Longitudinal Semicontinuous Data Research Articles

Modelling occurrence and quantity of longitudinal semicontinuous data simultaneously with nonparametric unobserved heterogeneity

AbstractSemicontinuous data frequently occur in longitudinal studies. The popular two‐part modelling approach deals with longitudinal semicontinuous data by analyzing the occurrence of positive values and the intensity of positive values separately; however, this separation may break down the natural sequence of semicontinuous data within a subject and destroy its serial dependence structure. In this article, we introduce a Tweedie compound Poisson mixed model to study the occurrence of positive values and the quantity of the semicontinuous response simultaneously. In our approach, covariate effects on the semicontinuous response are assessed directly. The correlation within a subject and the unobserved heterogeneity are incorporated with serially correlated nonparametric random effects. Our model unifies subject‐specific and population‐averaged interpretations. We illustrate the approach with applications to a Brief Symptom Inventory study and an infants' fluoride intake study.

Bayesian Analysis of Tweedie Compound Poisson Partial Linear Mixed Models with Nonignorable Missing Response and Covariates

Under the Bayesian framework, this study proposes a Tweedie compound Poisson partial linear mixed model on the basis of Bayesian P-spline approximation to nonparametric function for longitudinal semicontinuous data in the presence of nonignorable missing covariates and responses. The logistic regression model is simultaneously used to specify the missing response and covariate mechanisms. A hybrid algorithm combining the Gibbs sampler and the Metropolis-Hastings algorithm is employed to produce the joint Bayesian estimates of unknown parameters and random effects as well as nonparametric function. Several simulation studies and a real example relating to the osteoarthritis initiative data are presented to illustrate the proposed methodologies.

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.

Marginalized Two-Part Joint Modeling of Longitudinal Semi-Continuous Responses and Survival Data: With Application to Medical Costs

Non-negative continuous outcomes with a substantial number of zero values and incomplete longitudinal follow-up are quite common in medical costs data. It is thus critical to incorporate the potential dependence of survival status and longitudinal medical costs in joint modeling, where censorship is death-related. Despite the wide use of conventional two-part joint models (CTJMs) to capture zero-inflation, they are limited to conditional interpretations of the regression coefficients in the model’s continuous part. In this paper, we propose a marginalized two-part joint model (MTJM) to jointly analyze semi-continuous longitudinal costs data and survival data. We compare it to the conventional two-part joint model (CTJM) for handling marginal inferences about covariate effects on average costs. We conducted a series of simulation studies to evaluate the superior performance of the proposed MTJM over the CTJM. To illustrate the applicability of the MTJM, we applied the model to a set of real electronic health record (EHR) data recently collected in Iran. We found that the MTJM yielded a smaller standard error, root-mean-square error of estimates, and AIC value, with unbiased parameter estimates. With this MTJM, we identified a significant positive correlation between costs and survival, which was consistent with the simulation results.

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.

An imputation approach for fitting two-part mixed effects models for longitudinal semi-continuous data.

Two-part mixed effects models are often used for analyzing longitudinal data with many zeros. Typically, these models are formulated with binary and continuous components separately with random effects that are correlated between the two components. Researchers have developed maximum-likelihood and Bayesian approaches for fitting these models that often require using particular software packages or very specialized software. We propose an imputation approach that will allow practitioners to separately use standard linear and generalized linear mixed models to estimate the fixed effects for two-part mixed effects models with complex random effects structures. An approximation to the conditional distribution of positive measurements given an individual's pattern of non-zero measurements is proposed that can be easily estimated and then imputed from. We show that for a wide range of parameter values, the imputation approach results in nearly unbiased estimation and can be implemented with standard software. We illustrate the proposed imputation approach for the analysis of longitudinal clinical trial data with many zeros.

Two-part mixed-effects location scale models.

