Mixed-effects Models For Longitudinal Data Research Articles

Classification of longitudinal profiles using semi-parametric nonlinear mixed models with P-Splines and the SAEM algorithm.

In this work, we propose an extension of a semiparametric nonlinear mixed-effects model for longitudinal data that incorporates more flexibility with penalized splines (P-splines) as smooth terms. The novelty of the proposed approach consists of the formulation of the model within the stochastic approximation version of the EM algorithm for maximum likelihood, the so-called SAEM algorithm. The proposed approach takes advantage of the formulation of a P-spline as a mixed-effects model and the use of the computational advantages of the existing software for the SAEM algorithm for the estimation of the random effects and the variance components. Additionally, we developed a supervised classification method for these non-linear mixed models using an adaptive importance sampling scheme. To illustrate our proposal, we consider two studies on pregnant women where two biomarkers are used as indicators of changes during pregnancy. In both studies, information about the women's pregnancy outcomes is known. Our proposal provides a unified framework for the classification of longitudinal profiles that may have important implications for the early detection and monitoring of pregnancy-related changes and contribute to improved maternal and fetal health outcomes. We show that the proposed models improve the analysis of this type of data compared to previous studies. These improvements are reflected both in the fit of the models and in the classification of the groups.

Modeling biomarker variability in joint analysis of longitudinal and time-to-event data.

The role of visit-to-visit variability of a biomarker in predicting related disease has been recognized in medical science. Existing measures of biological variability are criticized for being entangled with random variability resulted from measurement error or being unreliable due to limited measurements per individual. In this article, we propose a new measure to quantify the biological variability of a biomarker by evaluating the fluctuation of each individual-specific trajectory behind longitudinal measurements. Given a mixed-effects model for longitudinal data with the mean function over time specified by cubic splines, our proposed variability measure can be mathematically expressed as a quadratic form of random effects. A Cox model is assumed for time-to-event data by incorporating the defined variability as well as the current level of the underlying longitudinal trajectory as covariates, which, together with the longitudinal model, constitutes the joint modeling framework in this article. Asymptotic properties of maximum likelihood estimators are established for the present joint model. Estimation is implemented via an Expectation-Maximization (EM) algorithm with fully exponential Laplace approximation used in E-step to reduce the computation burden due to the increase of the random effects dimension. Simulation studies are conducted to reveal the advantage of the proposed method over the two-stage method, as well as a simpler joint modeling approach which does not take into account biomarker variability. Finally, we apply our model to investigate the effect of systolic blood pressure variability on cardiovascular events in the Medical Research Council elderly trial, which is also the motivating example for this article.

Longitudinal Changes in Vision and Retinal Morphology in Wolfram Syndrome

To report long-term ophthalmic findings in Wolfram syndrome, including rates of visual decline, macular thinning, retinal nerve fiber layer (RNFL) thinning, and outer plexiform layer (OPL) lamination. Single-center, cohort study. A total of 38 participants were studied, who underwent a complete ophthalmic examination as well as optical coherence tomography imaging of the macula and nerve on an annual basis. Linear mixed-effects models for longitudinal data were used to examine both fixed and random effects related to visual acuity and optic nerve quadrants of RNFL and macula thickness. Participants completed a mean of 6.44 years of follow-up (range 2-10 years). Visual acuity declined over time in all participants, with a mean slope of 0.059 logMAR/y (95% CI=0.07-0.05 logMAR/y), although nearly 25% of participants experienced more rapid visual decline. RNFL thickness decreased in superior, inferior, and nasal quadrants (β=-0.5 µm/y, -0.98 µm/y, -0.28 µm/y, respectively). OPL lamination was noted in 3 study participants, 2 of whom had autosomal dominant mutations. Our study describes the longest and largest natural history study of visual acuity decline and retinal morphometry in Wolfram syndrome to date. Results suggest that there are slower and faster progressing subgroups and that OPL lamination is present in some individuals with this disease.

