Missing Covariate Research Articles

A comparison of different methods to handle missing data in the context of propensity score analysis

Propensity score analysis is a popular method to control for confounding in observational studies. A challenge in propensity methods is missing values in confounders. Several strategies for handling missing values exist, but guidance in choosing the best method is needed. In this simulation study, we compared four strategies of handling missing covariate values in propensity matching and propensity weighting. These methods include: complete case analysis, missing indicator method, multiple imputation and combining multiple imputation and missing indicator method. Concurrently, we aimed to provide guidance in choosing the optimal strategy. Simulated scenarios varied regarding missing mechanism, presence of effect modification or unmeasured confounding. Additionally, we demonstrated how missingness graphs help clarifying the missing structure. When no effect modification existed, complete case analysis yielded valid causal treatment effects even when data were missing not at random. In some situations, complete case analysis was also able to partially correct for unmeasured confounding. Multiple imputation worked well if the data were missing (completely) at random, and if the imputation model was correctly specified. In the presence of effect modification, more complex imputation models than default options of commonly used statistical software were required. Multiple imputation may fail when data are missing not at random. Here, combining multiple imputation and the missing indicator method reduced the bias as the missing indicator variable can be a proxy for unobserved confounding. The optimal way to handle missing values in covariates of propensity score models depends on the missing data structure and the presence of effect modification. When effect modification is present, default settings of imputation methods may yield biased results even if data are missing at random.

Parental non-alcoholic fatty liver disease increases risk of non-alcoholic fatty liver disease in offspring.

Little is known regarding the risk of hepatic steatosis (HS) among adult children of affected parents. We examined the association between parental and offspring HS in the multigenerational Framingham Heart Study, which characterized HS using computed tomography. We performed multivariable logistic regression models adjusted for age, sex, alcohol use, and body mass index to generate the odds of HS according to parental HS. We determined the proportion of participants with HS according to parental HS and the presence or absence of hypertension, diabetes, or obesity (BMI ≥30kg/m2 ). After excluding heavy alcohol use (n=126) and missing covariates (n=1), 785 offspring with at least one parent were included. Approximately 23% (183/785) had at least one parent with HS and 1.1% had two affected parents (9/785). In adjusted models, participants with at least one parent with HS had a nearly two-fold increased odds of HS compared to participants without a parental history of HS (OR 1.86, 95% confidence interval 1.15-3.03). Among participants without hypertension, diabetes, or obesity, a higher proportion had HS if they had a parental history of HS compared to those without (16.1% vs 5.2%, P<0.001). However, for participants with cardiometabolic risk factors, we did not observe a difference in HS among those with and without parental HS (30.3% vs 28.5%, P=0.78). Individuals with a parental history of HS are at increased risk for HS. Specifically, a parental history of HS may be an important factor among those that are otherwise metabolically healthy.

Propensity Score Estimation Using Classification and Regression Trees in the Presence of Missing Covariate Data

Abstract Data mining and machine learning techniques such as classification and regression trees (CART) represent a promising alternative to conventional logistic regression for propensity score estimation. Whereas incomplete data preclude the fitting of a logistic regression on all subjects, CART is appealing in part because some implementations allow for incomplete records to be incorporated in the tree fitting and provide propensity score estimates for all subjects. Based on theoretical considerations, we argue that the automatic handling of missing data by CART may however not be appropriate. Using a series of simulation experiments, we examined the performance of different approaches to handling missing covariate data; (i) applying the CART algorithm directly to the (partially) incomplete data, (ii) complete case analysis, and (iii) multiple imputation. Performance was assessed in terms of bias in estimating exposure-outcome effects among the exposed, standard error, mean squared error and coverage. Applying the CART algorithm directly to incomplete data resulted in bias, even in scenarios where data were missing completely at random. Overall, multiple imputation followed by CART resulted in the best performance. Our study showed that automatic handling of missing data in CART can cause serious bias and does not outperform multiple imputation as a means to account for missing data.

Estimation and test of restricted linear EV model with nonignorable missing covariates

This paper deals with estimation and test procedures for restricted linear errors-invariables (EV) models with nonignorable missing covariates. We develop a restricted weighted corrected least squares (WCLS) estimator based on the propensity score, which is fitted by an exponentially tilted likelihood method. The limiting distributions of the proposed estimators are discussed when tilted parameter is known or unknown. To test the validity of the constraints, we construct two test procedures based on corrected residual sum of squares and empirical likelihood method and derive their asymptotic properties. Numerical studies are conducted to examine the finite sample performance of our proposed methods.

