Nonrandom Dropout Research Articles

Male-female differences in the natural history of ascending thoracic aortic aneurysms: insights from the dissexion study

Abstract Background The natural history of ascending thoracic aortic aneurysms (ATAAs) remains poorly understood, including potential male-female differences. Purpose Therefore, this study sought to evaluate the natural history and outcomes of patients with ATAAs, with an emphasis on aortic growth and male-female differences. Methods Adult patients who received diagnostic and/or follow-up imaging of an ATAA (≥40mm) in two academic centers between 2007-2021 were included. Aortic growth was modeled non-linearly through mixed-effects models using repeated measurements, including age and sex through interaction terms. Joint models were constructed to adjust for the effect of nonrandom dropout due to surgical intervention. Secondary outcomes were male-female differences in presentation, aneurysm characteristics, time to intervention, acute aortic syndromes and survival relative to the age- and sex-matched general Dutch population. Cox models were employed to evaluate predictors of intervention and survival. Results In total, 1,858 patients were included (31.6% female). Median clinical follow-up was 5.9 [IQR: 3.0-10.0] years. Median age was 65.4 years [53.4-71.7y] for females and 59.0 years [IQR: 49.3-68.0y] for males (p<0.001). Males had more bicuspid aortic valve (BAV) disease (male: 27.5%, female: 18.2%; p<0.001) and were more likely to have had a previous dissection or aneurysm in a major artery other than the thoracic aorta (male: 8.9%, female: 4.9%; p=0.004) before diagnosis. Maximum absolute aortic diameter at diagnosis did not differ between males (46.5mm) and females (45.0mm; p=0.388). Unadjusted 10-year survival was 76% (68%-78%) for females (relative survival: 68.8%) and 78% (75%-81%) for males (relative survival: 78.7%). For females, tubular ascending aortic growth was faster compared to males (p=0.005), particularly in the first 6 years after diagnosis. Contrarily, SoV growth was faster in males compared to females (p=0.002). Surgical intervention occurred more quickly after diagnosis in females (HR: 2.5 (1.2-5.2), corrected for BAV, diameter and age. Twenty-three type A dissections (TAD) occurred with an incidence rate ratio between females and males of 3.4 (95%CI:1.5-8.0, p=0.004). The last known aortic diameter before TAD was comparable between males (47.5mm [47.0-50.8]) and females (47.0mm [43.0-51.0]; p=0.416). Conclusions Ascending aortic aneurysms show a strikingly different growth pattern between males and females at the level of the SoV and tubular ascending aorta, suggesting that although females present at an older age, their aortic disease progresses faster and with more TAD compared to males, and warranting an even more careful follow-up. Absolute known aortic diameter before TAD was comparable between males and females. These insights can be used to fuel discussion about male-female specific follow-up and treatment recommendations in future clinical guidelines development.Ascending aortic growth (A) & sinus (B)Freedom from intervention from diagnosis

#3434 A FREE OPEN-SOURCE FRAMEWORK TO INFER PATIENT-LEVEL EGFR SLOPES FOR ASSOCIATION WITH MOLECULAR DATA

Abstract Background and Aims Unfolding over years or decades, patient kidney disease trajectories can be characterised using routinely collected longitudinal data. One frequently measured marker of kidney function is serum creatinine, used in clinical settings to estimate the glomerular filtration rate (eGFR). Change in eGFR over time has been put forward as a proxy endpoint for clinical trials, and most work on inferring eGFR slopes has been done in this context. However, 'omics-based studies of the molecular mechanisms of kidney disease have so far largely neglected to make use of longitudinal data linked to tissue samples. Here, we build on existing approaches [1, 2] and provide an open-source, R framework to infer eGFR slopes from routinely available clinical data linked to biobank samples. Method In our framework, we calculate eGFR from serum creatinine measurements using the CKD-EPI 2021 formula. To aid data adjudication, we implement criteria outlined by Hapca et al. [1] for automatic flagging of acute kidney injury (AKI) events, and extend the criteria to detect potential in-patient stays. Since the biopsy procedure may arise as a result of loss of kidney function and/or coincide with a change in treatment regime, we use a two-spline linear mixed effects model similar to that proposed by Vonesh et al. [2] to differentiate between immediate-post-biopsy (acute) and chronic eGFR slopes. Additionally, to account for non-random dropout due to the onset of dialysis or occurrence of kidney transplant, we jointly model the longitudinal and dropout processes. Our framework is made available as free, open-source software in R. Results From routinely collected longitudinal clinical data, we generate patient-specific values characterising the post-biopsy disease trajectory. These values can then be incorporated into the analysis of 'omics data to find molecular patterns associated with treatment response or prognosis. Conclusion We have developed an open-source R framework for inferring patient-level eGFR slopes from serum creatinine measurements linked to biobank samples. Leveraging this data could improve 'omics studies of the molecular mechanisms of kidney disease, allowing us to incorporate information about disease trajectory and treatment response and thus ask questions of the molecular data which have real clinical relevance.

