A Bayesian analysis for longitudinal semicontinuous data with an application to an acupuncture clinical trial

Zero-augmented beta-prime model for multilevel semi-continuous data: a Bayesian inference

Semi-continuous data characterized by an excessive proportion of zeros and right-skewed continuous positive values appear frequently in medical research. One example would be the pharmaceutical expenditure (PE) data for which a substantial proportion of subjects investigated may report zero. Two-part mixed-effects models have been developed to analyse clustered measures of semi-continuous data from multilevel studies. In this study, we propose a new flexible two-part mixed-effects model with skew distributions for nested semi-continuous cost data under the framework of a Bayesian approach. The proposed model specification consists of two mixed-effects models linked by the correlated random effects: Part I) a model on the occurrence of positive values using a generalized logistic mixed model; and Part II) a model on the magnitude of positive values using a linear mixed model where the model errors follow skew distributions including beta-prime (BP). The proposed method is illustrated with pharmaceutical expenditure data from a multilevel observational study and the analytic results are reported by comparing potential models under different skew distributions. Simulation studies are conducted to assess the performance of the proposed model. The DIC₃, LPML, WAIC, and LOO as the Bayesian model selection criteria and measures of divergence used to compare the models.

A semi-parametric Bayesian model for semi-continuous longitudinal data

Semi-continuous data present challenges in both model fitting and interpretation. Parametric distributions may be inappropriate for extreme long right tails of the data. Mean effects of covariates, susceptible to extreme values, may fail to capture relevant information for most of the sample. We propose a two-component semi-parametric Bayesian mixture model, with the discrete component captured by a probability mass (typically at zero) and the continuous component of the density modeled by a mixture of B-spline densities that can be flexibly fit to any data distribution. The model includes random effects of subjects to allow for application to longitudinal data. We specify prior distributions on parameters and perform model inference using a Markov chain Monte Carlo (MCMC) Gibbs-sampling algorithm programmed in R. Statistical inference can be made for multiple quantiles of the covariate effects simultaneously providing a comprehensive view. Various MCMC sampling techniques are used to facilitate convergence. We demonstrate the performance and the interpretability of the model via simulations and analyses on the National Consortium on Alcohol and Neurodevelopment in Adolescence study (NCANDA) data on alcohol binge drinking.

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.

Sparse group regularization for semi-continuous transportation data

Motor vehicle crashes are a global public health concern. Most analysis have used zero-inflated count models for examining crash counts. However, few methods are available to account for safety metrics that have semi-continuous observations. This article considers the problem of variable selection for the semi-continuous zero-inflated (SCZI) models. These models include two parts: a zero-inflated part and a nonzero continuous part. A special group regularization is designed to accommodate the unique structure of two-part SCZI models, and a type of Bayesian information criterion is proposed to select tuning parameters. We illustrate the variable selection process of the proposed model using lane position data from a driving simulator study. In the study, drivers stay in the intended lane for the majority of their drive (zero-inflated part). On occasion, some drivers do drift out of their intended driving lane (nonzero continuous part). Our findings show that individual differences can be captured with the proposed model, which has implications for driving safety and the design of in-vehicle alerting systems.

Nonparametric Bayesian functional two-part random effects model for longitudinal semicontinuous data analysis

Longitudinal semicontinuous data, characterized by repeated measures of a large portion of zeros and continuous positive values, are frequently encountered in many applications including biomedical, epidemiological, and social science studies. Two-part random effects models (TPREM) have been used to investigate the association between such longitudinal semicontinuous data and covariates accounting for the within-subject correlation. The existing TPREM is, however, limited to incorporate a functional covariate, which is often available in a longitudinal study. Moreover, the existing TPREM typically assumes the normality of subject-specific random effects, which can be easily violated when there exists a subgroup structure. In this article, we propose a nonparametric Bayesian functional TPREM to assess the relationship between the longitudinal semicontinuous outcome and various types of covariates including a functional covariate. The proposed model also relaxes the normality assumption for the random effects through a Dirichlet process mixture of normals, which allows for identifying an underlying subgroup structure. The methodology is illustrated through an application to social insurance expenditure data collected by the Korean Welfare Panel Study and a simulation study.