Maternal mental health and childhood injury: evidence from the UK Millennium Cohort Study

ObjectiveWe assessed whether maternal mental health problems increased rates for child injury during the preschool years and mid-childhood, and the extent to which associations could be accounted for by a...

Foetal ultrasound measurement imputations based on growth curves versus multiple imputation chained equation (MICE)

Ultrasound measures are valuable for epidemiologic studies of risk factors for growth restriction. Longitudinal measurements enable investigation of rates of change and identification of windows where growth is impacted more acutely. However, missing data can be problematic in these studies, limiting sample size, ability to characterise windows of vulnerability, and in some instances creating bias. We sought to compare a parametric linear mixed model (LMM) approach to multiple imputation in this setting with multiple imputation by chained equation (MICE) methodology. Ultrasound scans performed for clinical purposes were abstracted from women in the LIFECODES birth cohort (n=1003) if they were close in time to three study visits (median 18, 26, and 35weeks' gestation). We created imputed datasets using LMM and MICE and calculated associations between demographic factors and ultrasound parameters cross-sectionally and longitudinally. Results were compared with a complete-case analysis. Most participants had ultrasounds at 18weeks' gestation, and ~50% had measurements at 26 and 35weeks; 100% had birthweight. Associations between demographic factors and ultrasound measures were similar in magnitude, but more precise, when either imputed datasets were used, compared with a complete-case analysis, in both the cross-sectional or longitudinal analyses. MICE, though ignoring the non-linear features of the trajectory and within subject correlation, is able to provide reasonable imputation of foetal growth data when compared to LMM. Because it simultaneously imputes missing covariate data and does not require specification of variance structure as in LMM, MICE may be preferable for imputation in this setting.

Multiple imputation in Cox regression when there are time-varying effects of covariates.

In Cox regression, it is important to test the proportional hazards assumption and sometimes of interest in itself to study time‐varying effects (TVEs) of covariates. TVEs can be investigated with log hazard ratios modelled as a function of time. Missing data on covariates are common and multiple imputation is a popular approach to handling this to avoid the potential bias and efficiency loss resulting from a “complete‐case” analysis. Two multiple imputation methods have been proposed for when the substantive model is a Cox proportional hazards regression: an approximate method (Imputing missing covariate values for the Cox model in Statistics in Medicine (2009) by White and Royston) and a substantive‐model‐compatible method (Multiple imputation of covariates by fully conditional specification: accommodating the substantive model in Statistical Methods in Medical Research (2015) by Bartlett et al). At present, neither accommodates TVEs of covariates. We extend them to do so for a general form for the TVEs and give specific details for TVEs modelled using restricted cubic splines. Simulation studies assess the performance of the methods under several underlying shapes for TVEs. Our proposed methods give approximately unbiased TVE estimates for binary covariates with missing data, but for continuous covariates, the substantive‐model‐compatible method performs better. The methods also give approximately correct type I errors in the test for proportional hazards when there is no TVE and gain power to detect TVEs relative to complete‐case analysis. Ignoring TVEs at the imputation stage results in biased TVE estimates, incorrect type I errors, and substantial loss of power in detecting TVEs. We also propose a multivariable TVE model selection algorithm. The methods are illustrated using data from the Rotterdam Breast Cancer Study. R code is provided.

Strategies for imputing missing covariates in accelerated failure time models.

Missing covariates often occur in biomedical studies with survival outcomes. Multiple imputation via chained equations (MICE) is a semi-parametric and flexible approach that imputes multivariate data by a series of conditional models, one for each incomplete variable. When applying MICE, practitioners tend to specify the conditional models in a simple fashion largely dictated by the software, which could lead to suboptimal results. Practical guidelines for specifying appropriate conditional models in MICE are lacking. Motivated by a study of time to hip fractures in the Women's Health Initiative Observational Study using accelerated failure time models, we propose and experiment with some rationales leading to appropriate MICE specifications. This strategy starts with specifying a joint model for the variables involved. We first derive the conditional distribution of each variable under the joint model, then approximate these conditional distributions to the extent which can be characterized by commonly used regression models. We propose to fit separate models to impute incomplete variables by the failure status, which is key to generating appropriate MICE specifications for survival outcomes. The proposed strategy can be conveniently implemented with all available imputation software that uses fully conditional specifications. Our simulation results show that some commonly used simple MICE specifications can produce suboptimal results, while those based on the proposed strategy appear to perform well and be robust toward model misspecifications. Hence, we warn against a mechanical use of MICE and suggest careful modeling of the conditional distributions of variables to ensure proper performance.