FocusedDropout for Convolutional Neural Network

In a convolutional neural network (CNN), dropout cannot work well because dropped information is not entirely obscured in convolutional layers where features are correlated spatially. Except for randomly discarding regions or channels, many approaches try to overcome this defect by dropping influential units. In this paper, we propose a non-random dropout method named FocusedDropout, aiming to make the network focus more on the target. In FocusedDropout, we use a simple but effective method to search for the target-related features, retain these features and discard others, which is contrary to the existing methods. We find that this novel method can improve network performance by making the network more target focused. Additionally, increasing the weight decay while using FocusedDropout can avoid overfitting and increase accuracy. Experimental results show that with a slight cost, 10% of batches employing FocusedDropout, can produce a nice performance boost over the baselines on multiple datasets of classification, including CIFAR10, CIFAR100 and Tiny ImageNet, and has a good versatility for different CNN models.

Shared parameter and copula models for analysis of semicontinuous longitudinal data with nonrandom dropout and informative censoring.

Analysis of longitudinal semicontinuous data characterized by subjects' attrition triggered by nonrandom dropout is complex and requires accounting for the within-subject correlation, and modeling of the dropout process. While methods that address the within-subject correlation and missing data are available, approaches that incorporate the nonrandom dropout, also referred to informative right censoring, in the modeling step are scarce due to the computational intensity and possible intractable integration needed for its implementation. Appreciating the complexity of this problem and the need for a new methodology that is feasible for implementation, we propose to extend a framework of likelihood-based marginalized two-part models to account for informative right censoring. The censoring process is modeled using two approaches: (1) Poisson censoring for the count of visits before dropout and (2) survival time to dropout. Novel consideration was given to the proposed joint modeling approaches for the semicontinuous and censoring components of the likelihood function which included (1) shared parameter, and (2) Clayton copula. The cross-part and within-part correlations were accounted for through a complex random effect structure that models correlated random intercepts and slopes. Feasibility of implementation, and accuracy of these approaches were investigated using extensive simulation studies and clinical application.

Linear Mixed Models for Longitudinal Data with Nonrandom Dropouts

Longitudinal studies represent one of the principal research strategies employed in medical and social research. These studies are the most appropriate for studying individual change over time. The prematurely withdrawal of some subjects from the study (dropout) is termed nonrandom when the probability of missingness depends on the missing value. Nonrandom dropout is common phenomenon associated with longitudinal data and it complicates statistical inference. Linear mixed effects model is used to fit longitudinal data in the presence of nonrandom dropout. The stochastic EM algorithm is developed to obtain the model parameter estimates. Also, parameter estimates of the dropout model have been obtained. Standard errors of estimates have been calculated using the developed Monte Carlo method. All these methods are applied to two data sets.

Cognitive reserve in amyotrophic lateral sclerosis (ALS): a population-based longitudinal study

BackgroundAmyotrophic lateral sclerosis (ALS) is often associated with cognitive and/or behavioural impairment. Cognitive reserve (CR) may play a protective role in offsetting cognitive impairment. This study examined the relationship between...

Investigating walnut consumption and cognitive trajectories in a representative sample of older US adults.