What contributes to medical debt? Evidence from patients in rural China

BACKGROUND

Rural households in developing countries usually have severe medical debt due to high out-of-pocket (OOP) payments, which contributes to bankruptcy. China implemented the critical illness insurance (CII) in 2012 to decrease patients' medical expenditure. This paper aimed to explore the medical debt of rural Chinese patients and its influencing factors.

METHODS

A questionnaire survey of health expenditures and medical debt was conducted in two counties of Central and Western China in 2017. Patients who received CII were used as the sample on the basis of multi-stage stratified cluster sampling. Descriptive statistics and multivariate analysis of variance were used in all data. A two-part model was used to evaluate the occurrence and extent of medical debt.

RESULTS

A total of 826 rural patients with CII were surveyed. The percentages of patients incurring medical debt exceeded 50% and the median debt load was 20,000 Chinese yuan (CNY, 650 CNY = US$100). Financial assistance from kin (P < 0.001) decreased the likelihood of medical debt. High inpatient expenses (IEs, P < 0.01), CII reimbursement ratio (P < 0.001), and non-direct medical costs (P < 0.001) resulted in increased medical debt load.

CONCLUSIONS

Medical debt is still one of the biggest problems in rural China. High IEs, CII reimbursement ratio, municipal or high-level hospitals were the risk determinants of medical debt load. Financial assistance from kin and household income were the protective factors. Increasing service capability of hospitals in counties could leave more patiemts in county-level and township hospitals. Improving CII with increased reimbursement rate may also be issues of concern.

A model-based approach for clustering of multivariate semicontinuous data with application to dietary pattern analysis and intervention

Semicontinuous data, characterized by a sizable number of zeros and observations from a continuous distribution, are frequently encountered in health research concerning food consumptions, physical activities, medical and pharmacy claims expenditures, and many others. In analyzing such semicontinuous data, it is imperative that the excessive zeros be adequately accounted for to obtain unbiased and efficient inference. Although many methods have been proposed in the literature for the modeling and analysis of semicontinuous data, little attention has been given to clustering of semicontinuous data to identify important patterns that could be indicative of certain health outcomes or intervention effects. We propose a Bernoulli-normal mixture model for clustering of multivariate semicontinuous data and demonstrate its accuracy as compared to the well-known clustering method with the conventional normal mixture model. The proposed method is illustrated with data from a dietary intervention trial to promote healthy eating behavior among children with type 1 diabetes. In the trial, certain diabetes friendly foods (eg, total fruit, whole fruit, dark green and orange vegetables and legumes, whole grain) were only consumed by a proportion of study participants, yielding excessive zero values due to nonconsumption of the foods. Baseline foods consumptions data in the trial are used to explore preintervention dietary patterns among study participants. While the conventional normal mixture model approach fails to do so, the proposed Bernoulli-normal mixture model approach has shown to be able to identify a dietary profile that significantly differentiates the intervention effects from others, as measured by the popular healthy eating index at the end of the trial.

Shared random parameter models: A legacy of the biostatistics program at the National Heart, Lung, and Blood Institute

Shared random parameter models (SRPMs) were first introduced by researchers at the National Heart Lung and Blood Institute (NHLBI) Biostatistics Branch for analyzing longitudinal data with informative dropout (Wu and Carroll, 1987; Wu and Bailey, 1988; Follmann and Wu, 1995; Albert and Follmann, 2000; Albert et al, 2002). This work was all focused on characterizing the longitudinal data process in the presence of an informative missing data mechanism that is treated as a nuisance. Shared random parameter modeling approaches have also been developed from the perspective of characterizing the relationship between longitudinal data and a subsequent outcome that may be an event time, a dichotomous measurement, or another longitudinal outcome. This article will review the early contributions of the NHLBI biostatisticians on SRPMs for analyzing longitudinal data with dropout and demonstrate how these ideas have, more recently, been applied in these other areas of biostatistics. Rather than focus on technical details or specific analyses, this article presents a conceptual framework for SRPMs within a historical context.

