Bayesian estimation of cognitive decline in patients with alzheimer's disease

Recently, there has been great interest in estimating the decline in cognitive ability in patients with Alzheimer's disease. Measuring decline is not straightforward, since one must consider the choice of scale to measure cognitive ability, possible floor and ceiling effects, between-patient variability, and the unobserved age of onset. The authors demonstrate how to account for the above features by modeling decline in scores on the Mini-Mental State Exam in two different data sets. To this end, they use hierarchical Bayesian models with change points, for which posterior distributions are calculated using the Gibbs sampler. They make comparisons between several such models using both prior and posterior Bayes factors, and compare the results from the models suggested by these two model selection criteria.     

A Bayesian random effects model for analyzing mixed negative binomial and continuous longitudinal responses

ABSTRACT

A general Bayesian random effects model for analyzing longitudinal mixed correlated continuous and negative binomial responses with and without missing data is presented. This Bayesian model, given some random effects, uses a normal distribution for the continuous response and a negative binomial distribution for the count response. A Markov Chain Monte Carlo sampling algorithm is described for estimating the posterior distribution of the parameters. This Bayesian model is illustrated by a simulation study. For sensitivity analysis to investigate the change of parameter estimates with respect to the perturbation from missing at random to not missing at random assumption, the use of posterior curvature is proposed. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the negative binomial response of joint damage and continuous response of body mass index. The simultaneous effects of some covariates on both responses are also investigated.     

Impact of missing data on type 1 error rates in non‐inferiority trials

In this paper, a simulation study is conducted to systematically investigate the impact of different types of missing data on six different statistical analyses: four different likelihood‐based linear mixed effects models and analysis of covariance (ANCOVA) using two different data sets, in non‐inferiority trial settings for the analysis of longitudinal continuous data. ANCOVA is valid when the missing data are completely at random. Likelihood‐based linear mixed effects model approaches are valid when the missing data are at random. Pattern‐mixture model (PMM) was developed to incorporate non‐random missing mechanism. Our simulations suggest that two linear mixed effects models using unstructured covariance matrix for within‐subject correlation with no random effects or first‐order autoregressive covariance matrix for within‐subject correlation with random coefficient effects provide well control of type 1 error (T1E) rate when the missing data are completely at random or at random. ANCOVA using last observation carried forward imputed data set is the worst method in terms of bias and T1E rate. PMM does not show much improvement on controlling T1E rate compared with other linear mixed effects models when the missing data are not at random but is markedly inferior when the missing data are at random. Copyright © 2009 John Wiley & Sons, Ltd.     

Skew-mixed effects model for multivariate longitudinal data with categorical outcomes and missingness

A longitudinal study commonly follows a set of variables, measured for each individual repeatedly over time, and usually suffers from incomplete data problem. A common approach for dealing with longitudinal categorical responses is to use the Generalized Linear Mixed Model (GLMM). This model induces the potential relation between response variables over time via a vector of random effects, assumed to be shared parameters in the non-ignorable missing mechanism. Most GLMMs assume that the random-effects parameters follow a normal or symmetric distribution and this leads to serious problems in real applications. In this paper, we propose GLMMs for the analysis of incomplete multivariate longitudinal categorical responses with a non-ignorable missing mechanism based on a shared parameter framework with the less restrictive assumption of skew-normality for the random effects. These models may contain incomplete data with monotone and non-monotone missing patterns. The performance of the model is evaluated using simulation studies and a well-known longitudinal data set extracted from a fluvoxamine trial is analyzed to determine the profile of fluvoxamine in ambulatory clinical psychiatric practice.     

Analysis of ordinal outcomes with longitudinal covariates subject to missingness

We propose a mixture model for data with an ordinal outcome and a longitudinal covariate that is subject to missingness. Data from a tailored telephone delivered, smoking cessation intervention for construction laborers are used to illustrate the method, which considers as an outcome a categorical measure of smoking cessation, and evaluates the effectiveness of the motivational telephone interviews on this outcome. We propose two model structures for the longitudinal covariate, for the case when the missing data are missing at random, and when the missing data mechanism is non-ignorable. A generalized EM algorithm is used to obtain maximum likelihood estimates.     

