Complex Longitudinal Data Research Articles

Segmental modeling of viral load changes for HIV longitudinal data with skewness and detection limits

Although it is a common practice to analyze complex HIV longitudinal data using nonlinear mixed-effects or nonparametric mixed-effects models in literature, the following issues may standout. (i) In clinical practice, the profile of each subject's viral response may follow a 'broken-stick'-like trajectory, indicating multiple phases of decline and increase in response. Such multiple phases (change points) may be an important indicator to help quantify treatment effect and improve management of patient care. To estimate change points, nonlinear mixed-effects or nonparametric mixed-effects models become a challenge because of complicated structures of model formulations. (ii) The commonly assumed distribution for model random errors is normal, but this assumption may unrealistically obscure important features of subject variations. (iii) The response observations (viral load) may be subject to left censoring due to a limit of detection. Inferential procedures can be complicated dramatically when data with asymmetric (skewed) characteristics and left censoring are observed in conjunction with change points as unknown parameters into models. There is relatively little work concerning all these features simultaneously. This article proposes segmental mixed-effects models with skew distributions for the response process (with left censoring) under a Bayesian framework. A real data example is used to illustrate the proposed methods.

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Model uncertainty and multimodel inference in reliability estimation within a longitudinal framework

Laenen, Alonso, and Molenberghs (2007) and Laenen, Alonso, Molenberghs, and Vangeneugden (2009) proposed a method to assess the reliability of rating scales in a longitudinal context. The methodology is based on hierarchical linear models, and reliability coefficients are derived from the corresponding covariance matrices. However, finding a good parsimonious model to describe complex longitudinal data is a challenging task. Frequently, several models fit the data equally well, raising the problem of model selection uncertainty. When model uncertainty is high one may resort to model averaging, where inferences are based not on one but on an entire set of models. We explored the use of different model building strategies, including model averaging, in reliability estimation. We found that the approach introduced by Laenen et al. (2007, 2009) combined with some of these strategies may yield meaningful results in the presence of high model selection uncertainty and when all models are misspecified, in so far as some of them manage to capture the most salient features of the data. Nonetheless, when all models omit prominent regularities in the data, misleading results may be obtained. The main ideas are further illustrated on a case study in which the reliability of the Hamilton Anxiety Rating Scale is estimated. Importantly, the ambit of model selection uncertainty and model averaging transcends the specific setting studied in the paper and may be of interest in other areas of psychometrics.

Mixed-effects regression modeling of real-time momentary pain assessments in osteoarthritis (OA) patients

Mixed-effects regression models have extraordinary capacity to capture the essence of complex longitudinal data. In this paper we present analyses from a longitudinal study on 157 patients with osteoarthritis (OA) where real-time momentary self-reported pain and physical activity level were collected. We illustrate the application of heterogeneous mixed effect (HME) models to examine the effect of time-varying activity level assessments (sedentary vs. moderate/vigorous) during the 30 min prior to the pain recording on mean pain levels and variability. We also examine the variation in pain associated with activity level and the degree that participant characteristics (weight-bearing vs. non-weight bearing joint w/OA, body mass index, white vs. non-white race, gender) and psychosocial factors (mood, tension, pain catastrophizing) influence pain variation. We fit separate HME models for each characteristic or factor. These HME models are fit with standard statistical software, but there are challenges to fitting and interpreting these models. Once fit the models provide a powerful tool to help guide researchers in understanding highly individualized pain trajectories.

Investigating Stage-Sequential Growth Mixture Models with Multiphase Longitudinal Data

This article investigates three types of stage-sequential growth mixture models in the structural equation modeling framework for the analysis of multiple-phase longitudinal data. These models can be important tools for situations in which a single-phase growth mixture model produces distorted results and can allow researchers to better understand population heterogeneity and growth over multiple phases. Through theoretical and empirical comparisons of the models, the authors discuss strategies with respect to model selection and interpreting outcomes. The unique attributes of each approach are illustrated using ecological momentary assessment data from a tobacco cessation study. Transitional discrepancy between phases as well as growth factors are examined to see whether they can give us useful information related to a distal outcome, abstinence at 6 months postquit. It is argued that these statistical models are powerful and flexible tools for the analysis of complex and detailed longitudinal data.

