Semiparametric Nonlinear Mixed-Effects Models and Their Applications

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.

Constrained Nonlinear and Mixed Effects Integral Differential Equation Models for Dynamic Cell Polarity Signaling

Polar cell growth is a process that couples the establishment of cell polarity with growth and is extremely important in the growth, development, and reproduction of eukaryotic organisms, such as pollen tube growth during plant fertilization and neuronal axon growth in animals. Pollen tube growth requires dynamic but polarized distribution and activation of a signaling protein named ROP1 to the plasma membrane via three processes: positive feedback and negative feedback regulation of ROP1 activation and its lateral diffusion along the plasma membrane. In this paper, we introduce a mechanistic integro-differential equation (IDE) along with constrained semiparametric regression to quantitatively describe the interplay among these three processes that lead to the polar distribution of active ROP1 at a steady state. Moreover, we introduce a population variability by a constrained nonlinear mixed model. Our analysis of ROP1 activity distributions from multiple pollen tubes revealed that the equilibrium between the positive and negative feedbacks for pollen tubes with similar shapes are remarkably stable, permitting us to infer an inherent quantitative relationship between the positive and negative feedback loops that defines the tip growth of pollen tubes and the polarity of tip growth.

A recipe for accurate estimation of lifespan brain trajectories, distinguishing longitudinal and cohort effects

We address the problem of estimating how different parts of the brain develop and change throughout the lifespan, and how these trajectories are affected by genetic and environmental factors. Estimation of these lifespan trajectories is statistically challenging, since their shapes are typically highly nonlinear, and although true change can only be quantified by longitudinal examinations, as follow-up intervals in neuroimaging studies typically cover less than 10% of the lifespan, use of cross-sectional information is necessary. Linear mixed models (LMMs) and structural equation models (SEMs) commonly used in longitudinal analysis rely on assumptions which are typically not met with lifespan data, in particular when the data consist of observations combined from multiple studies. While LMMs require a priori specification of a polynomial functional form, SEMs do not easily handle data with unstructured time intervals between measurements. Generalized additive mixed models (GAMMs) offer an attractive alternative, and in this paper we propose various ways of formulating GAMMs for estimation of lifespan trajectories of 12 brain regions, using a large longitudinal dataset and realistic simulation experiments. We show that GAMMs are able to more accurately fit lifespan trajectories, distinguish longitudinal and cross-sectional effects, and estimate effects of genetic and environmental exposures. Finally, we discuss and contrast questions related to lifespan research which strictly require repeated measures data and questions which can be answered with a single measurement per participant, and in the latter case, which simplifying assumptions that need to be made. The examples are accompanied with R code, providing a tutorial for researchers interested in using GAMMs.

Retrospective versus prospective score tests for genetic association with case-control data

Since the seminal work of Prentice and Pyke, the prospective logistic likelihood has become the standard method of analysis for retrospectively collected case-control data, in particular for testing the association between a single genetic marker and a disease outcome in genetic case-control studies. In the study of multiple genetic markers with relatively small effects, especially those with rare variants, various aggregated approaches based on the same prospective likelihood have been developed to integrate subtle association evidence among all the markers considered. Many of the commonly used tests are derived from the prospective likelihood under a common-random-effect assumption, which assumes a common random effect for all subjects. We develop the locally most powerful aggregation test based on the retrospective likelihood under an independent-random-effect assumption, which allows the genetic effect to vary among subjects. In contrast to the fact that disease prevalence information cannot be used to improve efficiency for the estimation of odds ratio parameters in logistic regression models, we show that it can be utilized to enhance the testing power in genetic association studies. Extensive simulations demonstrate the advantages of the proposed method over the existing ones. A real genome-wide association study is analyzed for illustration.

