Semiparametric Nonlinear Mixed-effects Models 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.

Predicting pregnancy outcomes using longitudinal information: a penalized splines mixed-effects model approach.

We propose a semiparametric nonlinear mixed-effects model (SNMM) using penalized splines to classify longitudinal data and improve the prediction of a binary outcome. The work is motivated by a study in which different hormone levels were measured during the early stages of pregnancy, and the challenge is using this information to predict normal versus abnormal pregnancy outcomes. The aim of this paper is to compare models and estimation strategies on the basis of alternative formulations of SNMMs depending on the characteristics of the data set under consideration. For our motivating example, we address the classification problem using a particular case of the SNMM in which the parameter space has a finite dimensional component (fixed effects and variance components) and an infinite dimensional component (unknown function) that need to be estimated. The nonparametric component of the model is estimated using penalized splines. For the parametric component, we compare the advantages of using random effects versus direct modeling of the correlation structure of the errors. Numerical studies show that our approach improves over other existing methods for the analysis of this type of data. Furthermore, the results obtained using our method support the idea that explicit modeling of the serial correlation of the error term improves the prediction accuracy with respect to a model with random effects, but independent errors. Copyright © 2017 John Wiley & Sons, Ltd.

The estimation of branching curves in the presence of subject-specific random effects.

Branching curves are a technique for modeling curves that change trajectory at a change (branching) point. Currently, the estimation framework is limited to independent data, and smoothing splines are used for estimation. This article aims to extend the branching curve framework to the longitudinal data setting where the branching point varies by subject. If the branching point is modeled as a random effect, then the longitudinal branching curve framework is a semiparametric nonlinear mixed effects model. Given existing issues with using random effects within a smoothing spline, we express the model as a B-spline based semiparametric nonlinear mixed effects model. Simple, clever smoothness constraints are enforced on the B-splines at the change point. The method is applied to Women's Health data where we model the shape of the labor curve (cervical dilation measured longitudinally) before and after treatment with oxytocin (a labor stimulant).

LASSO-type estimators for semiparametric nonlinear mixed-effects models estimation

Parametric nonlinear mixed effects models (NLMEs) are now widely used in biometrical studies, especially in pharmacokinetics research and HIV dynamics models, due to, among other aspects, the computational advances achieved during the last years. However, this kind of models may not be flexible enough for complex longitudinal data analysis. Semiparametric NLMEs (SNMMs) have been proposed as an extension of NLMEs. These models are a good compromise and retain nice features of both parametric and nonparametric models resulting in more flexible models than standard parametric NLMEs. However, SNMMs are complex models for which estimation still remains a challenge. Previous estimation procedures are based on a combination of log-likelihood approximation methods for parametric estimation and smoothing splines techniques for nonparametric estimation. In this work, we propose new estimation strategies in SNMMs. On the one hand, we use the Stochastic Approximation version of EM algorithm (SAEM) to obtain exact ML and REML estimates of the fixed effects and variance components. On the other hand, we propose a LASSO-type method to estimate the unknown nonlinear function. We derive oracle inequalities for this nonparametric estimator. We combine the two approaches in a general estimation procedure that we illustrate with simulations and through the analysis of a real data set of price evolution in on-line auctions.

Mixed-Effects Joint Models with Skew-Normal Distribution for HIV Dynamic Response with Missing and Mismeasured Time-Varying Covariate

Longitudinal data arise frequently in medical studies and it is a common practice to analyze such complex data with nonlinear mixed-effects (NLME) models, which enable us to account for between-subject and within-subject variations. To partially explain the variations, time-dependent covariates are usually introduced to these models. Some covariates, however, may be often measured with substantial errors and missing observations. It is often the case that model random error is assumed to be distributed normally, but the normality assumption may not always give robust and reliable results, particularly if the data exhibit skewness. In the literature, there has been considerable interest in accommodating either skewed response or covariate measured with error and missing data in such models, but there has been relatively little study concerning all these features simultaneously. This article is to address simultaneous impact of skewness in response and measurement error and missing data in covariate by jointly modeling the response and covariate processes under a framework of Bayesian semiparametric nonlinear mixed-effects models. In particular, we aim at exploring how mixed-effects joint models based on one-compartment model with one phase time-varying decay rate and two-compartment model with two phase time-varying decay rates contribute to modeling results and inference. The method is illustrated by an AIDS data example to compare potential models with different distributional specifications and various scenarios. The findings from this study suggest that the one-compartment model with a skew-normal distribution may provide more reasonable results if the data exhibit skewness in response and/or have measurement error and missing observations in covariates.