Bias correction for multiple covariate analysis using empirical bayesian estimation in mixed-effects models for longitudinal data

The naïve empirical Bayes method has been widely used as an ad hoc tool in fitting linear mixed-effect models, which is much computationally efficient than the maximum likelihood estimation method. However, the shrinkage effect of the empirical Bayes method causes bias in the estimates of the fixed effects. Bias-correction has been proposed for the mixed-effects model when only one covariate is present. In this paper, we derive the shrinkage factor of the empirical Bayes predictors of the random effects and the variance-covariance matrix of the corrected estimates when the model has more than one covariate. The empirical Bayes estimates and test statistics are then corrected using the derived factor. Theoretical derivations, simulation studies and a real data application demonstrate the validity of the proposed method in that the corrected estimates are unbiased and the corrected tests have correct p-values.

A semiparametric Bayesian approach to binomial distribution logistic mixed-effects models for longitudinal data

Logistic mixed-effects models are widely used to study the relationship between the binary response and covariates for longitudinal data analysis, where the random effects are typically assumed to have a fully parametric distribution. As this assumption is likely limited or unreasonable in a multitude of practical researches, a semiparametric Bayesian approach for relaxing it is developed in this paper. In the context of binomial distribution logistic mixed-effects models, a general Bayesian framework is presented in which a semiparametric hierarchical modelling with an approximate truncated Dirichlet process prior distribution is specified for the random effects. The stick-breaking prior and the blocked Gibbs sampler using Pólya-Gamma mixture are employed to efficiently sample in the posterior analysis. Besides, a procedure calculating DIC for Bayesian model comparison is addressed. The methodology is demonstrated through simulation studies and a real example.

Joint modelling of longitudinal response and time-to-event data using conditional distributions: a Bayesian perspective

Over the last 20 or more years a lot of clinical applications and methodological development in the area of joint models of longitudinal and time-to-event outcomes have come up. In these studies, patients are followed until an event, such as death, occurs. In most of the work, using subject-specific random-effects as frailty, the dependency of these two processes has been established. In this article, we propose a new joint model that consists of a linear mixed-effects model for longitudinal data and an accelerated failure time model for the time-to-event data. These two sub-models are linked via a latent random process. This model will capture the dependency of the time-to-event on the longitudinal measurements more directly. Using standard priors, a Bayesian method has been developed for estimation. All computations are implemented using OpenBUGS. Our proposed method is evaluated by a simulation study, which compares the conditional model with a joint model with local independence by way of calibration. Data on Duchenne muscular dystrophy (DMD) syndrome and a set of data in AIDS patients have been analysed.

Variable Selection in Semiparametric Mixed Effect Model for High-Dimension Longitudinal Data

This paper considers the problem of simultaneous variable selection and estimation in a semiparametric mixed-effects model for longitudinal data with normal errors. We approximate the nonparametric function by regression spline and simultaneously estimate and select the variables under the optimization of the penalized objective function. Under some regularity conditions, the asymptotic behaviour of the resulting estimators is established in a high-dimensional framework where the number of parametric covariates increases as the sample size increases. For practical implementation, we use an EM algorithm to selects the significant variables and estimates the nonzero coefficient functions. Simulation studies are carried out to assess the performance of our proposed method, and a real data set is analyzed to illustrate the proposed procedure.

Subgroup analysis based on structured mixed-effects models for longitudinal data

ABSTRACT In recent years, subgroup analysis has emerged as an important tool to identify unknown subgroup memberships. However, subgroup analysis is still under-studied for longitudinal data. In this paper, we propose a structured mixed-effects approach for longitudinal data to model subgroup distribution and identify subgroup membership simultaneously. In the proposed structured mixed-effects model, the heterogeneous treatment effect is modeled as a random effect from a two-component mixture model, while the membership of the mixture model is incorporated using a logistic model with respect to some covariates. One advantage of our approach is that we are able to derive the estimation of the treatment effects through an EM-type algorithm which keeps the subgroup membership unchanged over time. Our numerical studies and real data example demonstrate that the proposed model outperforms other competing methods.

Progressive left ventricular dysfunction and myocardial fibrosis in Duchenne and Becker muscular dystrophy: a longitudinal cardiovascular magnetic resonance study.