Joint generalized estimating equations for multivariate longitudinal binary outcomes with missing data: an application to acquired immune deficiency syndrome data

Summary.  In a large, prospective longitudinal study designed to monitor cardiac abnormalities in children born to women who are infected with the human immunodeficiency virus, instead of a single outcome variable, there are multiple binary outcomes (e.g. abnormal heart rate, abnormal blood pressure and 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 by using generalized estimating equations (GEEs), and consistent estimates can be obtained under the assumption of a missingness completely at random mechanism. When the missing data mechanism is missingness at random, i.e. 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 by using a single modified GEE based on an EM-type algorithm. The method proposed is motivated by the longitudinal study of cardiac abnormalities in children who were born to women infected with the human immunodeficiency virus, 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 a missingness at random mechanism in which missingness depends on all observed outcome variables, our joint estimation via the modified GEE produces almost unbiased estimates, provided that the correlation model has been correctly specified, whereas estimates from standard GEEs can lead to substantial bias.     

A multivariate finite mixture latent trajectory model with application to dementia studies

Dementia patients exhibit considerable heterogeneity in individual trajectories of cognitive decline, with some patients showing rapid decline following diagnoses while others exhibiting slower decline or remaining stable for several years. Dementia studies often collect longitudinal measures of multiple neuropsychological tests aimed to measure patients’ decline across a number of cognitive domains. We propose a multivariate finite mixture latent trajectory model to identify distinct longitudinal patterns of cognitive decline simultaneously in multiple cognitive domains, each of which is measured by multiple neuropsychological tests. EM algorithm is used for parameter estimation and posterior probabilities are used to predict latent class membership. We present results of a simulation study demonstrating adequate performance of our proposed approach and apply our model to the Uniform Data Set from the National Alzheimer's Coordinating Center to identify cognitive decline patterns among dementia patients.     

Mixed Correlated Bivariate Ordinal and Negative Binomial Longitudinal Responses with Nonignorable Missing Values

Regression models with random effects are proposed for joint analysis of negative binomial and ordinal longitudinal data with nonignorable missing values under fully parametric framework. The presented model simultaneously considers a multivariate probit regression model for the missing mechanisms, which provides the ability of examining the missing data assumptions and a multivariate mixed model for the responses. Random effects are used to take into account the correlation between longitudinal responses of the same individual. A full likelihood-based approach that allows yielding maximum likelihood estimates of the model parameters is used. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the ordinal response of osteoporosis of the spine and negative binomial response is the number of joint damage. A sensitivity of the results to the assumptions is also investigated. The effect of some covariates on all responses are investigated simultaneously.     

Variable selection for longitudinal data with high-dimensional covariates and dropouts

A new variable selection approach utilizing penalized estimating equations is developed for high-dimensional longitudinal data with dropouts under a missing at random (MAR) mechanism. The proposed method is based on the best linear approximation of efficient scores from the full dataset and does not need to specify a separate model for the missing or imputation process. The coordinate descent algorithm is adopted to implement the proposed method and is computational feasible and stable. The oracle property is established and extensive simulation studies show that the performance of the proposed variable selection method is much better than that of penalized estimating equations dealing with complete data which do not account for the MAR mechanism. In the end, the proposed method is applied to a Lifestyle Education for Activity and Nutrition study and the interaction effect between intervention and time is identified, which is consistent with previous findings.     

Latent class models for longitudinal studies of the elderly with data missing at random

The elderly population in the USA is expected to double in size by the year 2025, making longitudinal health studies of this population of increasing importance. The degree of loss to follow-up in studies of the elderly, which is often because elderly people cannot remain in the study, enter a nursing home or die, make longitudinal studies of this population problematic. We propose a latent class model for analysing multiple longitudinal binary health outcomes with multiple-cause non-response when the data are missing at random and a non-likelihood-based analysis is performed. We extend the estimating equations approach of Robins and co-workers to latent class models by reweighting the multiple binary longitudinal outcomes by the inverse probability of being observed. This results in consistent parameter estimates when the probability of non-response depends on observed outcomes and covariates (missing at random) assuming that the model for non-response is correctly specified. We extend the non-response model so that institutionalization, death and missingness due to failure to locate, refusal or incomplete data each have their own set of non-response probabilities. Robust variance estimates are derived which account for the use of a possibly misspecified covariance matrix, estimation of missing data weights and estimation of latent class measurement parameters. This approach is then applied to a study of lower body function among a subsample of the elderly participating in the 6-year Longitudinal Study of Aging.     

