A Bayesian conditional model for bivariate mixed ordinal and skew continuous longitudinal responses using quantile regression

In this paper, we develop a conditional model for analyzing mixed bivariate continuous and ordinal longitudinal responses. We propose a quantile regression model with random effects for analyzing continuous responses. For this purpose, an Asymmetric Laplace Distribution (ALD) is allocated for continuous response given random effects. For modeling ordinal responses, a cumulative logit model is used, via specifying a latent variable model, with considering other random effects. Therefore, the intra-association between continuous and ordinal responses is taken into account using their own exclusive random effects. But, the inter-association between two mixed responses is taken into account by adding a continuous response term in the ordinal model. We use a Bayesian approach via Markov chain Monte Carlo method for analyzing the proposed conditional model and to estimate unknown parameters, a Gibbs sampler algorithm is used. Moreover, we illustrate an application of the proposed model using a part of the British Household Panel Survey data set. The results of data analysis show that gender, age, marital status, educational level and the amount of money spent on leisure have significant effects on annual income. Also, the associated parameter is significant in using the best fitting proposed conditional model, thus it should be employed rather than analyzing separate models.     

Joint modeling of multivariate longitudinal mixed measurements and time to event data using a Bayesian approach

In many longitudinal studies multiple characteristics of each individual, along with time to occurrence of an event of interest, are often collected. In such data set, some of the correlated characteristics may be discrete and some of them may be continuous. In this paper, a joint model for analysing multivariate longitudinal data comprising mixed continuous and ordinal responses and a time to event variable is proposed. We model the association structure between longitudinal mixed data and time to event data using a multivariate zero-mean Gaussian process. For modeling discrete ordinal data we assume a continuous latent variable follows the logistic distribution and for continuous data a Gaussian mixed effects model is used. For the event time variable, an accelerated failure time model is considered under different distributional assumptions. For parameter estimation, a Bayesian approach using Markov Chain Monte Carlo is adopted. The performance of the proposed methods is illustrated using some simulation studies. A real data set is also analyzed, where different model structures are used. Model comparison is performed using a variety of statistical criteria.     

A Bayesian test of homogeneity of association parameter using transition modelling of longitudinal mixed responses

In this paper, a Bayesian framework using a joint transition model for analysing longitudinal mixed ordinal and continuous responses is considered. The joint model considers a multivariate mixed model for the responses in which a transitive cumulative logistic regression model and an autoregressive regression model are used to model ordinal and continuous responses, respectively. Also, to take into account the association between longitudinal ordinal and continuous responses, a dynamic association parameter is used. A test is conducted to see whether this parameter is time-invariant and another test is presented to see whether this parameter is equal to zero or significantly far from zero. Our approach is applied to longitudinal PIAT (Peabody Individual Achievement Test) data where the Bayesian estimates of parameters are obtained.     

General location multivariate latent variable models for mixed correlated bounded continuous,ordinal, and nominal responses with non-ignorable missing data

Using a multivariate latent variable approach, this article proposes some new general models to analyze the correlated bounded continuous and categorical (nominal or/and ordinal) responses with and without non-ignorable missing values. First, we discuss regression methods for jointly analyzing continuous, nominal, and ordinal responses that we motivated by analyzing data from studies of toxicity development. Second, using the beta and Dirichlet distributions, we extend the models so that some bounded continuous responses are replaced for continuous responses. The joint distribution of the bounded continuous, nominal and ordinal variables is decomposed into a marginal multinomial distribution for the nominal variable and a conditional multivariate joint distribution for the bounded continuous and ordinal variables given the nominal variable. We estimate the regression parameters under the new general location models using the maximum-likelihood method. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms of the model on the maximal normal curvature. The proposed models are applied to two data sets: BMI, Steatosis and Osteoporosis data and Tehran household expenditure budgets.     

Augmented mixed beta regression models with skew-normal independent distributions: Bayesian analysis of labor force data

Abstract

Augmented mixed beta regression models are suitable choices for modeling continuous response variables on the closed interval [0, 1]. The random eeceeects in these models are typically assumed to be normally distributed, but this assumption is frequently violated in some applied studies. In this paper, an augmented mixed beta regression model with skew-normal independent distribution for random effects are used. Next, we adopt a Bayesian approach for parameter estimation using the MCMC algorithm. The methods are then evaluated using some intensive simulation studies. Finally, the proposed models have applied to analyze a dataset from an Iranian Labor Force Survey.     

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.     