This study examined the progression of left ventricular dysfunction and myocardial fibrosis in patients with Duchenne muscular dystrophy (DMD) or Becker muscular dystrophy (BMD) to evaluate the effects of angiotensin-converting enzyme inhibitor (ACEI). Ninety-eight cardiovascular magnetic resonance (CMR) studies in 34 consecutive patients with DMD (n = 21) or BMD (n = 13) were retrospectively reviewed. Left ventricular ejection fraction (LVEF) and the extent of myocardial late gadolinium enhancement (LGE) were semiautomatically quantified. During the study period, five patients had already been treated with ACEI at the first CMR; five were started on ACEI at LVEF ≥ 55% and 10at LVEF < 55%. All patients had hyperenhanced myocardium on LGE images at the first CMR (median extent, 3.3%; interquartile range 0.1-14.3%). A mixed-effects model for longitudinal data of each patient, adjusted for age, type of muscular dystrophy, steroid use, and ACEI use showed that higher age (β=- 1.1%/year; 95% confidence interval [CI], - 1.8% to - 0.4%; p = 0.005) and no use of ACEI (β=- 3.1%; 95% CI, - 5.4% to - 0.8%; p = 0.009) were significantly associated with a lower LVEF. When ACEI use was stratified by time of initiation (LVEF ≥ 55% vs. < 55%), only ACEI initiation at LVEF < 55% had a beneficial effect on LVEF at each imaging examination (β = 3.7%; 95% CI, 0.9-6.4%; p = 0.010). ACEI use or the time of initiation of ACEI did not significantly affect age-related increase in LGE. ACEI attenuated the age-related decline in LVEF only in patients with DMD or BMD and reduced LVEF, suggesting that further investigation on prophylactic use of cardioprotective therapy in these patients is warranted.

Comparative effectiveness of tocilizumab versus TNF inhibitors as monotherapy or in combination with conventional synthetic disease-modifying antirheumatic drugs in patients with rheumatoid arthritis after the use of at least one biologic disease-modifying antirheumatic drug: analyses from the pan-European TOCERRA register collaboration

ObjectiveTo compare the effectiveness of tocilizumab (TCZ) and tumour necrosis factor (TNF) inhibitors (TNFi) as monotherapy or in combination with conventional synthetic disease-modifying antirheumatic drugs (DMARDs) in patients with rheumatoid...

Recursive Partitioning with Nonlinear Models of Change

ABSTRACTIn this article, we introduce nonlinear longitudinal recursive partitioning (nLRP) and the R package longRpart2 to carry out the analysis. This method implements recursive partitioning (also known as decision trees) in order to split data based on individual- (i.e., cluster) level covariates with the goal of predicting differences in nonlinear longitudinal trajectories. At each node, a user-specified linear or nonlinear mixed-effects model is estimated. This method is an extension of Abdolell et al.'s (2002) longitudinal recursive partitioning while permitting a nonlinear mixed-effects model in addition to a linear mixed-effects model in each node. We give an overview of recursive partitioning, nonlinear mixed-effects models for longitudinal data, describe nLRP, and illustrate its use with empirical data from the Early Childhood Longitudinal Study—Kindergarten Cohort.

Perturbation diagnostics of autocorrelation coefficients in non linear mixed-effects models with AR(1) errors based on M-estimation

ABSTRACTIn this work we propose and analyze non linear mixed-effects models for longitudinal data, which are widely used in the fields of economics, biopharmaceuticals, agriculture, and so on. A robust method to obtain maximum likelihood estimates for the parameters is presented, as well as perturbation diagnostics of autocorrelation coefficient in non linear models based on robust estimates and influence curvature. The obtained results are illustrated by plasma concentrations data presented in Davidian and Giltinan, which was analyzed under the non robust situation.

Effectiveness of tocilizumab with and without synthetic disease-modifying antirheumatic drugs in rheumatoid arthritis: results from a European collaborative study

ObjectivesTo examine the effectiveness of tocilizumab (TCZ) with and without synthetic disease-modifying antirheumatic drugs (sDMARDs) in a large observational study.MethodsPatients with rheumatoid arthritis treated with TCZ who had a baseline...