Longitudinal associations between psychotic experiences and disordered eating behaviours in adolescence: a UK population-based study

SummaryBackgroundPsychotic experiences might represent non-specific markers of poor mental health in adolescence. However, only a few predominantly cross-sectional studies have tested their association with disordered eating behaviours in adolescent and adult populations. The aim of this study was to explore the association between psychotic experiences at age 13 years, and disordered eating behaviours and body-mass index (BMI) at age 18 years.MethodsWe used data from the Avon Longitudinal Study of Parents and Children, a longitudinal birth cohort based in Avon (England, UK) including mothers with an expected delivery date between April 1, 1991, and Dec 31, 1992, and their children. Psychotic experiences (such as delusions and hallucinations) and BMI were measured at clinical assessments when children were nearly aged 13 years, and data on disordered eating behaviours (ie, presence of binge eating, purging, fasting, or excessive exercise for weight loss; any of these behaviours [included to increase statistical power]; and number of behaviours [included to investigate severity]) were obtained via a postal questionnaire that used adapted questions from the Youth Risk Behaviour Surveillance System questionnaire at approximately age 18 years. For each outcome, we ran a univariable model and four multivariable models (logistic, linear [for BMI], or negative binomial [for the number of behaviours] regression), progressively adjusting for child and maternal sociodemographic, physical, and mental health characteristics (including child's sex, and maternal age, marital status, and highest academic qualification); autistic traits at age 7 years (measured with the Social and Communication Disorder Checklist); baseline BMI at age 13 years, and depressive symptoms at baseline (ie, at age 13 years when psychotic experiences were measured: childs' symptoms measured with the Moods and feelings Questionnaire, and maternal symptoms measured at 32 weeks' gestation with the Edinburgh Postnatal Depression Scale). We imputed missing outcome and covariate data.FindingsOur sample included 6361 children, of whom 734 (12%) reported psychotic experiences at age 13 years. In univariable models, psychotic experiences were associated with greater odds of reporting any disordered eating behaviours (odds ratio [OR] 1·92, 95% CI 1·46–2·52; p<0·0001), and more severe symptoms (as measured by the number of disordered eating behaviours: 0·58, 0·32–0·84; p<0·0001) at age 18 years. These associations were slightly attenuated by adjustment for maternal and child characteristics (any disordered eating behaviours OR 1·82, 95% CI 1·35–2·44, p<0·0001; number of disordered eating behaviours 0·49, 95% CI 0·23–0·75, p<0·00001), autistic traits at age 7 years (any disordered eating behaviours OR 1·80, 95% CI 1·34–2·41, p<0·0001; number of disordered eating behaviours 0·48, 95% CI 0·22–0·74, p<0·00001), and BMI (any disordered-eating behaviours OR 1·83, 95% CI 1·36–2·46, p<0·0001; number of disordered-eating behaviours 0·32, 95% CI 0·06–0·57, p<0·00001) Adjusting for baseline depressive symptoms attenuated, but not removed, these associations (any disordered eating OR 1·50, 95% CI 1·10–2·03, p=0·010; more severe symptoms 0·32, 0·06–0·57, p=0·017). Psychotic experiences were also associated with greater binge eating, purging, and fasting behaviours, although some associations weakened after controlling for depressive symptoms. We noted no associations between psychotic experiences and excessive exercise or BMI in any of the models.InterpretationOur findings suggested that psychotic experiences are markers of increased risk for several disordered eating behaviours in late adolescence, possibly by indicating more severe psychopathology in early adolescence. More research investigating shared risk factors for psychotic experiences and eating disorders is warranted to elucidate shared and specific causal pathways.FundingWellcome Trust, the Royal Society, University College London Hospitals National Institute for Health Research Biomedical Research Centre, UK Medical Research Council, and the University of Bristol.

