Conditional Second-Order Generalized Estimating Equations for Generalized Linear and Nonlinear Mixed-Effects Models

Smoothed simulated pseudo-maximum likelihood estimation for nonlinear mixed effects models with censored responses

Nonlinear mixed effects models have been widely applied to analyses of data that arise from biological, agricultural, and environmental sciences. Estimation of and inference on parameters in nonlinear mixed effects models are often based on the specification of a likelihood function. Maximizing this likelihood function can be complicated by the specification of the random effects distribution, especially in the presence of multiple random effects. The implementation of nonlinear mixed effects models can be further complicated by left-censored responses, representing measurements from bioassays where the exact quantification below a certain threshold is not possible. Motivated by the need to characterize the nonlinear human immunodeficiency virus RNA viral load trajectories after the interruption of antiretroviral therapy, we propose a smoothed simulated pseudo-maximum likelihood estimation approach to fit nonlinear mixed effects models in the presence of left-censored observations. We establish the consistency and asymptotic normality of the resulting estimators. We develop testing procedures for the correlation among random effects and for testing the distributional assumptions on random effects against a specific alternative. In contrast to the existing variants of expectation-maximization approaches, the proposed methods offer flexibility in the specification of the random effects distribution and convenience in making inference about higher-order correlation parameters. We evaluate the finite-sample performance of the proposed methods through extensive simulation studies and illustrate them on a combined dataset from six AIDS Clinical Trials Group treatment interruption studies.

Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and Applications

Nonlinear mixed effects models have become a standard platform for analysis when data is in the form of continuous and repeated measurements of subjects from a population of interest, while temporal profiles of subjects commonly follow a nonlinear tendency. While frequentist analysis of nonlinear mixed effects models has a long history, Bayesian analysis of the models has received comparatively little attention until the late 1980s, primarily due to the time-consuming nature of Bayesian computation. Since the early 1990s, Bayesian approaches for the models began to emerge to leverage rapid developments in computing power, and have recently received significant attention due to (1) superiority to quantify the uncertainty of parameter estimation; (2) utility to incorporate prior knowledge into the models; and (3) flexibility to match exactly the increasing complexity of scientific research arising from diverse industrial and academic fields. This review article presents an overview of modeling strategies to implement Bayesian approaches for the nonlinear mixed effects models, ranging from designing a scientific question out of real-life problems to practical computations.

Nonlinear mixed-effects models for HIV viral load trajectories before and after antiretroviral therapy interruption, incorporating left censoring

OBJECTIVES

Characterizing features of the viral rebound trajectories and identifying host, virological, and immunological factors that are predictive of the viral rebound trajectories are central to HIV cure research. We investigate if key features of HIV viral decay and CD4 trajectories during antiretroviral therapy (ART) are associated with characteristics of HIV viral rebound following ART interruption.

METHODS

Nonlinear mixed effect (NLME) models are used to model viral load trajectories before and following ART interruption, incorporating left censoring due to lower detection limits of viral load assays. A stochastic approximation EM (SAEM) algorithm is used for parameter estimation and inference. To circumvent the computational intensity associated with maximizing the joint likelihood, we propose an easy-to-implement three-step method.

RESULTS

We evaluate the performance of the proposed method through simulation studies and apply it to data from the Zurich Primary HIV Infection Study. We find that some key features of viral load during ART (e.g., viral decay rate) are significantly associated with important characteristics of viral rebound following ART interruption (e.g., viral set point).

CONCLUSIONS

The proposed three-step method works well. We have shown that key features of viral decay during ART may be associated with important features of viral rebound following ART interruption.