Bayesian estimation and influence diagnostics of generalized partially linear mixed-effects models for longitudinal data

This paper develops a Bayesian approach to obtain the joint estimates of unknown parameters, nonparametric functions and random effects in generalized partially linear mixed models (GPLMMs), and presents three case deletion influence measures to identify influential observations based on the φ-divergence, Cook's posterior mean distance and Cook's posterior mode distance of parameters. Fisher's iterative scoring algorithm is developed to evaluate the posterior modes of parameters in GPLMMs. The first-order approximation to Cook's posterior mode distance is presented. The computationally feasible formulae for the φ-divergence diagnostic and Cook's posterior mean distance are given. Several simulation studies and an example are presented to illustrate our proposed methodologies.

Quantile regression-based Bayesian semiparametric mixed-effects models for longitudinal data with non-normal, missing and mismeasured covariate

Quantile regression (QR) models have received increasing attention recently for longitudinal data analysis. When continuous responses appear non-centrality due to outliers and/or heavy-tails, commonly used mean regression models may fail to produce efficient estimators, whereas QR models may perform satisfactorily. In addition, longitudinal outcomes are often measured with non-normality, substantial errors and non-ignorable missing values. When carrying out statistical inference in such data setting, it is important to account for the simultaneous treatment of these data features; otherwise, erroneous or even misleading results may be produced. In the literature, there has been considerable interest in accommodating either one or some of these data features. However, there is relatively little work concerning all of them simultaneously. There is a need to fill up this gap as longitudinal data do often have these characteristics. Inferential procedure can be complicated dramatically when these data features arise in longitudinal response and covariate outcomes. In this article, our objective is to develop QR-based Bayesian semiparametric mixed-effects models to address the simultaneous impact of these multiple data features. The proposed models and method are applied to analyse a longitudinal data set arising from an AIDS clinical study. Simulation studies are conducted to assess the performance of the proposed method under various scenarios.

Problem adaptation therapy for older adults with major depression and cognitive impairment: a randomized clinical trial.

Problem adaptation therapy (PATH) is a treatment for older adults with major depression, cognitive impairment (from mild cognitive deficits to moderate dementia), and disability. Antidepressants have limited efficacy in this population and psychosocial interventions are inadequately investigated. To test the efficacy of 12-week PATH vs supportive therapy for cognitively impaired patients (ST-CI) in reducing depression and disability in 74 older adults with major depression, cognitive impairment, and disability. A randomized clinical trial at the Weill Cornell Institute of Geriatric Psychiatry from April 1, 2006, to September 31, 2011. Interventions were administered at the participants' homes. Participants included 74 older individuals (age ≥ 65 years) with major depression and cognitive impairment to the level of moderate dementia. They were recruited through collaborating community agencies of Weill Cornell Institute of Geriatric Psychiatry and were randomly assigned to 12 weekly sessions of PATH or ST-CI (14.8% attrition rate). Home-delivered PATH vs home-delivered ST-CI. Problem adaptation therapy integrates a problem-solving approach with compensatory strategies, environmental adaptations, and caregiver participation to improve patients' emotion regulation. Supportive therapy for cognitively impaired patients focuses on expression of affect, understanding, and empathy. Mixed-effects models for longitudinal data compared the efficacy of PATH with that of ST-CI in reducing depression (Montgomery-Asberg Depression Rating Scale) and disability (World Health Organization Disability Assessment Schedule II) during 12 weeks of treatment. Participants in PATH had significantly greater reduction in depression (Cohen d, 0.60; 95% CI, 0.13-1.06; treatment × time, F(1,179) = 8.03; P = .005) and disability (Cohen d, 0.67; 95% CI, 0.20-1.14; treatment × time, F(1,169) = 14.86; P = .001) than ST-CI participants during the 12-week period (primary outcomes). Furthermore, PATH participants had significantly greater depression remission rates than ST-CI participants (37.84% vs 13.51%; χ(2) = 5.74; P = .02; number needed to treat = 4.11) (secondary outcome). Problem adaptation therapy was more efficacious than ST-CI in reducing depression and disability. Problem adaptation therapy may provide relief to a large group of depressed and cognitively impaired older adults who have few treatment options. Clinicaltrials.gov Identifier: NCT00368940.