Simultaneous Bayesian Inference for Skew-Normal Semiparametric Nonlinear Mixed-Effects Models with Covariate Measurement Errors

Longitudinal data arise frequently in medical studies and it is a common practice to analyze such complex data with nonlinear mixed-effects (NLME) models which enable us to account for between-subject and within-subject variations. To partially explain the variations, covariates are usually introduced to these models. Some covariates, however, may be often measured with substantial errors. It is often the case that model random error is assumed to be distributed normally, but the normality assumption may not always give robust and reliable results, particularly if the data exhibit skewness. Although there has been considerable interest in accommodating either skewness or covariate measurement error in the literature, there is relatively little work that considers both features simultaneously. In this article, our objectives are to address simultaneous impact of skewness and covariate measurement error by jointly modeling the response and covariate processes under a general framework of Bayesian semiparametric nonlinear mixed-effects models. The method is illustrated in an AIDS data example to compare potential models which have different distributional specifications. The findings from this study suggest that the models with a skew-normal distribution may provide more reasonable results if the data exhibit skewness and/or have measurement errors in covariates.

A B-Spline Based Semiparametric Nonlinear Mixed Effects Model

The Semiparametric Nonlinear Mixed Effects Model (SNMM) (Ke and Wang 2001) provides a flexible framework for longitudinal comparisons of curve shapes between groups. In this article, we develop an alternative method for fitting the SNMM by reformulating Ke and Wang’s smoothing spline based model in terms of B-splines. The existing algorithm is based on a backfitting procedure that iterates between two mixed models whose corresponding likelihoods are not equivalent to the likelihood of all model parameters. The consequence is a lack of reliable convergence and statistical inference. Using B-splines, however, overcomes these disadvantages by simplifying the likelihood computations without sacrificing model flexibility. Therefore, the algorithm can be expressed in terms of existing, accurate techniques based on Adaptive Gaussian Quadrature. The model is applied to labor curves, cervical dilation measured longitudinally, from women attempting a vaginal birth after cesarean. Only partial curves were measured on cases of uterine rupture given the need for emergency c-section while controls completed delivery naturally. The model allowed us to estimate and compare the average labor curve shape between cases and controls and also determine the earliest time at which clinicians could distinguish between the average labor curves in different groups. Supplemental materials are available online.

Modeling HIV Dynamics Using Unified Mixed-Effects Models

SYNOPTIC ABSTRACTStudies of HIV dynamics play a crucial role in understanding the pathogenesis of HIV infection and in evaluating antiviral therapies. To accelerate AIDS clinical trials, viral load (HIV-1 RNA copies) is now used as a surrogate marker. The time range of collecting the viral load data may be divided into three successive stages. There are several existing models for the first and second stage viral load data. Recently a semiparametric nonlinear mixed-effects (NLME) model has been proposed for the complete viral load data which include the third stage viral load data, i.e., the data of those patients who fail the therapies. An important and challenging problem is if the existing models that are good only for the first one or two-stage viral load data can be generalized so that they are applicable for the complete viral load data. Another question is which model is the most preferred. In this paper, we propose a unified mixed-effects model, which models population characteristics and individual variations semiparametrically, so that all existing and recently proposed models are its special cases. We employ a basis-based approach to solve this model. We also discuss and generalize the existing and recently proposed models, including uniexponential, biexponential, multiexponential and biexponential semiparametric models. We employed AIC and BIC to compare these models and found that for the complete viral load data, the recently proposed biexponential semiparametric NLME model is the best.

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.