Longitudinal time use data afford the opportunity to study within- and between-individual differences, but can present challenges in data analysis. Often the response set includes a large number of zeros representing those who did not engage in the target behavior. Coupled with this is a continuous measure of time use for those who did engage. The latter is strictly positive and skewed to the right if relatively few individuals engage in the behavior to a greater extent. Data analysis is further complicated for repeated measures, because within-individual responses are typically correlated, and some respondents may have missing data. This combination of zeros and positive responses is characteristic of a type of semicontinuous data in which the response is equal to a discrete value and is otherwise continuous. Two-part models have been successfully applied to cross-sectional time use data when the research goals distinguish between a respondent's likelihood to engage in a behavior and the time spent conditional on any time being spent, as these models allow different covariates to relate to each distinct aspect of a behavior. Two-part mixed-effects models extend two-part models for analysis of longitudinal semicontinuous data to simultaneously address longitudinal decisions to engage in a behavior and time spent conditional on any time spent. Heterogeneity between and within individuals can be studied in unique ways. This paper presents applications of these models to daily diary data to study individual differences in time spent relaxing or engaged in leisure activities for an adult sample.

Analysis of longitudinal semicontinuous data using marginalized two-part model

BackgroundConnective tissue growth factor (CTGF), is a secreted matricellular factor that has been linked to increased risk of cardiovascular disease in diabetic subjects. Despite the biological role of CTGF in diabetes, it still remains unclear how CTGF expression is regulated. In this study, we aim to identify the clinical parameters that modulate plasma CTGF levels measured longitudinally in type 1 diabetic patients over a period of 10 years. A number of patients had negligible measured values of plasma CTGF that formed a point mass at zero, whereas others had high positive values of CTGF that were measured on a continuous scale. The observed combination of excessive zero and continuous positively distributed non-zero values in the CTGF outcome is referred to as semicontinuous data.MethodsWe propose a novel application of a marginalized two-part model (mTP) extended to accommodate longitudinal semicontinuous data in which the marginal mean is expressed in terms of the covariates and estimates of their effect on the mean responses are generated. The continuous component is assumed to follow distributions that stem from the generalized gamma family whereas the binary measure is analyzed using logistic model and both have correlated random effects. Other approaches including the one- and two-part with uncorrelated and correlated random effects models were also applied and their estimates were all compared.ResultsOur results using the mTP model identified intensive glucose control treatment and smoking as clinical factors that were associated with decreased and increased odds of observing non-zero CTGF values respectively. In addition, hemoglobin A1c, systolic blood pressure, and high density lipoprotein were all shown to be significant risk factors that contribute to increasing CTGF levels. These findings were consistently observed under the mTP model but varied with the distributions for the other models. Accuracy and precision of the mTP model was further validated using simulation studies.ConclusionThe mTP model identified new clinical determinants that modulate the levels of CTGF in diabetic subjects. Applicability of this approach can be extended to other biomarkers measured in patient populations that display a combination of negligible zero and non-zero values.

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.

A corrected formulation for marginal inference derived from two-part mixed models for longitudinal semi-continuous data

For semi-continuous data which are a mixture of true zeros and continuously distributed positive values, the use of two-part mixed models provides a convenient modelling framework. However, deriving population-averaged (marginal) effects from such models is not always straightforward. Su et al. presented a model that provided convenient estimation of marginal effects for the logistic component of the two-part model but the specification of marginal effects for the continuous part of the model presented in that paper was based on an incorrect formulation. We present a corrected formulation and additionally explore the use of the two-part model for inferences on the overall marginal mean, which may be of more practical relevance in our application and more generally.

A marginalized two-part model for longitudinal semicontinuous data.

In health services research, it is common to encounter semicontinuous data, characterized by a point mass at zero followed by a right-skewed continuous distribution with positive support. Examples include health expenditures, in which the zeros represent a subpopulation of patients who do not use health services, while the continuous distribution describes the level of expenditures among health services users. Longitudinal semicontinuous data are typically analyzed using two-part random-effect mixtures with one component that models the probability of health services use, and a second component that models the distribution of log-scale positive expenditures among users. However, because the second part conditions on a non-zero response, obtaining interpretable effects of covariates on the combined population of health services users and non-users is not straightforward, even though this is often of greatest interest to investigators. Here, we propose a marginalized two-part model for longitudinal data that allows investigators to obtain the effect of covariates on the overall population mean. The model additionally provides estimates of the overall population mean on the original, untransformed scale, and many covariates take a dual population average and subject-specific interpretation. Using a Bayesian estimation approach, this model maintains the flexibility to include complex random-effect structures and easily estimate functions of the overall mean. We illustrate this approach by evaluating the effect of a copayment increase on health care expenditures in the Veterans Affairs health care system over a four-year period.