Tree-Structured Mixed-Effects Regression Modeling for Longitudinal Data

Tree-structured models have been widely used because they function as interpretable prediction models that offer easy data visualization. A number of tree algorithms have been developed for univariate response data and can be extended to analyze multivariate response data. We propose a tree algorithm by combining the merits of a tree-based model and a mixed-effects model for longitudinal data. We alleviate variable selection bias through residual analysis, which is used to solve problems that exhaustive search approaches suffer from, such as undue preference to split variables with more possible splits, expensive computational cost, and end-cut preference. Most importantly, our tree algorithm discovers trends over time on each of the subspaces from recursive partitioning, while other tree algorithms predict responses. We investigate the performance of our algorithm with both simulation and real data studies. We also develop an R package melt that can be used conveniently and freely. Additional results are provided as online supplementary material.

Kinematics of a highly congruent mobile-bearing total knee prosthesis

Limited or absent axial rotation of the mobile insert of total knee prostheses could lead to high contact stresses and stresses at the bone-implant interface, which in turn might lead to implant loosening. The aim of this study was to assess knee kinematics and muscle activation and their possible change over time in patients with a highly congruent, mobile-bearing total knee prosthesis. A prospective series of 11 rheumatoid arthritis patients was included to participate in this fluoroscopic and EMG study; only 7 patients completed the study. Kinematic evaluations took place 7 months, 1 and 2 years post-operatively. Repeated measurements ANOVA and linear mixed-effects model for longitudinal data were used to compare the differences between the follow-ups. There are no significant changes in axial rotations between follow-up moments for the femoral component as well as the mobile insert. The insert remained mobile and followed the femoral component from 0° until approximately 60° of knee flexion. Diverging and reversed axial rotations and translations were seen during the dynamic motions. Knee kinematics and muscle activation do not appear to change in the first 2 post-operative years. Reversed and divergent axial rotations with increasing knee flexion indicate that as soon as the congruency decreases, the femoral component is no longer forced in a certain position by the insert and moves to a self-imposed position. At lower knee flexion angles, the femoral component might be obstructed by the highly congruent insert and therefore might not be able to move freely. Therapeutic study, Level IV.

A Bayesian approach for modelling non-linear longitudinal/hierarchical data with random effects in forestry

Summary Longitudinal or hierarchical data are often observed in forestry, which can pose both challenges and opportunities when performing statistical analyses. The current standard approach for analysing these types of data is mixed-effects models under the frequentist paradigm. Bayesian techniques have several advantages when compared with traditional approaches, but their use in forestry has been relatively limited. In this paper, we propose a Bayesian solution to nonlinear mixed-effects models for longitudinal data in forestry. We demonstrate the Bayesian modelling process using individual tree height–age data for balsam fir ( Abies balsamea (L.)) collected from eastern Maine. Due to its frequent utilization in modelling dominant tree height growth over time, we choose to examine models based on the Chapman– Richards function. We established four different model formulations, each having varying subject-specific parameters, which we estimated using both frequentist and Bayesian approaches. We found the estimation results to be quite close between the two methods. In addition, an important feature of the Bayesian method is the unified manner in which estimation and prediction are handled. Specifically, local parameters can be predicted for a new dataset after setting the posterior distributions from the estimation stage as new priors in the prediction phase.

Predictive Cross‐validation for the Choice of Linear Mixed‐Effects Models with Application to Data from the Swiss HIV Cohort Study

Model choice in linear mixed-effects models for longitudinal data is a challenging task. Apart from the selection of covariates, also the choice of the random effects and the residual correlation structure should be possible. Application of classical model choice criteria such as Akaike information criterion (AIC) or Bayesian information criterion is not obvious, and many versions do exist. In this article, a predictive cross-validation approach to model choice is proposed based on the logarithmic and the continuous ranked probability score. In contrast to full cross-validation, the model has to be fitted only once, which enables fast computations, even for large data sets. Relationships to the recently proposed conditional AIC are discussed. The methodology is applied to search for the best model to predict the course of CD4+ counts using data obtained from the Swiss HIV Cohort Study.