The parametric g‐formula to estimate the effect of highly active antiretroviral therapy on incident AIDS or death

The parametric g-formula can be used to contrast the distribution of potential outcomes under arbitrary treatment regimes. Like g-estimation of structural nested models and inverse probability weighting of marginal structural models, the parametric g-formula can appropriately adjust for measured time-varying confounders that are affected by prior treatment. However, there have been few implementations of the parametric g-formula to date. Here, we apply the parametric g-formula to assess the impact of highly active antiretroviral therapy on time to acquired immune deficiency syndrome (AIDS) or death in two US-based human immunodeficiency virus cohorts including 1498 participants. These participants contributed approximately 7300 person-years of follow-up (49% exposed to highly active antiretroviral therapy) during which 382 events occurred and 259 participants were censored because of dropout. Using the parametric g-formula, we estimated that antiretroviral therapy substantially reduces the hazard of AIDS or death (hazard ratio = 0.55; 95% confidence limits [CL]: 0.42, 0.71). This estimate was similar to one previously reported using a marginal structural model, 0.54 (95% CL: 0.38, 0.78). The 6.5-year difference in risk of AIDS or death was 13% (95% CL: 8%, 18%). Results were robust to assumptions about temporal ordering, and extent of history modeled, for time-varying covariates. The parametric g-formula is a viable alternative to inverse probability weighting of marginal structural models and g-estimation of structural nested models for the analysis of complex longitudinal data.

Simultaneous Bayesian Inference for Longitudinal Data with Asymmetry, Left-censoring and Covariates Measured with Errors

It is a common practice to analyze complex longitudinal data using flexible nonlinear mixed-effects (NLME) models with normality assumption. However, a serious departure of normality may cause lack of robustness and subsequently lead to invalid inference and unreasonable estimates. Covariates are usually introduced in such models to partially explain inter-subject variations, but some covariates may be often measured with substantial errors. Moreover, the response observations may be subject to left-censoring due to a detection limit. Inferential procedures can be complicated dramatically when data with asymmetric (skewed) characteristics, left-censoring and measurement errors are observed. In the literature, there has been considerable interest in accommodating either skewness, censoring or covariate measurement errors in such models, but there is relatively little work concerning all of the three features simultaneously. In this article, we jointly investigate a skew-t NLME model for response (with left-censoring) process and a skew-t nonparametric mixed-effects model for covariate (with measurement errors) process. We propose a robust skew-t Bayesian modeling approach in a general form to analyze data in capturing the effects of skewness, censoring and measurement errors in covariates simultaneously. A real data example is offered to illustrate the methodologies. The proposed modeling alternative offers important advantages in the sense that the model can be easily fitted in freely available software and the computational effort for the model with a skew-t distribution is almost equivalent to that of the model with a standard normal distribution.

Bayesian Semiparametric Nonlinear Mixed‐Effects Joint Models for Data with Skewness, Missing Responses, and Measurement Errors in Covariates

It is a common practice to analyze complex longitudinal data using semiparametric nonlinear mixed-effects (SNLME) models with a normal distribution. Normality assumption of model errors may unrealistically obscure important features of subject variations. To partially explain between- and within-subject variations, covariates are usually introduced in such models, but some covariates may often be measured with substantial errors. Moreover, the responses may be missing and the missingness may be nonignorable. Inferential procedures can be complicated dramatically when data with skewness, missing values, and measurement error are observed. In the literature, there has been considerable interest in accommodating either skewness, incompleteness or covariate measurement error in such models, but there has been relatively little study concerning all three features simultaneously. In this article, our objective is to address the simultaneous impact of skewness, missingness, and covariate measurement error by jointly modeling the response and covariate processes based on a flexible Bayesian SNLME model. The method is illustrated using a real AIDS data set to compare potential models with various scenarios and different distribution specifications.