On the use of a penalized quasilikelihood information criterion for generalized linear mixed models

Information criteria are commonly used for joint fixed and random effects selection in mixed models. While information criteria are straightforward to implement, a major difficulty in applying them is that they are typically based on maximum likelihood estimates, but calculating such estimates for one candidate mixed model, let alone multiple models, presents a major computational challenge. To overcome this hurdle, we study penalized quasilikelihood estimation and use it as the basis for performing fast joint selection. Under a general framework, we show that penalized quasilikelihood estimation produces consistent estimates of the true parameters. We then propose a new penalized quasilikelihood information criterion whose distinguishing feature is the way it accounts for model complexity in the random effects, since penalized quasilikelihood estimation effectively treats the random effects as fixed. We demonstrate that the criterion asymptotically identifies the true set of important fixed and random effects. Simulations show that the quasilikelihood information criterion performs competitively with and sometimes better than common maximum likelihood information criteria for joint selection, while offering substantial reductions in computation time.

Predation Pressure on Sentinel Insect Prey Along a Riverside Urbanization Gradient in Hungary

Urbanization is one of the most important global trends which causes habitat reduction and alteration which are, in turn, the main reasons for the well-documented reduction in structural and functional diversity in urbanized environments. In contrast, effects on ecological mechanisms are less known. Predation is one of the most important ecological functions because of its community-structuring effects. We studied six forest habitats along a riverside urbanization gradient in Szeged, a major city in southern Hungary, crossed by the river Tisza, to describe how extreme events (e.g., floods) as primary selective pressure act on adaptation in riparian habitats. We found a generally decreasing predation pressure from rural to urban habitats as predicted by the increasing disturbance hypothesis (higher predator abundances in rural than in urban habitats). The only predators that reacted differently to urbanization were ground active arthropods, where results conformed to the prediction of the intermediate disturbance hypothesis (higher abundance in moderately disturbed suburban habitats). We did not find any evidence that communities exposed to extreme flood events were preadapted to the effects of urbanization. The probable reason is that changes accompanied by urbanization are much faster than natural landscape change, so the communities cannot adapt to them.

Modelling and estimation of nonlinear quantile regression with clustered data

In regression applications, the presence of nonlinearity and correlation among observations offer computational challenges not only in traditional settings such as least squares regression, but also (and especially) when the objective function is nonsmooth as in the case of quantile regression. Methods are developed for the modelling and estimation of nonlinear conditional quantile functions when data are clustered within two-level nested designs. The proposed estimation algorithm is a blend of a smoothing algorithm for quantile regression and a second order Laplacian approximation for nonlinear mixed models. This optimization approach has the appealing advantage of reducing the original nonsmooth problem to an approximated L 2 problem. While the estimation algorithm is iterative, the objective function to be optimized has a simple analytic form. The proposed methods are assessed through a simulation study and two applications, one in pharmacokinetics and one related to growth curve modelling in agriculture.

Three-part joint modeling methods for complex functional data mixed with zero-and-one-inflated proportions and zero-inflated continuous outcomes with skewness

We take a functional data approach to longitudinal studies with complex bivariate outcomes. This work is motivated by data from a physical activity study that measured 2 responses over time in 5-minute intervals. One response is the proportion of time active in each interval, a continuous proportions with excess zeros and ones. The other response, energy expenditure rate in the interval, is a continuous variable with excess zeros and skewness. This outcome is complex because there are 3 possible activity patterns in each interval (inactive, partially active, and completely active), and those patterns, which are observed, induce both nonrandom and random associations between the responses. More specifically, the inactive pattern requires a zero value in both the proportion for active behavior and the energy expenditure rate; a partially active pattern means that the proportion of activity is strictly between zero and one and that the energy expenditure rate is greater than zero and likely to be moderate, and the completely active pattern means that the proportion of activity is exactly one, and the energy expenditure rate is greater than zero and likely to be higher. To address these challenges, we propose a 3-part functional data joint modeling approach. The first part is a continuation-ratio model to reorder the ordinal valued 3 activity patterns. The second part models the proportions when they are in interval (0,1). The last component specifies the skewed continuous energy expenditure rate with Box-Cox transformations when they are greater than zero. In this 3-part model, the regression structures are specified as smooth curves measured at various time points with random effects that have a correlation structure. The smoothed random curves for each variable are summarized using a few important principal components, and the association of the 3 longitudinal components is modeled through the association of the principal component scores. The difficulties in handling the ordinal and proportional variables are addressed using a quasi-likelihood type approximation. We develop an efficient algorithm to fit the model that also involves the selection of the number of principal components. The method is applied to physical activity data and is evaluated empirically by a simulation study.