Plug-in marginal estimation under a general regression model with missing responses and covariates

In this paper, we consider a general regression model where missing data occur in the response and in the covariates. Our aim is to estimate the marginal distribution function and a marginal functional, such as the mean, the median or any α -quantile of the response variable. A missing at random condition is assumed in order to prevent from bias in the estimation of the marginal measures under a non-ignorable missing mechanism. We give two different approaches for the estimation of the responses distribution function and of a given marginal functional, involving inverse probability weighting and the convolution of the distribution function of the observed residuals and that of the observed estimated regression function. Through a Monte Carlo study and two real data sets, we illustrate the behaviour of our proposals.

Multiple imputation of missing data in nested case-control and case-cohort studies.

SummaryThe nested case-control and case-cohort designs are two main approaches for carrying out a substudy within a prospective cohort. This article adapts multiple imputation (MI) methods for handling missing covariates in full-cohort studies for nested case-control and case-cohort studies. We consider data missing by design and data missing by chance. MI analyses that make use of full-cohort data and MI analyses based on substudy data only are described, alongside an intermediate approach in which the imputation uses full-cohort data but the analysis uses only the substudy. We describe adaptations to two imputation methods: the approximate method (MI-approx) of White and Royston (2009) and the “substantive model compatible” (MI-SMC) method of Bartlett et al. (2015). We also apply the “MI matched set” approach of Seaman and Keogh (2015) to nested case-control studies, which does not require any full-cohort information. The methods are investigated using simulation studies and all perform well when their assumptions hold. Substantial gains in efficiency can be made by imputing data missing by design using the full-cohort approach or by imputing data missing by chance in analyses using the substudy only. The intermediate approach brings greater gains in efficiency relative to the substudy approach and is more robust to imputation model misspecification than the full-cohort approach. The methods are illustrated using the ARIC Study cohort. Supplementary Materials provide R and Stata code.

A semiparametric imputation approach for regression with censored covariate with application to an AMD progression study.

This research is motivated by studying the progression of age-related macular degeneration where both a covariate and the response variable are subject to censoring. We develop a general framework to handle regression with censored covariate where the response can be different types and the censoring can be random or subject to (constant) detection limits. Multiple imputation is a popular technique to handle missing data that requires compatibility between the imputation model and the substantive model to obtain valid estimates. With censored covariate, we propose a novel multiple imputation-based approach, namely, the semiparametric two-step importance sampling imputation (STISI) method, to impute the censored covariate. Specifically, STISI imputes the missing covariate from a semiparametric accelerated failure time model conditional on fully observed covariates (Step 1) with the acceptance probability derived from the substantive model (Step 2). The 2-step procedure automatically ensures compatibility and takes full advantage of the relaxed semiparametric assumption in the imputation. Extensive simulations demonstrate that the STISI method yields valid estimates in all scenarios and outperforms some existing methods that are commonly used in practice. We apply STISI on data from the Age-related Eye Disease Study, to investigate the association between the progression time of the less severe eye and that of the more severe eye. We also illustrate the method by analyzing the urine arsenic data for patients from National Health and Nutrition Examination Survey (2003-2004) where the response is binary and 1 covariate is subject to detection limit.

Cox model with interval-censored covariate in cohort studies.

In cohort studies the outcome is often time to a particular event, and subjects are followed at regular intervals. Periodic visits may also monitor a secondary irreversible event influencing the event of primary interest, and a significant proportion of subjects develop the secondary event over the period of follow-up. The status of the secondary event serves as a time-varying covariate, but is recorded only at the times of the scheduled visits, generating incomplete time-varying covariates. While information on a typical time-varying covariate is missing for entire follow-up period except the visiting times, the status of the secondary event are unavailable only between visits where the status has changed, thus interval-censored. One may view interval-censored covariate of the secondary event status as missing time-varying covariates, yet missingness is partial since partial information is provided throughout the follow-up period. Current practice of using the latest observed status produces biased estimators, and the existing missing covariate techniques cannot accommodate the special feature of missingness due to interval censoring. To handle interval-censored covariates in the Cox proportional hazards model, we propose an available-data estimator, a doubly robust-type estimator as well as the maximum likelihood estimator via EM algorithm and present their asymptotic properties. We also present practical approaches that are valid. We demonstrate the proposed methods using our motivating example from the Northern Manhattan Study.

A fast imputation algorithm in quantile regression.