Mixed-Effects Tobit Joint Models for Longitudinal Data with Skewness, Detection Limits, and Measurement Errors

Complex longitudinal data are commonly analyzed using nonlinear mixed-effects (NLME) models with a normal distribution. However, a departure from normality may lead to invalid inference and unreasonable parameter estimates. Some covariates may be measured with substantial errors, and the response observations may also be subjected to left-censoring due to a detection limit. Inferential procedures can be complicated dramatically when such data with asymmetric characteristics, left censoring, and measurement errors are analyzed. There is relatively little work concerning all of the three features simultaneously. In this paper, we jointly investigate a skew-tNLME Tobit model for response (with left censoring) process and a skew-tnonparametric mixed-effects model for covariate (with measurement errors) process under a Bayesian framework. A real data example is used to illustrate the proposed methods.

Making predictions from complex longitudinal data, with application to planning monitoring intervals in a national screening programme.

When biological or physiological variables change over time, we are often interested in making predictions either of future measurements or of the time taken to reach some threshold value. On the basis of longitudinal data for multiple individuals, we develop Bayesian hierarchical models for making these predictions together with their associated uncertainty. Particular aspects addressed, which include some novel components, are handling curvature in individuals’ trends over time, making predictions for both underlying and measured levels, making predictions from a single baseline measurement, making predictions from a series of measurements, allowing flexibility in the error and random-effects distributions, and including covariates. In the context of data on the expansion of abdominal aortic aneurysms over time, where reaching a certain threshold leads to referral for surgery, we discuss the practical application of these models to the planning of monitoring intervals in a national screening programme. Prediction of the time to reach a threshold was too imprecise to be practically useful, and we focus instead on limiting the probability of exceeding the threshold after given time intervals. Although more complex models can be shown to fit the data better, we find that relatively simple models seem to be adequate for planning monitoring intervals.

Simultaneous Bayesian Inference for Linear, Nonlinear and Semiparametric Mixed-Effects Models with Skew-Normality and Measurement Errors in Covariates

In recent years, various mixed-effects models have been suggested for estimating viral decay rates in HIV dynamic models for complex longitudinal data. Among those models are linear mixed-effects (LME), nonlinear mixed-effects (NLME), and semiparametric nonlinear mixed-effects (SNLME) models. However, a critical question is whether these models produce coherent estimates of viral decay rates, and if not, which model is appropriate and should be used in practice. In addition, one often assumes that a model random error is normally distributed, but the normality assumption may be unrealistic, particularly if the data exhibit skewness. Moreover, some covariates such as CD4 cell count may be often measured with substantial errors. This paper addresses these issues simultaneously by jointly modeling the response variable with skewness and a covariate process with measurement errors using a Bayesian approach to investigate how estimated parameters are changed or different under these three models. A real data set from an AIDS clinical trial study was used to illustrate the proposed models and methods. It was found that there was a significant incongruity in the estimated decay rates in viral loads based on the three mixed-effects models, suggesting that the decay rates estimated by using Bayesian LME or NLME joint models should be interpreted differently from those estimated by using Bayesian SNLME joint models. The findings also suggest that the Bayesian SNLME joint model is preferred to other models because an arbitrary data truncation is not necessary; and it is also shown that the models with a skew-normal distribution and/or measurement errors in covariate may achieve reliable results when the data exhibit skewness.

A Bayesian Approach to Joint Mixed-Effects Models with a Skew-Normal Distribution and Measurement Errors in Covariates

In recent years, nonlinear mixed-effects (NLME) models have been proposed for modeling complex longitudinal data. Covariates are usually introduced in the models to partially explain intersubject variations. However, one often assumes that both model random error and random effects are normally distributed, which may not always give reliable results if the data exhibit skewness. Moreover, some covariates such as CD4 cell count may be often measured with substantial errors. In this article, we address these issues simultaneously by jointly modeling the response and covariate processes using a Bayesian approach to NLME models with covariate measurement errors and a skew-normal distribution. A real data example is offered to illustrate the methodologies by comparing various potential models with different distribution specifications. It is showed that the models with skew-normality assumption may provide more reasonable results if the data exhibit skewness and the results may be important for HIV/AIDS studies in providing quantitative guidance to better understand the virologic responses to antiretroviral treatment.