A joint modeling and estimation method for multivariate longitudinal data with mixed types of responses to analyze physical activity data generated by accelerometers

A mixed effect model is proposed to jointly analyze multivariate longitudinal data with continuous, proportion, count, and binary responses. The association of the variables is modeled through the correlation of random effects. We use a quasi-likelihood type approximation for nonlinear variables and transform the proposed model into a multivariate linear mixed model framework for estimation and inference. Via an extension to the EM approach, an efficient algorithm is developed to fit the model. The method is applied to physical activity data, which uses a wearable accelerometer device to measure daily movement and energy expenditure information. Our approach is also evaluated by a simulation study.

Estimation Methods for Mixed Logistic Models with Few Clusters

For mixed models generally, it is well known that modeling data with few clusters will result in biased estimates, particularly of the variance components and fixed effect standard errors. In linear mixed models, small sample bias is typically addressed through restricted maximum likelihood estimation (REML) and a Kenward-Roger correction. Yet with binary outcomes, there is no direct analog of either procedure. With a larger number of clusters, estimation methods for binary outcomes that approximate the likelihood to circumvent the lack of a closed form solution such as adaptive Gaussian quadrature and the Laplace approximation have been shown to yield less-biased estimates than linearization estimation methods that instead linearly approximate the model. However, adaptive Gaussian quadrature and the Laplace approximation are approximating the full likelihood rather than the restricted likelihood; the full likelihood is known to yield biased estimates with few clusters. On the other hand, linearization methods linearly approximate the model, which allows for restricted maximum likelihood and the Kenward-Roger correction to be applied. Thus, the following question arises: Which is preferable, a better approximation of a biased function or a worse approximation of an unbiased function? We address this question with a simulation and an illustrative empirical analysis.

Non-linear Models for Longitudinal Data

While marginal models, random-effects models, and conditional models are routinely considered to be the three main modeling families for continuous and discrete repeated measures with linear and generalized linear mean structures, respectively, it is less common to consider non-linear models, let alone frame them within the above taxonomy. In the latter situation, indeed, when considered at all, the focus is often exclusively on random-effects models. In this paper, we consider all three families, exemplify their great flexibility and relative ease of use, and apply them to a simple but illustrative set of data on tree circumference growth of orange trees.

Estimating variance parameters from multivariate normal variables subject to limit of detection: MLE, REML, or Bayesian approaches?

Likelihood-based approaches, which naturally incorporate left censoring due to limit of detection, are commonly utilized to analyze censored multivariate normal data. However, the maximum likelihood estimator (MLE) typically underestimates variance parameters. The restricted maximum likelihood estimator (REML), which corrects the underestimation of variance parameters, cannot be easily extended to analyze censored multivariate normal data. In the light of the connection between the REML and a Bayesian approach discovered in 1974 by Dr Harville, this paper describes a Bayesian approach to censored multivariate normal data. This Bayesian approach is justified through its link to the REML via Laplace's approximation and its performance is evaluated through a simulation study. We consider the Bayesian approach as a valuable alternative because it yields less biased variance parameter estimates than the MLE, and because a solid REML is technically difficult when data are left censored.