Design of Long‐Term HIV Dynamic Studies Using Semiparametric Mixed‐Effects Models

Studies of HIV dynamics in AIDS research are very important in understanding the pathogenesis of HIV-1 infection and also in assessing the effectiveness of antiviral therapies. There are many AIDS clinical trials on HIV dynamics currently in development worldwide, giving rise to many design issues yet to be addressed. For example, most studies are focused on short-term viral dynamics and the existing models may not be applicable to describe long-term virologic response. In this paper, we use a simulation-based approach to study the designs of long-term viral dynamics under semiparametric nonlinear mixed-effects models. These models not only can preserve the meaningful interpretation of the short-term HIV dynamics, but also characterize the long-term virologic responses to antiretroviral (ARV) treatment. We investigate a number of feasible clinical protocol designs similar to those currently used in AIDS clinical trials. In particular, we evaluate whether earlier samplings can result in more useful information about the viral response trajectory; we also evaluate the effectiveness of two strategies: more frequent samplings per subject with fewer subjects versus fewer samplings per subject with more subjects while keeping the total number of samplings constant. The results of our investigation provide quantitative guidance for designing and selecting ARV therapy.

A semiparametric nonlinear mixed-effects model with non-ignorable missing data and measurement errors for HIV viral data

Semiparametric nonlinear mixed-effects (NLME) models are very flexible in modeling long-term HIV viral dynamics. In practice, statistical analyses are often complicated due to measurement errors and missing data in covariates and non-ignorable missing data in the responses. We consider likelihood methods which simultaneously address measurement error and missing data problems. A real dataset is analyzed in detail, and a simulation study is conducted to evaluate the methods.

The study of long-term HIV dynamics using semi-parametric non-linear mixed-effects models.

Modelling HIV dynamics has played an important role in understanding the pathogenesis of HIV infection in the past several years. Non-linear parametric models, derived from the mechanisms of HIV infection and drug action, have been used to fit short-term clinical data from AIDS clinical trials. However, it is found that the parametric models may not be adequate to fit long-term HIV dynamic data. To preserve the meaningful interpretation of the short-term HIV dynamic models as well as to characterize the long-term dynamics, we introduce a class of semi-parametric non-linear mixed-effects (NLME) models. The models are non-linear in population characteristics (fixed effects) and individual variations (random effects), both of which are modelled semi-parametrically. A basis-based approach is proposed to fit the models, which transforms a general semi-parametric NLME model into a set of standard parametric NLME models, indexed by the bases used. The bases that we employ are natural cubic splines for easy implementation. The resulting standard NLME models are low-dimensional and easy to solve. Statistical inferences that include testing parametric against semi-parametric mixed-effects are investigated. Innovative bootstrap procedures are developed for simulating the empirical distributions of the test statistics. Small-scale simulation and bootstrap studies show that our bootstrap procedures work well. The proposed approach and procedures are applied to long-term HIV dynamic data from an AIDS clinical study.

Semiparametric Nonlinear Mixed-Effects Models and Their Applications

Nonlinear mixed effects models (NLMMs) and self-modeling nonlinear regression (SEMOR) models are often used to fit repeated measures data. They use a common function shared by all subjects to model variation within each subject and some fixed and/or random parameters to model variation between subjects. The parametric NLMM may be too restrictive, and the semiparametric SEMOR model ignores correlations within each subject. In this article we propose a class of semiparametric nonlinear mixed effects models (SNMMs) that extend NLMMs, SEMOR models, and many other existing models in a natural way. A SNMM assumes that the mean function depends on some parameters and nonparametric functions. The parameters provide an interpretable data summary. The nonparametric functions provide flexibility to allow the data to decide some unknown or uncertain components, such as the shape of the mean response over time. A second-stage model with fixed and random effects is used to model the parameters. Smoothing splines are used to model the nonparametric functions. Covariate effects on parameters can be built into the second-stage model, and covariate effects on nonparametric functions can be constructed using smoothing spline ANOVA decompositions. Laplace approximations to the marginal likelihood and penalized marginal likelihood are used to estimate all parameters and nonparametric functions. We propose and compare two estimation procedures, and also show how to construct approximate Bayesian confidence intervals for the nonparametric functions based on a Bayesian formulation of SNMMs. We evaluate the proposed estimation and inference procedures through a simulation study. Applications of SNMMs are illustrated with analyses of Canadian temperature data.