Existing research suggests walnut intake may be associated with better cognitive function in older adults, yet few studies utilise longitudinal data from observational studies of ageing populations. Our objective was to estimate the association between whole walnut intake and cognitive change in a representative sample of older Americans. Secondary analysis of the Health and Retirement Study and Health Care and Nutrition Study. Walnut consumption was defined as a categorical measure (none, low intake (0·01-0·08 1 oz. servings per day) and moderate intake (>0·08 1 oz. servings per day)) and cognitive function was measured using the Telephone Interview for Cognitive Status. Latent growth modelling estimated the association between walnut consumption and trajectories of cognitive status over a 4-year observational period. Sensitivity analyses assessing non-random dropout and Monte Carlo power analyses were conducted to contextualise results. The USA. A sample of 3632 US adults aged 65 years and older. Those reporting any walnut consumption had greater cognitive scores at baseline than those not consuming walnuts (low walnut consumption, b = 1·53, se = 0·21, P < 0·001; moderate walnut consumption, b = 2·22, se = 0·27, P < 0·001), but walnut consumption was not associated with cognitive change. Walnut consumption was positively associated with socioeconomic status and health behaviours as well as intake of nutrients identified to have neuroprotective benefits. We identified an association between walnut consumption and cognitive function in older adults, although we did not find that walnut consumption was protective against age-related cognitive decline.

Impact of Smoking in Response to Tumor Necrosis Factor Inhibitors in Axial Spondyloarthritis: Methodologic Considerations for Longitudinal Observational Studies

Observational data facilitate examination of treatment-effect heterogeneity, but the risk of bias is substantial. The present study was undertaken to highlight methodologic considerations through an analysis of whether smoking affects response to tumor necrosis factor inhibitors (TNFi) in axial spondyloarthritis (SpA). We used longitudinal data from the British Society for Rheumatology Biologics Register for Ankylosing Spondylitis. Participants fulfilling the Assessment of SpondyloArthritis international Society criteria for axial SpA who started their first TNFi were eligible for analysis. In comparing the impact of smoking status, weighted generalized estimating equations were used to examine changes in several continuous outcome measures, including the Bath Ankylosing Spondylitis Disease Activity Index (BASDAI) and the Ankylosing Spondylitis Disease Activity Score (ASDAS). Inverse probability weights were used to account for differences in baseline covariates and excluded participants. We separately assessed response in the first 3 months to account for nonrandom dropout. For 840 participants who started on TNFi, 1,641 assessments from 627 individuals were analyzed (69% male, mean age 46 years). A total of 33% were current smokers and 30% ex-smokers. Ex-smokers and current smokers had worse disease than never smokers at baseline. Accounting for these differences, response did not differ according to smoking status. Compared to never smokers, ex-smokers (β = -0.6, 95% confidence interval [95% CI] -1.4, 0.3) and current smokers (β = -0.4, 95% CI -1.1, 0.4) had a similar response according to the BASDAI and ASDAS (ex-smokers β = -0.1, 95% CI -0.5, 0.3; current smokers β = -0.01, 95% CI -0.4, 0.4) at 3 months. TNFi response did not differ according to baseline smoking status in this UK cohort. Conflicting results from previous studies were likely due to methodologic differences. This analysis highlights potential sources of bias that should be addressed in future studies.

Trajectories of posttraumatic stress disorders among adolescents in the area worst-hit by the Wenchuan earthquake

BackgroundThis study examined the trajectories of posttraumatic stress disorder (PTSD) among adolescents in the area worst-hit by the Wenchuan earthquake, and assessed the role of trauma exposure in differentiating distinct PTSD trajectories. MethodsIn total, 391 adolescents were selected from the Wenchuan and Maoxian counties in Sichuan province (the area worst-hit by the earthquake). Participants completed a trauma exposure questionnaire at 1 year, and the Child PTSD Symptom Scale at 1 year, 1.5 years, 2 years, and 2.5 years after the earthquake. ResultsThree latent PTSD trajectories were found in adolescents: moderate-stable (81.6%), decreasing (8.7%), and increasing trajectories (9.7%). T1 trauma exposure was more likely to be related to the decreasing trajectory group. LimitationsAll variables were measured by self-report scales. There was high non-random dropout. ConclusionThe course of PTSD was heterogeneous among adolescents following the same natural disaster.

On an enhanced rank-preserving structural failure time model to handle treatment switch, crossover, and dropout.

It is very challenging to estimate the comparative treatment effect between a treatment therapy and a control therapy on overall survival in the presence of treatment crossover, switch to an alternative non-study therapy, and non-random patient dropout. Existing methods (e.g., intent-to-treat and per-protocol) are known to be biased. We proposed two new estimators to address these analytical challenges and evaluated their performance via a comprehensive simulation study. The new estimators were constructed by combining an enhanced rank-preserving structural failure time model and the inverse probability censoring weighting approach. In the simulation study, we assessed and compared the performance of the two new estimators with four estimators from existing methods. The simulation results show that the new estimators have much better performance in almost all considered settings compared with the existing estimators. Copyright © 2017 John Wiley & Sons, Ltd.