Penalized nonlinear mixed effects model to identify biomarkers that predict disease progression

Precise modeling of disease progression in neurodegenerative disorders may enable early intervention before clinical manifestation of a disease, which is crucial since early intervention at the premanifest stage is expected to be more effective. Neuroimaging biomarkers are indicative of the underlying disease pathology and may be used to predict future disease occurrence at the premanifest stage. As observed in many pivotal studies, longitudinal measurements of clinical outcomes, such as motor or cognitive symptoms, often present nonlinear sigmoid shapes over time, where the inflection points of the trajectories mark a meaningful time in disease progression. Therefore, to identify neuroimaging biomarkers predicting disease progression, we propose a nonlinear mixed effects model based on a sigmoid function to predict longitudinal clinical outcomes, and associate a linear combination of neuroimaging biomarkers with subject-specific inflection points. Based on an expectation-maximization (EM) algorithm, we propose a method that can fit a nonlinear model with many potentially correlated biomarkers for random inflection points while achieving computational stability. Variable selection is introduced in the algorithm in order to identify important biomarkers of disease progression and to reduce prediction variability. We apply the proposed method to the data from the Predictors of Huntington's Disease study to select brain subcortical regional volumes predictive of the inflection points of the motor and cognitive function trajectories. Our results reveal that brain atrophy in the striatum and expansion of the ventricular system are highly predictive of the inflection points. Furthermore, these inflection points may precede clinically defined disease onset by as early as a decade and thus may be useful biomarkers as early signs of Huntington's Disease onset.

Model selection in multivariate semiparametric regression

Variable selection in semiparametric mixed models for longitudinal data remains a challenge, especially in the presence of multiple correlated outcomes. In this paper, we propose a model selection procedure that simultaneously selects fixed and random effects using a maximum penalized likelihood method with the adaptive least absolute shrinkage and selection operator penalty. Through random effects selection, we determine the correlation structure among multiple outcomes and therefore address whether a joint model is necessary. Additionally, we include a bivariate nonparametric component, as approximated by tensor product splines, to accommodate the joint nonlinear effects of two independent variables. We use an adaptive group least absolute shrinkage and selection operator to determine whether the bivariate nonparametric component can be reduced to additive components. To implement the selection and estimation method, we develop a two-stage expectation-maximization procedure. The operating characteristics of the proposed method are assessed through simulation studies. Finally, the method is illustrated in a clinical study of blood pressure development in children.

Regression analysis of case K interval-censored failure time data in the presence of informative censoring

Interval-censored failure time data occur in many fields such as demography, economics, medical research, and reliability and many inference procedures on them have been developed (Sun, 2006; Chen, Sun, and Peace, 2012). However, most of the existing approaches assume that the mechanism that yields interval censoring is independent of the failure time of interest and it is clear that this may not be true in practice (Zhang et al., 2007; Ma, Hu, and Sun, 2015). In this article, we consider regression analysis of case K interval-censored failure time data when the censoring mechanism may be related to the failure time of interest. For the problem, an estimated sieve maximum-likelihood approach is proposed for the data arising from the proportional hazards frailty model and for estimation, a two-step procedure is presented. In the addition, the asymptotic properties of the proposed estimators of regression parameters are established and an extensive simulation study suggests that the method works well. Finally, we apply the method to a set of real interval-censored data that motivated this study.

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).

Enriched Stick Breaking Processes for Functional Data

In many applications involving functional data, prior information is available about the proportion of curves having different attributes. It is not straightforward to include such information in existing procedures for functional data analysis. Generalizing the functional Dirichlet process (FDP), we propose a class of stick-breaking priors for distributions of functions. These priors incorporate functional atoms drawn from constrained stochastic processes. The stick-breaking weights are specified to allow user-specified prior probabilities for curve attributes, with hyperpriors accommodating uncertainty. Compared with the FDP, the random distribution is enriched for curves having attributes known to be common. Theoretical properties are considered, methods are developed for posterior computation, and the approach is illustrated using data on temperature curves in menstrual cycles.

Composite growth model applied to human oral and pharyngeal structures and identifying the contribution of growth types

The growth patterns of different anatomic structures in the human body vary in terms of growth amount over time, growth rate and growth periods. The oral and pharyngeal structures, also known as vocal tract structures, are housed in the craniofacial complex where the cranium/brain follows a distinct neural growth pattern, and the face follows a distinct somatic or skeletal growth pattern. Thus, it is reasonable to expect the oral and pharyngeal structures to follow a combined or mixed growth pattern. Existing parametric growth models are limited in that they are mainly focused on modeling one particular type of growth pattern. In this paper, we propose a novel composite growth model using neural and somatic baseline curves to fit the combined growth pattern of select vocal tract structures. The method can also determine the overall percent contribution of each of the growth types.