Two-part models with stochastic processes for modelling longitudinal semicontinuous data: Computationally efficient inference and modelling the overall marginal mean

Several researchers have described two-part models with patient-specific stochastic processes for analysing longitudinal semicontinuous data. In theory, such models can offer greater flexibility than the standard two-part model with patient-specific random effects. However, in practice, the high dimensional integrations involved in the marginal likelihood (i.e. integrated over the stochastic processes) significantly complicates model fitting. Thus, non-standard computationally intensive procedures based on simulating the marginal likelihood have so far only been proposed. In this paper, we describe an efficient method of implementation by demonstrating how the high dimensional integrations involved in the marginal likelihood can be computed efficiently. Specifically, by using a property of the multivariate normal distribution and the standard marginal cumulative distribution function identity, we transform the marginal likelihood so that the high dimensional integrations are contained in the cumulative distribution function of a multivariate normal distribution, which can then be efficiently evaluated. Hence, maximum likelihood estimation can be used to obtain parameter estimates and asymptotic standard errors (from the observed information matrix) of model parameters. We describe our proposed efficient implementation procedure for the standard two-part model parameterisation and when it is of interest to directly model the overall marginal mean. The methodology is applied on a psoriatic arthritis data set concerning functional disability.

Two-Part and Related Regression Models for Longitudinal Data

Statistical models that involve a two-part mixture distribution are applicable in a variety of situations. Frequently, the two parts are a model for the binary response variable and a model for the outcome variable that is conditioned on the binary response. Two common examples are zero-inflated or hurdle models for count data and two-part models for semicontinuous data. Recently, there has been particular interest in the use of these models for the analysis of repeated measures of an outcome variable over time. The aim of this review is to consider motivations for the use of such models in this context and to highlight the central issues that arise with their use. We examine two-part models for semicontinuous and zero-heavy count data, and we also consider models for count data with a two-part random effects distribution.

Bayesian inference for two-part mixed-effects model using skew distributions, with application to longitudinal semicontinuous alcohol data

Semicontinuous data featured with an excessive proportion of zeros and right-skewed continuous positive values arise frequently in practice. One example would be the substance abuse/dependence symptoms data for which a substantial proportion of subjects investigated may report zero. Two-part mixed-effects models have been developed to analyze repeated measures of semicontinuous data from longitudinal studies. In this paper, we propose a flexible two-part mixed-effects model with skew distributions for correlated semicontinuous alcohol data under the framework of a Bayesian approach. The proposed model specification consists of two mixed-effects models linked by the correlated random effects: (i) a model on the occurrence of positive values using a generalized logistic mixed-effects model (Part I); and (ii) a model on the intensity of positive values using a linear mixed-effects model where the model errors follow skew distributions including skew- t and skew-normal distributions (Part II). The proposed method is illustrated with an alcohol abuse/dependence symptoms data from a longitudinal observational study, and the analytic results are reported by comparing potential models under different random-effects structures. Simulation studies are conducted to assess the performance of the proposed models and method.

A spatial augmented beta regression model for periodontal proportion data

Clinical dental research generates large amounts of data with a potentially complex correlation structure from measurements recorded at several sites throughout the mouth. Clinical attachment level (CAL) is one such measure popularly used to assess the periodontal disease (PD) status. We model the proportion of sites for each tooth-type (i.e., incisor, canine, pre-molar and molar) per subject that exhibit moderate to severe PD. Disease free and highly diseased tooth-sites cause these proportion responses to lie in the closed interval [0, 1]. In addition, PD may be spatially referenced, i.e., the disease status of a site is influenced by its neighbours. While beta regression can assess the covariate-response relationship for proportion data, its support in the interval (0, 1) impairs its ability to account for the observed proportions at zero and one. In contrast to ad hoc transformations that confine responses to (0, 1), we develop a framework that augments the beta density with non-zero masses at zero and one while also controlling for spatial referencing. Our approach is Bayesian and is computationally amenable to available software. A simulation study evaluates estimation of regression effects in scenarios of varying sample size, degree of spatial dependence and response transformations. Application to real PD data provide insights into assessing covariate effects on proportion responses.

Augmented mixed beta regression models for periodontal proportion data

Continuous (clustered) proportion data often arise in various domains of medicine and public health where the response variable of interest is a proportion (or percentage) quantifying disease status for the cluster units, ranging between zero and one. However, because of the presence of relatively disease-free as well as heavily diseased subjects in any study, the proportion values can lie in the interval [0,1]. While beta regression can be adapted to assess covariate effects in these situations, its versatility is often challenged because of the presence/excess of zeros and ones because the beta support lies in the interval (0,1). To circumvent this, we augment the probabilities of zero and one with the beta density, controlling for the clustering effect. Our approach is Bayesian with the ability to borrow information across various stages of the complex model hierarchy and produces a computationally convenient framework amenable to available freeware. The marginal likelihood is tractable and can be used to develop Bayesian case-deletion influence diagnostics based on q-divergence measures. Both simulation studies and application to a real dataset from a clinical periodontology study quantify the gain in model fit and parameter estimation over other ad hoc alternatives and provide quantitative insight into assessing the true covariate effects on the proportion responses.