Two-Part Pattern-Mixture Model for Longitudinal Incomplete Semi-Continuous Toenail Data

Longitudinal data with true zero values, known as longitudinal semi-continuous data, frequently occur in medical, environmental and biological studies. To model longitudinal semi-continuous data, two-part modelling approaches have been widely used in literature. In the first part of the two-part model, binary logistic regression is commonly used after converting the semi-continuous responses to binary responses. In the second part, the semi-continuous data are converted to positive continuous data after removing the true zero values from the responses. Although positive continuous or non-zero values tend to show a positively skewed distribution pattern, in the literature the normal distribution is commonly used to model them. Also, in longitudinal studies, data often suffer individual dropouts as they are collected overtime. In this paper, we propose a two-part pattern-mixture model to analyze longitudinal semi-continuous data with dropouts. In the proposed approach, we use pattern-mixture binary mixed models for the first part and positively continuous pattern-mixture gamma mixed models for the second part. Our approach can accommodate both subject- and time-specific correlation as well as dropout pattern. We also incorporate a computationally efficient estimation method for our models using a penalize quasi-likelihood approach. The proposed method is illustrated with an application to the longitudinal incomplete toenail data

A likelihood-based two-part marginal model for longitudinal semicontinuous data

Two-part models are an attractive approach for analysing longitudinal semicontinuous data consisting of a mixture of true zeros and continuously distributed positive values. When the population-averaged (marginal) covariate effects are of interest, two-part models that provide straightforward interpretation of the marginal effects are desirable. Presently, the only available approaches for fitting two-part marginal models to longitudinal semicontinuous data are computationally difficult to implement. Therefore, there exists a need to develop two-part marginal models that can be easily implemented in practice. We propose a fully likelihood-based two-part marginal model that satisfies this need by using the bridge distribution for the random effect in the binary part of an underlying two-part mixed model; and its maximum likelihood estimation can be routinely implemented via standard statistical software such as the SAS NLMIXED procedure. We illustrate the usage of this new model by investigating the marginal effects of pre-specified genetic markers on physical functioning, as measured by the Health Assessment Questionnaire, in a cohort of psoriatic arthritis patients from the University of Toronto Psoriatic Arthritis Clinic. An added benefit of our proposed marginal model when compared to a two-part mixed model is the robustness in regression parameter estimation when departure from the true random effects structure occurs. This is demonstrated through simulation.

Bias in 2-part mixed models for longitudinal semicontinuous data

Semicontinuous data in the form of a mixture of zeros and continuously distributed positive values frequently arise in biomedical research. Two-part mixed models with correlated random effects are an attractive approach to characterize the complex structure of longitudinal semicontinuous data. In practice, however, an independence assumption about random effects in these models may often be made for convenience and computational feasibility. In this article, we show that bias can be induced for regression coefficients when random effects are truly correlated but misspecified as independent in a 2-part mixed model. Paralleling work on bias under nonignorable missingness within a shared parameter model, we derive and investigate the asymptotic bias in selected settings for misspecified 2-part mixed models. The performance of these models in practice is further evaluated using Monte Carlo simulations. Additionally, the potential bias is investigated when artificial zeros, due to left censoring from some detection or measuring limit, are incorporated. To illustrate, we fit different 2-part mixed models to the data from the University of Toronto Psoriatic Arthritis Clinic, the aim being to examine whether there are differential effects of disease activity and damage on physical functioning as measured by the health assessment questionnaire scores over the course of psoriatic arthritis. Some practical issues on variance component estimation revealed through this data analysis are considered.

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

In many biomedical applications, researchers encounter semicontinuous data where data are either continuous or zero. When the data are collected over time the observations may be correlated. Analysis of this kind of longitudinal semicontinuous data is challenging due to the presence of strong skewness in the data. A flexible class of zero-inflated models in a longitudinal setting is developed. A Bayesian approach is used to analyze longitudinal data from an acupuncture clinical trial, in which the effects of active acupuncture, sham acupuncture and standard medical care is compared on chemotherapy-induced nausea in patients who were treated for advanced breast cancer. A spline model is introduced into the linear predictor of the model to explore the possibility of a nonlinear treatment effect. Possible serial correlation between successive observations is also accounted using the Brownian motion. Thus, the approach taken in this paper provides for a more flexible modeling framework and, with the use of WinBUGS, provides for a computationally simpler approach than direct maximum-likelihood. The Bayesian methodology is illustrated with the acupuncture clinical trial data.