments in naturalistic settings. Although the two complications often occur simultaneously, statistical methods that address both complications have not been routinely considered in data analysis practice in the prevention research field. This paper shows that identification and estimation of causal treatment effects considering both noncompliance and missing outcomes can be relatively easily conducted under various missing data assumptions. We review a few assumptions on missing data in the presence of noncompliance, including the latent ignorability proposed by Frangakis and Rubin (Biometrika 86:365–379, 1999), and show how these assumptions can be used in the parametric complier average causal effect (CACE) estimation framework. As an easy way of sensitivity analysis, we propose the use of alternative missing data assumptions, which will provide a range of causal effect estimates. In this way, we are less likely to settle with a possibly biased causal effect estimate based on a single assumption. We demonstrate how alternative missing data assumptions affect identification of causal effects, focusing on the CACE. The data from the Johns Hopkins School Intervention Study (Ialongo et al., Am J Community Psychol 27:599–642, 1999) will be used as an example.
Causal inference; Complier average causal effect; Latent ignorability; Missing at random; Missing data; Noncompliance
We use the framework of coarsened data to motivate performing sensitivity analysis in the presence of incomplete data. To perform the sensitivity analysis, we specify pattern-mixture models to allow departures from the assumption of coarsening at random, a generalization of missing at random and independent censoring. We apply the concept of coarsening to address potential bias from missing data and interval-censored data in a randomized controlled trial of an herbal treatment for acute hepatitis. Computer code using SAS PROC NLMIXED for fitting the models is provided.
Coarsened data; Interval censoring; Missing data; Nonignorable missingness; Sensitivity analysis
Many randomized trials involve missing binary outcomes. Although many previous adjustments for missing binary outcomes have been proposed, none of these makes explicit use of randomization to bound the bias when the data are not missing at random.
We propose a novel approach that uses the randomization distribution to compute the anticipated maximum bias when missing at random does not hold due to an unobserved binary covariate (implying that missingness depends on outcome and treatment group). The anticipated maximum bias equals the product of two factors: (a) the anticipated maximum bias if there were complete confounding of the unobserved covariate with treatment group among subjects with an observed outcome and (b) an upper bound factor that depends only on the fraction missing in each randomization group. If less than 15% of subjects are missing in each group, the upper bound factor is less than .18.
We illustrated the methodology using data from the Polyp Prevention Trial. We anticipated a maximum bias under complete confounding of .25. With only 7% and 9% missing in each arm, the upper bound factor, after adjusting for age and sex, was .10. The anticipated maximum bias of .25 × .10 =.025 would not have affected the conclusion of no treatment effect.
This approach is easy to implement and is particularly informative when less than 15% of subjects are missing in each arm.
In a large, prospective longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women, instead of a single outcome variable, there are multiple binary outcomes (e.g., abnormal heart rate, abnormal blood pressure, abnormal heart wall thickness) considered as joint measures of heart function over time. In the presence of missing responses at some time points, longitudinal marginal models for these multiple outcomes can be estimated using generalized estimating equations (GEE) (Liang and Zeger, 1986), and consistent estimates can be obtained under the assumption of a missing completely at random (MCAR) mechanism. When the missing data mechanism is missing at random (MAR), that is the probability of missing a particular outcome at a time-point depends on observed values of that outcome and the remaining outcomes at other time points, we propose joint estimation of the marginal models using a single modified GEE based on an EM-type algorithm. The proposed method is motivated by the longitudinal study of cardiac abnormalities in children born to HIV-infected women and analyses of these data are presented to illustrate the application of the method. Further, in an asymptotic study of bias, we show that under an MAR mechanism in which missingness depends on all observed outcome variables, our joint estimation via the modified GEE produces almost unbiased estimates, provided the correlation model has been correctly specified, whereas estimates from standard GEE can lead to substantial bias.
EM-type algorithm; generalized estimating equations; missing at random; missing completely at random
This article proposes a joint model for longitudinal measurements and competing risks survival data. The model consists of a linear mixed effects sub-model with t-distributed measurement errors for the longitudinal outcome, a proportional cause-specific hazards frailty sub-model for the survival outcome, and a regression sub-model for the variance-covariance matrix of the multivariate latent random effects based on a modified Cholesky decomposition. A Bayesian MCMC procedure is developed for parameter estimation and inference. Our method is insensitive to outlying longitudinal measurements in the presence of non-ignorable missing data due to dropout. Moreover, by modeling the variance-covariance matrix of the latent random effects, our model provides a useful framework for handling high-dimensional heterogeneous random effects and testing the homogeneous random effects assumption which is otherwise untestable in commonly used joint models. Finally, our model enables analysis of a survival outcome with intermittently measured time-dependent covariates and possibly correlated competing risks and dependent censoring, as well as joint analysis of the longitudinal and survival outcomes. Illustrations are given using a real data set from a lung study and simulation.
Joint model; Competing risks; Bayesian analysis; Cholesky decomposition; Mixed effects model; MCMC; Modeling random effects covariance matrix; Outlier