Joint modeling of multiple longitudinal patient-reported outcomes and survival

Researchers often include patient-reported outcomes (PROs) in Phase III clinical trials to demonstrate the value of treatment from the patient's perspective. These data are collected as longitudinal repeated measures and are often censored by occurrence of a clinical event that defines a survival time. Hierarchical Bayesian models having latent individual-level trajectories provide a flexible approach to modeling such multiple outcome types simultaneously. We consider the case of many zeros in the longitudinal data motivating a mixture model, and demonstrate several approaches to modeling multiple longitudinal PROs with survival in a cancer clinical trial. These joint models may enhance Phase III analyses and better inform health care decision makers.

A bayesian two-part latent class model for longitudinal medical expenditure data: assessing the impact of mental health and substance abuse parity

In 2001, the U.S. Office of Personnel Management required all health plans participating in the Federal Employees Health Benefits Program to offer mental health and substance abuse benefits on par with general medical benefits. The initial evaluation found that, on average, parity did not result in either large spending increases or increased service use over the four-year observational period. However, some groups of enrollees may have benefited from parity more than others. To address this question, we propose a Bayesian two-part latent class model to characterize the effect of parity on mental health use and expenditures. Within each class, we fit a two-part random effects model to separately model the probability of mental health or substance abuse use and mean spending trajectories among those having used services. The regression coefficients and random effect covariances vary across classes, thus permitting class-varying correlation structures between the two components of the model. Our analysis identified three classes of subjects: a group of low spenders that tended to be male, had relatively rare use of services, and decreased their spending pattern over time; a group of moderate spenders, primarily female, that had an increase in both use and mean spending after the introduction of parity; and a group of high spenders that tended to have chronic service use and constant spending patterns. By examining the joint 95% highest probability density regions of expected changes in use and spending for each class, we confirmed that parity had an impact only on the moderate spender class.

Longitudinal patterns of viremia and oral shedding of rhesus rhadinovirus and retroperitoneal fibromatosis herpesviruses in age-structured captive breeding populations of rhesus Macaques (Macaca mulatta)

Rhesus rhadinovirus (RRV) and retroperitoneal fibromatosis herpesvirus (RFHV), 2 closely related γ2 herpesviruses, are endemic in breeding populations of rhesus macaques at our institution. We previously reported significantly different prevalence levels, suggesting the transmission dynamics of RRV and RFHV differ with regard to viral shedding and infectivity. We designed a longitudinal study to further examine the previously observed differences between RRV and RFHV prevalence and the potential influence of age, season, and housing location on the same 90 rhesus macaques previously studied. Virus- and host-genome-specific real-time PCR assays were used to determine viral loads for both RRV and RFHV in blood and saliva samples collected at 6 time points over an 18-mo period. Proportions of positive animals and viral load in blood and saliva were compared between and within viruses by age group, location, and season by using 2-part longitudinal modeling with Bayesian inferences. Our results demonstrate that age and season are significant determinants, with age as the most significant factor analyzed, of viremia and oral shedding for both RRV and RFHV, and these pathogens exhibit distinctly different patterns of viremia and oral shedding over time within a single population.

Unified Computational Methods for Regression Analysis of Zero-Inflated and Bound-Inflated Data

Bounded data with excess observations at the boundary are common in many areas of application. Various individual cases of inflated mixture models have been studied in the literature for bound-inflated data, yet the computational methods have been developed separately for each type of model. In this article we use a common framework for computing these models, and expand the range of models for both discrete and semi-continuous data with point inflation at the lower boundary. The quasi-Newton and EM algorithms are adapted and compared for estimation of model parameters. The numerical Hessian and generalized Louis method are investigated as means for computing standard errors after optimization. Correlated data are included in this framework via generalized estimating equations. The estimation of parameters and effectiveness of standard errors are demonstrated through simulation and in the analysis of data from an ultrasound bioeffect study. The unified approach enables reliable computation for a wide class of inflated mixture models and comparison of competing models.