A Bayesian shared parameter model for joint modeling of longitudinal continuous and binary outcomes

Joint modeling of associated mixed biomarkers in longitudinal studies leads to a better clinical decision by improving the efficiency of parameter estimates. In many clinical studies, the observed time for two biomarkers may not be equivalent and one of the longitudinal responses may have recorded in a longer time than the other one. In addition, the response variables may have different missing patterns. In this paper, we propose a new joint model of associated continuous and binary responses by accounting different missing patterns for two longitudinal outcomes. A conditional model for joint modeling of the two responses is used and two shared random effects models are considered for intermittent missingness of two responses. A Bayesian approach using Markov Chain Monte Carlo (MCMC) is adopted for parameter estimation and model implementation. The validation and performance of the proposed model are investigated using some simulation studies. The proposed model is also applied for analyzing a real data set of bariatric surgery.     

A statistical algorithm for detecting cognitive plateaus in Alzheimer’s disease

Repeated neuropsychological measurements, such as mini-mental state examination (MMSE) scores, are frequently used in Alzheimer’s disease (AD) research to study change in cognitive function of AD patients. A question of interest among dementia researchers is whether some AD patients exhibit transient “plateaus” of cognitive function in the course of the disease. We consider a statistical approach to this question, based on irregularly spaced repeated MMSE scores. We propose an algorithm that formalizes the measurement of an apparent cognitive plateau, and a procedure to evaluate the evidence of plateaus in AD using this algorithm based on applying the algorithm to the observed data and to data sets simulated from a linear mixed model. We apply these methods to repeated MMSE data from the Michigan Alzheimer’s Disease Research Center, finding a high rate of apparent plateaus and also a high rate of false discovery. Simulation studies are also conducted to assess the performance of the algorithm. In general, the false discovery rate of the algorithm is high unless the rate of decline is high compared with the measurement error of the cognitive test. It is argued that the results are not a problem of the specific algorithm chosen, but reflect a lack of information concerning the presence of plateaus in the data.     

A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects

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.     

A latent variable model for analyzing mixed longitudinal (k,l)-inflated count and ordinal responses

A random effects model for analyzing mixed longitudinal count and ordinal data is presented where the count response is inflated in two points (k and l) and an (k,l)-Inflated Power series distribution is used as its distribution. A full likelihood-based approach is used to obtain maximum likelihood estimates of parameters of the model. For data with non-ignorable missing values models with probit model for missing mechanism are used.The dependence between longitudinal sequences of responses and inflation parameters are investigated using a random effects approach. Also, to investigate the correlation between mixed ordinal and count responses of each individuals at each time, a shared random effect is used. In order to assess the performance of the model, a simulation study is performed for a case that the count response has (k,l)-Inflated Binomial distribution. Performance comparisons of count-ordinal random effect model, Zero-Inflated ordinal random effects model and (k,l)-Inflated ordinal random effects model are also given. The model is applied to a real social data set from the first two waves of the national longitudinal study of adolescent to adult health (Add Health study). In this data set, the joint responses are the number of days in a month that each individual smoked as the count response and the general health condition of each individual as the ordinal response. For the count response there is incidence of excess values of 0 and 30.     

Inference methods for saturated models in longitudinal clinical trials with incomplete binary data

In the longitudinal studies with binary response, it is often of interest to estimate the percentage of positive responses at each time point and the percentage of having at least one positive response by each time point. When missing data exist, the conventional method based on observed percentages could result in erroneous estimates. This study demonstrates two methods of using expectation-maximization (EM) and data augmentation (DA) algorithms in the estimation of the marginal and cumulative probabilities for incomplete longitudinal binary response data. Both methods provide unbiased estimates when the missingness mechanism is missing at random (MAR) assumption. Sensitivity analyses have been performed for cases when the MAR assumption is in question.     