Bayesian Inference for Skew-normal Linear Mixed Models

Linear mixed models (LMM) are frequently used to analyze repeated measures data, because they are more flexible to modelling the correlation within-subject, often present in this type of data. The most popular LMM for continuous responses assumes that both the random effects and the within-subjects errors are normally distributed, which can be an unrealistic assumption, obscuring important features of the variations present within and among the units (or groups). This work presents skew-normal liner mixed models (SNLMM) that relax the normality assumption by using a multivariate skew-normal distribution, which includes the normal ones as a special case and provides robust estimation in mixed models. The MCMC scheme is derived and the results of a simulation study are provided demonstrating that standard information criteria may be used to detect departures from normality. The procedures are illustrated using a real data set from a cholesterol study.     

Latent variable model for mixed correlated power series and ordinal longitudinal responses with non ignorable missing values

We propose a joint model based on a latent variable for analyzing mixed power series and ordinal longitudinal data with and without missing values. A bivariate probit regression model is used for the missing mechanisms. Random effects are used to take into account the correlation between longitudinal responses. A full likelihood-based approach is used to yield maximum-likelihood estimates of the model parameters. Our model is applied to a medical data set, obtained from an observational study on women where the correlated responses are the ordinal response of osteoporosis of the spine and the power series response of the number of joint damages. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms and overdispersion of the model on likelihood displacement.     

Latent variable models with ordinal categorical covariates

We propose a general latent variable model for multivariate ordinal categorical variables, in which both the responses and the covariates are ordinal, to assess the effect of the covariates on the responses and to model the covariance structure of the response variables. A?fully Bayesian approach is employed to analyze the model. The Gibbs sampler is used to simulate the joint posterior distribution of the latent variables and the parameters, and the parameter expansion and reparameterization techniques are used to speed up the convergence procedure. The proposed model and method are demonstrated by simulation studies and a real data example.     

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.     

A shared parameter model of longitudinal measurements and survival time with heterogeneous random-effects distribution

Typical joint modeling of longitudinal measurements and time to event data assumes that two models share a common set of random effects with a normal distribution assumption. But, sometimes the underlying population that the sample is extracted from is a heterogeneous population and detecting homogeneous subsamples of it is an important scientific question. In this paper, a finite mixture of normal distributions for the shared random effects is proposed for considering the heterogeneity in the population. For detecting whether the unobserved heterogeneity exits or not, we use a simple graphical exploratory diagnostic tool proposed by Verbeke and Molenberghs [34] to assess whether the traditional normality assumption for the random effects in the mixed model is adequate. In the joint modeling setting, in the case of evidence against normality (homogeneity), a finite mixture of normals is used for the shared random-effects distribution. A Bayesian MCMC procedure is developed for parameter estimation and inference. The methodology is illustrated using some simulation studies. Also, the proposed approach is used for analyzing a real HIV data set, using the heterogeneous joint model for this data set, the individuals are classified into two groups: a group with high risk and a group with moderate risk.     

A linear mixed model for analyzing longitudinal skew-normal responses with random dropout

In this paper, a linear mixed effects model is used to fit skewed longitudinal data in the presence of dropout. Two distributional assumptions are considered to produce background for heavy tailed models. One is the linear mixed model with skew-normal random effects and normal errors and the other one is the linear mixed model with skew-normal errors and normal random effects. An ECM algorithm is developed to obtain the parameter estimates. Also an empirical Bayes approach is used for estimating random effects. A simulation study is implemented to investigate the performance of the presented algorithm. Results of an application are also reported where standard errors of estimates are calculated using the Bootstrap 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.     

A multiple group item response theory model with centered skew-normal latent trait distributions under a Bayesian framework

Very often, in psychometric research, as in educational assessment, it is necessary to analyze item response from clustered respondents. The multiple group item response theory (IRT) model proposed by Bock and Zimowski [12] provides a useful framework for analyzing such type of data. In this model, the selected groups of respondents are of specific interest such that group-specific population distributions need to be defined. The usual assumption for parameter estimation in this model, which is that the latent traits are random variables following different symmetric normal distributions, has been questioned in many works found in the IRT literature. Furthermore, when this assumption does not hold, misleading inference can result. In this paper, we consider that the latent traits for each group follow different skew-normal distributions, under the centered parameterization. We named it skew multiple group IRT model. This modeling extends the works of Azevedo et al. [4], Bazán et al. [11] and Bock and Zimowski [12] (concerning the latent trait distribution). Our approach ensures that the model is identifiable. We propose and compare, concerning convergence issues, two Monte Carlo Markov Chain (MCMC) algorithms for parameter estimation. A simulation study was performed in order to evaluate parameter recovery for the proposed model and the selected algorithm concerning convergence issues. Results reveal that the proposed algorithm recovers properly all model parameters. Furthermore, we analyzed a real data set which presents asymmetry concerning the latent traits distribution. The results obtained by using our approach confirmed the presence of negative asymmetry for some latent trait distributions.     