Some asymptotic results for semiparametric nonlinear mixed-effects models with incomplete data

In modeling complex longitudinal data, semiparametric nonlinear mixed-effects (SNLME) models are very flexible and useful. Covariates are often introduced in the models to partially explain the inter-individual variations. In practice, data are often incomplete in the sense that there are often measurement errors and missing data in longitudinal studies. The likelihood method is a standard approach for inference for these models but it can be computationally very challenging, so computationally efficient approximate methods are quite valuable. However, the performance of these approximate methods is often based on limited simulation studies, and theoretical results are unavailable for many approximate methods. In this article, we consider a computationally efficient approximate method for a class of SNLME models with incomplete data and investigate its theoretical properties. We show that the estimates based on the approximate method are consistent and asymptotically normally distributed.

Simultaneous Inference for Semiparametric Nonlinear Mixed‐Effects Models with Covariate Measurement Errors and Missing Responses

Semiparametric nonlinear mixed-effects (NLME) models are flexible for modeling complex longitudinal data. Covariates are usually introduced in the models to partially explain interindividual variations. Some covariates, however, may be measured with substantial errors. Moreover, the responses may be missing and the missingness may be nonignorable. We propose two approximate likelihood methods for semiparametric NLME models with covariate measurement errors and nonignorable missing responses. The methods are illustrated in a real data example. Simulation results show that both methods perform well and are much better than the commonly used naive method.

Statistical Analysis of Longitudinal Network Data With Changing Composition

Markov chains can be used for the modeling of complex longitudinal network data. One class of probability models to model the evolution of social networks are stochastic actor-oriented models for network change proposed by Snijders. These models are continuous-time Markov chain models that are implemented as simulation models. The authors propose an extension of the simulation algorithm of stochastic actor-oriented models to include networks of changing composition. In empirical research, the composition of networks may change due to actors joining or leaving the network at some point in time. The composition changes are modeled as exogenous events that occur at given time points and are implemented in the simulation algorithm. The estimation of the network effects, as well as the effects of actor and dyadic attributes that influence the evolution of the network, is based on the simulation of Markov chains.

Causal Inference for Complex Longitudinal Data: The Continuous Case

We extend Robins’ theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments. In particular we establish versions of the key results of the discrete theory: the $g$-computation formula and a collection of powerful characterizations of the $g$-null hypothesis of no treatment effect. This is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the covariates given the past. We also show that our assumptions concerning counterfactual variables place no restriction on the joint distribution of the observed variables: thus in a precise sense, these assumptions are “for free,” or if you prefer, harmless.

SOME MODELS FOR OVERDISPERSED BINOMIAL DATA

SummaryVarious models are currently used to model overdispersed binomial data. It is not always clear which model is appropriate for a given situation. Here we examine the assumptions and discuss the problems and pitfalls of some of these models. We focus on clustered data with one level of nesting, briefly touching on more complex strata and longitudinal data. The estimation procedures are illustrated and some critical comments are made about the various models. We indicate which models are restrictive and how and which can be extended to model more complex situations. In addition some inadequacies in testing procedures are noted. Recommendations as to which models should be used, and when, are made.

Registration of longitudinal data by means of musical notation

There is a lack of good registration methods for complex longitudinal data. Cumulating longitudinal data into cross-sections, even repeated ones, in order to allow for statistical analysis means considerable distortion of data; if they interact with each other, this is not possible to observe directly and the time factor is not taken into account to its full value, if at all. In this context, music may be defined as a complicated development in the course of time. Music is usually represented by musical notation which, if not ideal, means that we already possess a high precision registration method for complicated temporal developments. Musical notation used for registration of other longitudinal variables than musical ones allows for a multitude of qualitative variables being taken into account simultaneously and a high precision regarding the time factor. It should be no more difficult to analyse statistically a longitudinal pattern than a surface pattern.

Multilevel mixed linear model analysis using iterative generalized least squares

SUMMARY Models for the analysis of hierarchically structured data are discussed. An iterative generalized least squares estimation procedure is given and shown to be equivalent to maximum likelihood in the normal case. There is a discussion of applications to complex surveys, longitudinal data, and estimation in multivariate models with missing responses. An example is given using educational data.