A generalized linear mixed model for longitudinal binary data with a marginal logit link function

Longitudinal studies of a binary outcome are common in the health, social, and behavioral sciences. In general, a feature of random effects logistic regression models for longitudinal binary data is that the marginal functional form, when integrated over the distribution of the random effects, is no longer of logistic form. Recently, Wang and Louis (2003) proposed a random intercept model in the clustered binary data setting where the marginal model has a logistic form. An acknowledged limitation of their model is that it allows only a single random effect that varies from cluster to cluster. In this paper, we propose a modification of their model to handle longitudinal data, allowing separate, but correlated, random intercepts at each measurement occasion. The proposed model allows for a flexible correlation structure among the random intercepts, where the correlations can be interpreted in terms of Kendall's τ. For example, the marginal correlations among the repeated binary outcomes can decline with increasing time separation, while the model retains the property of having matching conditional and marginal logit link functions. Finally, the proposed method is used to analyze data from a longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women.     

Longitudinal conditional models with intermittent missingness: SAS code and applications

This work provides a set of macros performed with SAS (Statistical Analysis System) for Windows, which can be used to fit conditional models under intermittent missingness in longitudinal data. A formalized transition model, including random effects for individuals and measurement error, is presented. Model fitting is based on the missing completely at random or missing at random assumptions, and the separability condition. The problem translates to maximization of the marginal observed data density only, which for Gaussian data is again Gaussian, meaning that the likelihood can be expressed in terms of the mean and covariance matrix of the observed data vector. A simulation study is presented and misspecification issues are considered. A practical application is also given, where conditional models are fitted to the data from a clinical trial that assessed the effect of a Cuban medicine on a disease of the respiratory system.     

Bias of factor loadings from questionnaire data with imputed scores

This study investigated the bias of factor loadings obtained from incomplete questionnaire data with imputed scores. Three models were used to generate discrete ordered rating scale data typical of questionnaires, also known as Likert data. These methods were the multidimensional polytomous latent trait model, a normal ogive item response theory model, and the discretized normal model. Incomplete data due to nonresponse were simulated using either missing completely at random or not missing at random mechanisms. Subsequently, for each incomplete data matrix, four imputation methods were applied for imputing item scores. Based on a completely crossed six-factor design, it was concluded that in general, bias was small for all data simulation methods and all imputation methods, and under all nonresponse mechanisms. Imputation method, two-way-plus-error, had the smallest bias in the factor loadings. Bias based on the discretized normal model was greater than that based on the other two models.     

Segmental modeling of changing immunologic response for CD4 data with skewness,missingness and dropout

In clinical practice, the profile of each subject's CD4 response from a longitudinal study may follow a ‘broken stick’ like trajectory, indicating multiple phases of increase and/or decline in response. Such multiple phases (changepoints) may be important indicators to help quantify treatment effect and improve management of patient care. Although it is a common practice to analyze complex AIDS longitudinal data using nonlinear mixed-effects (NLME) or nonparametric mixed-effects (NPME) models in the literature, NLME or NPME models become a challenge to estimate changepoint due to complicated structures of model formulations. In this paper, we propose a changepoint mixed-effects model with random subject-specific parameters, including the changepoint for the analysis of longitudinal CD4 cell counts for HIV infected subjects following highly active antiretroviral treatment. The longitudinal CD4 data in this study may exhibit departures from symmetry, may encounter missing observations due to various reasons, which are likely to be non-ignorable in the sense that missingness may be related to the missing values, and may be censored at the time of the subject going off study-treatment, which is a potentially informative dropout mechanism. Inferential procedures can be complicated dramatically when longitudinal CD4 data with asymmetry (skewness), incompleteness and informative dropout are observed in conjunction with an unknown changepoint. Our objective is to address the simultaneous impact of skewness, missingness and informative censoring by jointly modeling the CD4 response and dropout time processes under a Bayesian framework. The method is illustrated using a real AIDS data set to compare potential models with various scenarios, and some interested results are presented.     

An extension of the placebo‐based pattern‐mixture model

Pattern‐mixture models provide a general and flexible framework for sensitivity analyses of nonignorable missing data in longitudinal studies. The placebo‐based pattern‐mixture model handles missing data in a transparent and clinically interpretable manner. We extend this model to include a sensitivity parameter that characterizes the gradual departure of the missing data mechanism from being missing at random toward being missing not at random under the standard placebo‐based pattern‐mixture model. We derive the treatment effect implied by the extended model. We propose to utilize the primary analysis based on a mixed‐effects model for repeated measures to draw inference about the treatment effect under the extended placebo‐based pattern‐mixture model. We use simulation studies to confirm the validity of the proposed method. We apply the proposed method to a clinical study of major depressive disorders. Copyright © 2013 John Wiley & Sons, Ltd.