An ECM Estimation Approach for Analyzing Multivariate Skew-Normal Data with Dropout

In this article, an ECM algorithm is developed to obtain the maximum likelihood estimates of parameters where multivariate skew-normal distribution is used for analyzing longitudinal skewed normal regression data with dropout. A simulation study is performed to investigate the performance of the presented algorithm. Also, the methodology is illustrated through two applications and the results of proposed methodology are compared with ECM under multivariate normal assumption using AIC and BIC criteria. Standard errors of parameter estimates are obtained by asymptotic observed information matrix.     

Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective

Linear mixed models were developed to handle clustered data and have been a topic of increasing interest in statistics for the past 50 years. Generally, the normality (or symmetry) of the random effects is a common assumption in linear mixed models but it may, sometimes, be unrealistic, obscuring important features of among-subjects variation. In this article, we utilize skew-normal/independent distributions as a tool for robust modeling of linear mixed models under a Bayesian paradigm. The skew-normal/independent distributions is an attractive class of asymmetric heavy-tailed distributions that includes the skew-normal distribution, skew-t, skew-slash and the skew-contaminated normal distributions as special cases, providing an appealing robust alternative to the routine use of symmetric distributions in this type of models. The methods developed are illustrated using a real data set from Framingham cholesterol study.     

A distance-based rounding strategy for post-imputation ordinal data

Multiple imputation has emerged as a widely used model-based approach in dealing with incomplete data in many application areas. Gaussian and log-linear imputation models are fairly straightforward to implement for continuous and discrete data, respectively. However, in missing data settings which include a mix of continuous and discrete variables, correct specification of the imputation model could be a daunting task owing to the lack of flexible models for the joint distribution of variables of different nature. This complication, along with accessibility to software packages that are capable of carrying out multiple imputation under the assumption of joint multivariate normality, appears to encourage applied researchers for pragmatically treating the discrete variables as continuous for imputation purposes, and subsequently rounding the imputed values to the nearest observed category. In this article, I introduce a distance-based rounding approach for ordinal variables in the presence of continuous ones. The first step of the proposed rounding process is predicated upon creating indicator variables that correspond to the ordinal levels, followed by jointly imputing all variables under the assumption of multivariate normality. The imputed values are then converted to the ordinal scale based on their Euclidean distances to a set of indicators, with minimal distance corresponding to the closest match. I compare the performance of this technique to crude rounding via commonly accepted accuracy and precision measures with simulated data sets.     

Instrumental variable estimation in ordinal probit models with mismeasured predictors

Researchers in the medical, health, and social sciences routinely encounter ordinal variables such as self‐reports of health or happiness. When modelling ordinal outcome variables, it is common to have covariates, for example, attitudes, family income, retrospective variables, measured with error. As is well known, ignoring even random error in covariates can bias coefficients and hence prejudice the estimates of effects. We propose an instrumental variable approach to the estimation of a probit model with an ordinal response and mismeasured predictor variables. We obtain likelihood‐based and method of moments estimators that are consistent and asymptotically normally distributed under general conditions. These estimators are easy to compute, perform well and are robust against the normality assumption for the measurement errors in our simulation studies. The proposed method is applied to both simulated and real data. The Canadian Journal of Statistics 47: 653–667; 2019 © 2019 Statistical Society of Canada     

A mixture latent variable model for modeling mixed data in heterogeneous populations and its applications

Latent variable models are widely used for jointly modeling of mixed data including nominal, ordinal, count and continuous data. In this paper, we consider a latent variable model for jointly modeling relationships between mixed binary, count and continuous variables with some observed covariates. We assume that, given a latent variable, mixed variables of interest are independent and count and continuous variables have Poisson distribution and normal distribution, respectively. As such data may be extracted from different subpopulations, consideration of an unobserved heterogeneity has to be taken into account. A mixture distribution is considered (for the distribution of the latent variable) which accounts the heterogeneity. The generalized EM algorithm which uses the Newton–Raphson algorithm inside the EM algorithm is used to compute the maximum likelihood estimates of parameters. The standard errors of the maximum likelihood estimates are computed by using the supplemented EM algorithm. Analysis of the primary biliary cirrhosis data is presented as an application of the proposed model.     

Bayesian inference for joint location and scale nonlinear models with skew-normal errors

A regression model with skew-normal errors provides a useful extension for ordinary normal regression models when the dataset under consideration involves asymmetric outcomes. In this article, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis for joint location and scale nonlinear models with skew-normal errors, which relax the normality assumption and include the normal one as a special case. The main advantage of these class of distributions is that they have a nice hierarchical representation that allows the implementation of MCMC methods to simulate samples from the joint posterior distribution. Finally, simulation studies and a real example are used to illustrate the proposed methodology.