Evaluation of anxiety sensitivity among daily adult smokers using item response theory analysis

The present investigation applied Item Response Theory (IRT) methodology to the 16-item Anxiety Sensitivity Index (ASI) [Reiss, S., Peterson, R. A., Gursky, M., & McNally, R. J. (1986). Anxiety sensitivity, anxiety frequency, and the prediction of fearfulness. Behaviour Research and Therapy, 24, 1-8] for a sample of 475 daily adult smokers (52% women; M(age)=26.9, S.D.=11.1, range=18-65). Using non-parametric item response analysis, all 16 ASI items were evaluated. Evaluation of the option characteristic curves for each item revealed 4 poorly discriminating ASI items (1: "It is important not to appear nervous;" 5: "It is important to me to stay in control of my emotions;" 7: "It embarrasses me when my stomach growls;" 9: "When I notice my heart beating rapidly, I worry that I might be having a heart attack"), which were dropped from analysis. Upon repeat analysis, the remaining items appeared to make adequate separations within levels of anxiety sensitivity in this sample. Graded response modeling data indicated important differences in ASI items' capacity to discriminate between, and provide information about, latent levels of anxiety sensitivity. Specifically, three items best discriminated and provided the most information regarding latent levels of AS-items 3, 15, and 16. Items 1, 5, 7, and 9 were omitted due to their limited capacity to discriminate between latent levels of anxiety sensitivity; items 8, 12, and 13 also performed poorly. Overall, current findings suggest that evaluation of anxiety sensitivity among adult smokers using the 16-item ASI may usefully choose to focus on items that performed well in these IRT analyses (items: 2, 3, 4, 6, 10, 11, 14, 15, and 16).

GENERALIZED PARTIALLY LINEAR MIXED-EFFECTS MODELS INCORPORATING MISMEASURED COVARIATES

In this article we consider a semiparametric generalized mixed-effects model, and propose combining local linear regression, and penalized quasilikelihood and local quasilikelihood techniques to estimate both population and individual parameters and nonparametric curves. The proposed estimators take into account the local correlation structure of the longitudinal data. We establish normality for the estimators of the parameter and asymptotic expansion for the estimators of the nonparametric part. For practical implementation, we propose an appropriate algorithm. We also consider the measurement error problem in covariates in our model, and suggest a strategy for adjusting the effects of measurement errors. We apply the proposed models and methods to study the relation between virologic and immunologic responses in AIDS clinical trials, in which virologic response is classified into binary variables. A dataset from an AIDS clinical study is analyzed.

Identifying significant covariates for anti-HIV treatment response: mechanism-based differential equation models and empirical semiparametric regression models

In this paper, the mechanism-based ordinary differential equation (ODE) model and the flexible semiparametric regression model are employed to identify the significant covariates for antiretroviral response in AIDS clinical trials. We consider the treatment effect as a function of three factors (or covariates) including pharmacokinetics, drug adherence and susceptibility. Both clinical and simulated data examples are given to illustrate these two different kinds of modeling approaches. We found that the ODE model is more powerful to model the mechanism-based nonlinear relationship between treatment effects and virological response biomarkers. The ODE model is also better in identifying the significant factors for virological response, although it is slightly liberal and there is a trend to include more factors (or covariates) in the model. The semiparametric mixed-effects regression model is very flexible to fit the virological response data, but it is too liberal to identify correct factors for the virological response; sometimes it may miss the correct factors. The ODE model is also biologically justifiable and good for predictions and simulations for various biological scenarios. The limitations of the ODE models include the high cost of computation and the requirement of biological assumptions that sometimes may not be easy to validate. The methodologies reviewed in this paper are also generally applicable to studies of other viruses such as hepatitis B virus or hepatitis C virus.