Maximum Likelihood Approach for Longitudinal Models with Nonignorable Missing Data Mechanism Using Fractional Imputation

In longitudinal studies data are collected for the same set of units for two or more occasions. This is in contrast to cross-sectional studies where a single outcome is measured for each individual. Some intended measurements might not be available for some units resulting in a missing data setting. When the probability of missing depends on the missing values, missing mechanism is termed nonrandom. One common type of the missing patterns is the dropout where the missing values never followed by an observed value. In nonrandom dropout, missing data mechanism must be included in the analysis to get unbiased estimates. The parametric fractional imputation method is proposed to handle the missingness problem in longitudinal studies and to get unbiased estimates in the presence of nonrandom dropout mechanism. Also, in this setting the jackknife replication method is used to find the standard errors for the fractionally imputed estimates. Finally, the proposed method is applied to a real data (mastitis data) in addition to a simulation study.

An Analysis of Selection Models for Incomplete Longitudinal Clinical Trials Due to Dropout: An Application to Multi-centre Trial Data

A common problem encountered in statistical analysis is that of missing data, which occurs when some variables have missing values in some units. The present paper deals with the analysis of longitudinal continuous measurements with incomplete data due to non-ignorable dropout. In repeated measurements data, as one solution to a problem, the selection model assumes a mechanism of outcome-dependent dropout and jointly both the measurement together with dropout process of repeated measures. We consider the construction of a particular type of selection model that uses a logistic regression model to describe the dependency of dropout indicators on the longitudinal measurement. We focus on the use of the Diggle-Kenward model as a tool for assessing the sensitivity of a selection model in terms of the modeling assumptions. Our main objective here is to investigate the influence on inference that might be exerted on the considered data by the dropout process. We restrict attention to a model for repeated Gaussian measures, subject to potentially non-random dropout. To investigate this, we carry out an application for analyzing incomplete longitudinal clinical trial with dropout by using a practical example in the form of a multi-centre clinical trial data.

The effect of events between waves on panel attrition

Panel surveys suffer from attrition. Most panel studies use propensity models or weighting class approaches to correct for non-random dropout. These models draw on variables measured in a previous wave or from paradata of the study. While it is plausible that they affect contactability and cooperativeness, panel studies usually cannot assess the impact of events between waves on attrition. The amount of change in the population could be seriously underestimated if such events had an effect on participation in subsequent waves. The panel study PASS is a novel dataset for labour market and poverty research. In PASS, survey data on (un)employment histories, income and education of participants are linked to corresponding data from respondents' administrative records. Thus, change can be observed for attritors as well as for continued participants. These data are used to show that change in household composition, employment status or receipt of benefits has an influence on contact and cooperation rates in the following wave. A large part of the effect is due to lower contactability of households who moved. Nevertheless, this effect can lead to biased estimates for the amount of change. After applying the survey’s longitudinal weights this bias is reduced, but not entirely eliminated.

A Bayesian approach for sensitivity analysis of incomplete multivariate longitudinal data with potential nonrandom dropout

Experiments involving repeated observations of multivariate outcomes are common in biomedical and public health researches which lead to multivariate longitudinal data. These kinds of data have a unique property in the sense that they allow the researcher to study the joint evolution of the multiple outcomes over the time. Recently, there has been a considerable amount of interest on using Bayesian modelling of longitudinal data, data which commonly suffer from incomplete observations. Those Bayesian models for longitudinal data that rely on the ignorability assumption of the dropout mechanism might give misleading inferences. Hence, there is a need to further study the impact of departures from the ignorability assumption on the Bayesian estimates of the model parameters. Current methodology for Bayesian sensitivity analysis mostly involves single response variable in both cross-sectional and longitudinal studies. In this paper, we propose a multivariate extension of the Bayesian index of sensitivity to non-ignorability for the general case of multivariate longitudinal studies with the possibility of having mixed correlated outcomes and a vector of multiple non-ignorability parameters in the missing mechanism. To simultaneously model the mixed responses over the time, we use a random effect latent variable approach. We illustrate the method conducting some simulation studies and analyzing a real data set from a longitudinal study for the comparison of two oral treatments for toenail dermatophyte onychomycosis.