In many applications, some covariates could be missing for various reasons. Regression quantiles could be either biased or under-powered when ignoring the missing data. Multiple imputation and EM-based augment approach have been proposed to fully utilize the data with missing covariates for quantile regression. Both methods however are computationally expensive. We propose a fast imputation algorithm (FI) to handle the missing covariates in quantile regression, which is an extension of the fractional imputation in likelihood based regressions. FI and modified imputation algorithms (FIIPW and MIIPW) are compared to existing MI and IPW approaches in the simulation studies, and applied to part of of the National Collaborative Perinatal Project study.

Using the Latent Correlations to Measure the Association Between Continuous and Binary Markers With Repeated Measurements

ABSTRACTThis study was motivated by one of the important challenges in sleep research: estimating not only an overall association between objective and subjective sleep measures but at within- and between-subject levels as well. Comparing to the literature on the association between two continuous or two binary outcomes, to our knowledge, fewer works were published for the case when one outcome is continuous and the other is binary while both are measured repeatedly. We address this problem by considering a latent correlation (LC) that is the correlation between two continuous outcomes. The first is the original observed continuous outcome while the second is an underlying latent variable, dichotomization of which produces the original binary outcome. Building upon a flexible Bayesian generalized linear mixed model fitted by the MCMC, our proposal allows estimation of the total correlation, and the between- and within-subject correlations simultaneously. The goal of joint estimation of all these correlations, to our knowledge, was not fulfilled in previous proposals (either the between- or within-subject correlation needed to be fixed at zero). Moreover, the approach avoids bias of a two-step procedure and does not rely on approximation of the covariance matrix and hence the correlations. It provides accurate estimates covering a wide range of values. We can also incorporate the cases of missing data, covariate adjustment, and comparison between subgroups. Furthermore, these advantages are achieved at a low computational cost. We demonstrate properties and implementation of our proposal with a large simulation study and with real data from two clinical trials of an insomnia drug where one of the questions is how continuous biomarkers derived from the polysomnographic measurements are associated with the binary patient-reported sleep quality. Supplementary materials for this article are available online.

Subgroups from regression trees with adjustment for prognostic effects and postselection inference.

Identification of subgroups with differential treatment effects in randomized trials is attracting much attention. Many methods use regression tree algorithms. This article addresses 2 important questions arising from the subgroups: how to ensure that treatment effects in subgroups are not confounded with effects of prognostic variables and how to determine the statistical significance of treatment effects in the subgroups. We address the first question by selectively including linear prognostic effects in the subgroups in a regression tree model. The second question is more difficult because it falls within the subject of postselection inference. We use a bootstrap technique to calibrate normal-theory t intervals so that their expected coverage probability, averaged over all the subgroups in a fitted model, approximates the desired confidence level. It can also provide simultaneous confidence intervals for all subgroups. The first solution is implemented in the GUIDE algorithm and is applicable to data with missing covariate values, 2 or more treatment arms, and outcomes subject to right censoring. Bootstrap calibration is applicable to any subgroup identification method; it is not restricted to regression tree models. Two real examples are used for illustration: a diabetes trial where the outcomes are completely observed but some covariate values are missing and a breast cancer trial where the outcome is right censored.

Dual diagnosis of mental illness and substance use disorder and injury in adults recently released from prison: a prospective cohort study

People with mental illness and substance use disorder are over-represented in prisons. Injury-related mortality is elevated in people released from prison, and both mental illness and substance use disorder are risk factors for injury. Effective care coordination during the transition between criminal justice and community service providers improves health outcomes for people released from prison. However, the health outcomes and support needs of people with dual diagnosis (co-occurring mental illness and substance use disorder) released from prison are poorly understood. Here we aim to examine the association between dual diagnosis and non-fatal injury in adults released from prison. Pre-release interview data collected between Aug 1, 2008, and July 31, 2010, from a representative sample of sentenced adults (≥18 years) in Queensland, Australia, were linked, retrospectively and prospectively, to person-level, state-wide emergency department and hospital records. We identified dual diagnoses from inpatient, emergency department, and prison medical records. We modelled the association between mental health status and all injury resulting in hospital contact by fitting a multivariate Cox regression, adjusting for sociodemographic, health, and criminogenic covariates, and replacing missing covariate data by multiple imputation. In 1307 adults released from prison, there were 2056 person-years of follow-up (median 495 days, IQR 163-958). The crude injury rates were 996 (95% CI 893-1112) per 1000 person-years for the dual diagnosis group, 538 (441-657) per 1000 person-years for the mental illness only group, 413 (354-482) per 1000 person-years for the substance use disorder only group, and 275 (247-307) per 1000 person-years for the no mental disorder group. After adjusting for model covariates, the dual diagnosis (adjusted hazard rate ratio 3·27, 95% CI 2·30-4·64; p<0·0001) and mental illness only (1·87, 1·19-2·95; p=0·0071) groups were at increased risk of injury after release from prison compared with the group with no mental health disorders. People released from prison experience high rates of injury compared with the general population. Among people released from prison, dual diagnosis is associated with an increased risk of injury. Contact with the criminal justice system is a key opportunity to prevent subsequent injury morbidity in people with co-occurring mental health disorders. Engagement with integrated psychiatric and addiction treatment delivered without interruption during the transition from prison into the community might prevent the injury-related disparities experienced by this vulnerable group. The development of targeted injury prevention strategies for people with dual diagnosis released from prison is warranted. National Health and Medical Research Council.