Laplace's approximation for relative risk frailty models

Relative risk frailty models are used extensively in analyzing clustered and/or recurrent time-to-event data. In this paper, Laplace's approximation for integrals is applied to marginal distributions of data arising from parametric relative risk frailty models. Under regularity conditions, the approximate maximum likelihood estimators (MLE) are consistent with a rate of convergence that depends on both the number of subjects and number of members per subject. We compare the approximate MLE against alternative estimators using limited simulation and demonstrate the utility of Laplace's approximation approach by analyzing U.S. patient waiting time to deceased kidney transplant data.

A nonlinear mixed-effect mixture model for functional mapping of dynamic traits

Functional mapping has emerged as a next-generation statistical tool for mapping quantitative trait loci (QTL) that affect complex dynamic traits. In this article, we incorporated the idea of nonlinear mixed-effect (NLME) models into the mixture-based framework of functional mapping, aimed to generalize the spectrum of applications for functional mapping. NLME-based functional mapping, implemented with the linearization algorithm based on the first-order Taylor expansion, can provide reasonable estimates of QTL genotypic-specific curve parameters (fixed effect) and the between-individual variation of these parameters (random effect). Results from simulation studies suggest that the NLME-based model is more general than traditional functional mapping. The new model can be useful for the identification of the ontogenetic patterns of QTL genetic effects during time course.

Generalized linear mixed models: a review and some extensions

Breslow and Clayton (J Am Stat Assoc 88:9-25,1993) was, and still is, a highly influential paper mobilizing the use of generalized linear mixed models in epidemiology and a wide variety of fields. An important aspect is the feasibility in implementation through the ready availability of related software in SAS (SAS Institute, PROC GLIMMIX, SAS Institute Inc., URL http://www.sas.com , 2007), S-plus (Insightful Corporation, S-PLUS 8, Insightful Corporation, Seattle, WA, URL http://www.insightful.com , 2007), and R (R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, URL http://www.R-project.org , 2006) for example, facilitating its broad usage. This paper reviews background to generalized linear mixed models and the inferential techniques which have been developed for them. To provide the reader with a flavor of the utility and wide applicability of this fundamental methodology we consider a few extensions including additive models, models for zero-heavy data, and models accommodating latent clusters.

HIV viral dynamic models with dropouts and missing covariates

In recent years HIV viral dynamic models have received great attention in AIDS studies. Often, subjects in these studies may drop out for various reasons such as drug intolerance or drug resistance, and covariates may also contain missing data. Statistical analyses ignoring informative dropouts and missing covariates may lead to misleading results. We consider appropriate methods for HIV viral dynamic models with informative dropouts and missing covariates and evaluate these methods via simulations. A real data set is analysed, and the results show that the initial viral decay rate, which may reflect the efficacy of the anti-HIV treatment, may be over-estimated if dropout patients are ignored. We also find that the current or immediate previous viral load values may be most predictive for patients' dropout. These results may be important for HIV/AIDS studies.

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.

Shared parameter models for the joint analysis of longitudinal data and event times

Longitudinal studies often gather joint information on time to some event (survival analysis, time to dropout) and serial outcome measures (repeated measures, growth curves). Depending on the purpose of the study, one may wish to estimate and compare serial trends over time while accounting for possibly non-ignorable dropout or one may wish to investigate any associations that may exist between the event time of interest and various longitudinal trends. In this paper, we consider a class of random-effects models known as shared parameter models that are particularly useful for jointly analysing such data; namely repeated measurements and event time data. Specific attention will be given to the longitudinal setting where the primary goal is to estimate and compare serial trends over time while adjusting for possible informative censoring due to patient dropout. Parametric and semi-parametric survival models for event times together with generalized linear or non-linear mixed-effects models for repeated measurements are proposed for jointly modelling serial outcome measures and event times. Methods of estimation are based on a generalized non-linear mixed-effects model that may be easily implemented using existing software. This approach allows for flexible modelling of both the distribution of event times and of the relationship of the longitudinal response variable to the event time of interest. The model and methods are illustrated using data from a multi-centre study of the effects of diet and blood pressure control on progression of renal disease, the modification of diet in renal disease study.