An Analysis of Selection Models For Non-ignorable Dropout: An Application to Multi-centre Trial Data

A common problem encountered in statistical analysis is that of missing data, which occurs when some variables have missing values in some units. The present paper deals with the analysis of longitudinal continuous measurements with incomplete data due to non-ignorable dropout. In repeated measurements data, as one solution to a problem, the selection model assumes a mechanism of outcome-dependent dropout and jointly both the measurement together with dropout process of repeated measures. We consider the construction of a particular type of selection model that uses a logistic regression model to describe the dependency of dropout indicators on the longitudinal measurement. We focus on the use of the Diggle-Kenward model as a tool for assessing the sensitivity of a selection model in terms of the modeling assumptions. Our main objective here is to investigate the influence on inference that might be exerted on the considered data by the dropout process. We restrict attention to a model for repeated Gaussian measures, subject to potentially non-random dropout. To investigate this, we carry out an application for analyzing incomplete longitudinal clinical trial with dropout by using a practical example in the form of a multi-center clinical trial data.

Analysis of Multivariate Longitudinal Data Subject to Nonrandom Dropout

Longitudinal data are collected for studying changes across time. We consider multivariate longitudinal data where multiple observed variables, measured at each time point, are used as indicators for theoretical constructs (latent variables) of interest. A common problem in longitudinal studies is dropout, where subjects exit the study prematurely. Ignoring the dropout mechanism can lead to biased estimates, especially when the dropout is nonrandom. Our proposed approach uses latent variable models to capture the evolution of the latent phenomenon over time while also accounting for possibly nonrandom dropout. The dropout mechanism is modeled with a hazard function that depends on the latent variables and observed covariates. Different relationships among these variables and the dropout mechanism are studied via 2 model specifications. The proposed models are used to study people’s perceptions of women’s work using 3 questions from 5 waves from the British Household Panel Survey.

Pseudo-empirical Likelihood-Based Method Using Calibration for Longitudinal Data with Drop-Out.

In observational studies, interest mainly lies in estimation of the population-level relationship between the explanatory variables and dependent variables, and the estimation is often undertaken using a sample of longitudinal data. In some situations, the longitudinal data sample features biases and loss of estimation efficiency due to non-random drop-out. However, inclusion of population-level information can increase estimation efficiency. In this paper we propose an empirical likelihood-based method to incorporate population-level information in a longitudinal study with drop-out. The population-level information is incorporated via constraints on functions of the parameters, and non-random drop-out bias is corrected by using a weighted generalized estimating equations method. We provide a three-step estimation procedure that makes computation easier. Some commonly used methods are compared in simulation studies, which demonstrate that our proposed method can correct the non-random drop-out bias and increase the estimation efficiency, especially for small sample size or when the missing proportion is high. In some situations, the efficiency improvement is substantial. Finally, we apply this method to an Alzheimer's disease study.

A Shared Parameter Model for Longitudinal Data with Missing Values

Longitudinal studies represent one of the principal research strategies employed in medical and social research. These studies are the most appropriate for studying individual change over time. The prematurely withdrawal of some subjects from the study (dropout) is termed nonrandom when the probability of missingness depends on the missing value. Nonrandom dropout is common phenomenon associated with longitudinal data and it complicates statistical inference. The shared parameter model is used to fit longitudinal data in the presence of nonrandom dropout. The stochastic EM algorithm is developed to obtain the model parameter estimates. Also, parameter estimates of the dropout model have been obtained. Standard errors of estimates have been calculated using the developed Monte Carlo method. The proposed approach performance is evaluated through a simulation study. Also, the proposed approach is applied to a real data set.

Response to Letter Regarding Article, “Longitudinal Changes in Ejection Fraction in Heart Failure Patients With Preserved and Reduced Ejection Fraction”