Spatial regression with an informatively missing covariate: Application to mapping fine particulate matter

The United States Environmental Protection Agency has established a large network of stations to monitor fine particulate matter of &lt;2.5 µm (PM2.5) that is known to be harmful to human health. Unfortunately, the network has limited spatial coverage, and stations often only measure PM2.5 every few days. Satellite‐measured aerosol optical depth (AOD) is a low‐cost surrogate with greater spatiotemporal coverage, and spatial regression models have established that including AOD as a covariate improves the spatial interpolation of PM2.5. However, AOD is often missing, and our analysis reveals that the conditions that lead to missing AOD are also conducive to high AOD. Therefore, naïve interpolation that ignores informative missingness may lead to bias. We propose a joint hierarchical model for PM2.5 and AOD that accounts for informatively missing AOD. We conduct a simulation study of the effects of ignoring informative missingness in the covariate and evaluate the performance of the proposed model. We apply the method to map daily PM2.5 in the Southeastern United States. Our analysis reveals statistically significant informative missingness and relationships between PM2.5 and AOD in many seasons after accounting for meteorological and land‐use variables.

EM algorithms for fitting multistate cure models.

Multistate cure models are multistate models in which transitions into one or more of the states cannot occur for a fraction of the population. In the study of cancer, multistate cure models can be used to identify factors related to the rate of cancer recurrence, the rate of death before and after recurrence, and the probability of being cured by initial treatment. However, the previous method for fitting multistate cure models requires substantial custom programming, making these valuable models less accessible to analysts. In this article, we present an Expectation-Maximization (EM) algorithm for fitting the multistate cure model using maximum likelihood. The proposed algorithm makes use of a weighted likelihood representation allowing it to be easily implemented with standard software and can incorporate either parametric or non-parametric baseline hazards for the state transition rates. A common complicating feature in cancer studies is that the follow-up times for recurrence and death may differ. Additionally, we may have missingness in the covariates. We propose a Monte Carlo EM (MCEM) algorithm for fitting the multistate cure model in the presence of covariate missingness and/or unequal follow-up of the two outcomes, we describe a novel approach for obtaining standard errors, and we provide some software. Simulations demonstrate good algorithmic performance as long as the modeling assumptions are sufficiently restrictive. We apply the proposed algorithm to a study of recurrence and death in patients with head and neck cancer.

Analysis of ordinal longitudinal data under nonignorable missingness and misreporting: An application to Alzheimer’s disease study

In many epidemiological and clinical studies, observations on individuals are recorded longitudinally on a Likert-type scale. In the process of recording, or due to some other causes, a proportion of outcomes and time-dependent covariates may be missing in one or more follow-up visits (non monotone missing). Even when the number of patients with intermittent missing data is small, exclusion of those patients from the study seems unsatisfactory. This apart, often due to misreporting, miscategorization of response can occur that results in potentially invalid inference when no correction is made. We propose a joint mixed model that corrects the likelihood function to account for missing response and/or covariates and adjusts the likelihood to tackle miscategorization of response. Under this extreme complex but useful setup, we seek to estimate the parameters of the proposed model that accounts for baseline and/or time dependent covariates. Monte Carlo expectation–maximization (MCEM) is a convenient approach for estimating the parameters in the model. A simulation study was carried out to assess the approach. We also analyzed Alzheimer’s Disease Neuroimaging Initiative (ADNI) data where some responses and covariates are missing and some responses are possibly miscategorized. Our investigation reveals that apolipo-protein plays a significant role in Alzheimer’s disease progression. This was not visible in earlier analyses of ADNI data.