HomeCirculation: Heart FailureVol. 6, No. 2Response to Letter Regarding Article, “Longitudinal Changes in Ejection Fraction in Heart Failure Patients With Preserved and Reduced Ejection Fraction” Free AccessResearch ArticlePDF/EPUBAboutView PDFView EPUBSections ToolsAdd to favoritesDownload citationsTrack citationsPermissions ShareShare onFacebookTwitterLinked InMendeleyReddit Jump toFree AccessResearch ArticlePDF/EPUBResponse to Letter Regarding Article, “Longitudinal Changes in Ejection Fraction in Heart Failure Patients With Preserved and Reduced Ejection Fraction” Shannon M. Dunlay, MD, MSc and Véronique L. Roger, MD, MPH Susan A. Weston, MS and Ruoxiang Jiang, BS Margaret M. Redfield, MD Shannon M. DunlayShannon M. Dunlay Division of Cardiovascular DiseasesDepartment of Internal MedicineMayo ClinicRochester, MN Search for more papers by this author and Véronique L. RogerVéronique L. Roger Division of Cardiovascular DiseasesDepartment of Internal MedicineMayo ClinicRochester, MN Search for more papers by this author Susan A. WestonSusan A. Weston Department of Health Sciences ResearchMayo ClinicRochester, MN Search for more papers by this author and Ruoxiang JiangRuoxiang Jiang Department of Health Sciences ResearchMayo ClinicRochester, MN Search for more papers by this author Margaret M. RedfieldMargaret M. Redfield Division of Cardiovascular DiseasesDepartment of Internal MedicineMayo ClinicRochester, MN Search for more papers by this author Originally published1 Mar 2013https://doi.org/10.1161/CIRCHEARTFAILURE.113.000154Circulation: Heart Failure. 2013;6:e26We appreciate the insightful comments from Nunez et al1 on our recently published article “Longitudinal changes in ejection fraction in heart failure patients with preserved and reduced ejection fraction.”2 To address their concerns on the effect of nonrandom or informative dropout on our findings, we repeated our analyses using a joint modeling approach as they recommended,3 using R Version 2.15.0. Overall, our findings were similar compared with those we previously published. Among community patients with heart failure followed over time, in patients with heart failure and reduced ejection fraction (EF), EF increased on average 5.0% over 5 years, using the new joint modeling approach, compared with 6.9%, using mixed effects models (our published approach). Similarly, in patients with preserved EF, EF decreased an average of 5.9% using the new approach, compared with a 5.8% decrease, as we previously published. When we reanalyzed the change in EF over time according to various factors, although some point estimates differed slightly, the overall findings were similar. For example, in patients with heart failure and preserved EF, greater declines in EF over time were still noted in those who were older at diagnosis (age 80 years or older showed decline in EF of 7.8% in 5 years with joint modeling, compared with 7.4% using mixed effects modeling). In patients with heart failure and reduced EF, greater increases in EF at 5 years were still noted in patients without coronary artery disease (CAD; 6.9% increase in those without CAD, 0.4% decrease in those with CAD using joint modeling, 9.4% increase in those without CAD, 0.3% increase in those with CAD using mixed effects modeling). These results show that our findings are robust and yield similar results when analyzed using different methods.Shannon M. Dunlay, MD, MScVéronique L. Roger, MD, MPHDivision of Cardiovascular DiseasesDepartment of Internal MedicineMayo ClinicRochester, MNSusan A. Weston, MSRuoxiang Jiang, BSDepartment of Health Sciences ResearchMayo ClinicRochester, MNMargaret M. Redfield, MDDivision of Cardiovascular DiseasesDepartment of Internal MedicineMayo ClinicRochester, MNDisclosuresNone.FootnotesCorrespondence to Margaret M. Redfield, MD, Division of Cardiovascular Diseases, Department of Internal Medicine, Mayo Clinic, 200 First St, SW, Rochester, MN 55905. E-mail [email protected]

Generalized Partially Linear Models for Incomplete Longitudinal Data In the Presence of Population‐Level Information

In observational studies, interest often lies in estimation of the population-level relationship between the explanatory variables and dependent variables, and the estimation is often done using longitudinal data. Longitudinal data often feature sampling error and bias due to nonrandom drop-out. However, inclusion of population-level information can increase estimation efficiency. In this article, we consider a generalized partially linear model for incomplete longitudinal data in the presence of the population-level information. A pseudo-empirical likelihood-based method is introduced to incorporate population-level information, and nonrandom drop-out bias is corrected by using a weighted generalized estimating equations method. A three-step estimation procedure is proposed, which makes the computation easier. Several methods that are often used in practice are compared in simulation studies, which demonstrate that our proposed method can correct the nonrandom drop-out bias and increase the estimation efficiency, especially for small sample size or when the missing proportion is high. We apply this method to an Alzheimer's disease study.