Investigating Determinants of Multiple Sclerosis in Longitunal Studies: A Bayesian Approach

Modelling data from Multiple Sclerosis longitudinal studies is a challenging topic since the phenotype of interest is typically ordinal; time intervals between two consecutive measurements are nonconstant and they can vary among individuals. Due to these unobservable sources of heterogeneity statistical models for analysis of Multiple Sclerosis severity evolve as a difficult feature. A few proposals have been provided in the biostatistical literature (Heijtan (1991); Albert, (1994)) to address the issue of investigating Multiple Sclerosis course. In this paper Bayesian P-Splines (Brezger and Lang, (2006); Fahrmeir and Lang (2001)) are indicated as an appropriate tool since they account for nonlinear smooth effects of covariates on the change in Multiple Sclerosis disability. By means of Bayesian P-Spline model we investigate both the randomness affecting Multiple Sclerosis data as well as the ordinal nature of the response variable.

Nonparametric estimation and testing of fixed effects panel data models

In this paper we consider the problem of estimating nonparametric panel data models with fixed effects. We introduce an iterative nonparametric kernel estimator. We also extend the estimation method to the case of a semiparametric partially linear fixed effects model. To determine whether a parametric, semiparametric or nonparametric model is appropriate, we propose test statistics to test between the three alternatives in practice. We further propose a test statistic for testing the null hypothesis of random effects against fixed effects in a nonparametric panel data regression model. Simulations are used to examine the finite sample performance of the proposed estimators and the test statistics.

Infant growth modelling using a shape invariant model with random effects

Models for infant growth have usually been based on parametric forms, commonly an exponential or similar model, which have been shown to fit poorly especially during the first year of life. An alternative approach is to use a non-parametric model, based on a shape invariant model (SIM), where a single function is transformed by shifting and scaling to fit each subject. In the model a regression spline is used as the function, with log transformation of the data and a simplification of the SIM, obtained from the relationship with the exponential model. All subjects are fitted as a nonlinear mixed effects model, allowing the variation in the parameters between subjects to be determined. Methods for the inclusion of covariates in growth models based on SIM are developed, with parameters for time independent covariates included in the model by varying either the shape, the size parameter or the growth parameter and time-dependent co-variates included by transforming the time axis, to either increase or decrease the growth rate dependent on the co-variate, similar to methods used for accelerated failure-time models. The model is used to fit weight data for 602 infants, measured from 0 to 2 years as part of the Childhood Asthma Prevention Study (CAPS) trial, and to determine the effect of breastfeeding on infant weight.

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.

Detecting pulsatile hormone secretions using nonlinear mixed effects partial spline models

Neuroendocrine ensembles communicate with their remote and proximal target cells via an intermittent pattern of chemical signaling. The identification of episodic releases of hormonal pulse signals constitutes a major emphasis of endocrine investigation. Estimating the number, temporal locations, secretion rate, and elimination rate from hormone concentration measurements is of critical importance in endocrinology. In this article, we propose a new flexible statistical method for pulse detection based on nonlinear mixed effects partial spline models. We model pulsatile secretions using biophysical models and investigate biological variation between pulses using random effects. Pooling information from different pulses provides more efficient and stable estimation for parameters of interest. We combine all nuisance parameters including a nonconstant basal secretion rate and biological variations into a baseline function that is modeled nonparametrically using smoothing splines. We develop model selection and parameter estimation methods for the general nonlinear mixed effects partial spline models and an R package for pulse detection and estimation. We evaluate performance and the benefit of shrinkage by simulations and apply our methods to data from a medical experiment.

Alterations in the dynamics of circulating ghrelin, adiponectin, and leptin in human obesity

Ghrelin plays a key role in the regulation of growth hormone secretion and energy homeostasis. Adiponectin is exclusively secreted by adipose tissue and is abundantly present in the circulation, with important effects on metabolism. We studied five lean and five obese young men [ages: 24.2 +/- 1.0 (lean) and 21.8 +/- 1.6 (obese) years (difference not significant); body mass indexes: 35.0 +/- 1.3 and 23.0 +/- 0.3 kg/m2 (P = 0.01)], sampled blood every 7 min over 24 h, and measured ghrelin, adiponectin, and leptin in 2,070 samples for a total of 6,210 data points. Circulating 24-h ghrelin showed significant ultradian fluctuations and an orderly pattern of release in lean and obese subjects with similar pulsatility characteristics. Plasma adiponectin concentrations were significantly lower in the obese group, with lower pulse height. In contrast to leptin, which is secreted in an orderly manner, the 24-h patterns of adiponectin were not significantly different from random in both the lean and obese groups. We show here that adipocytes can simultaneously secrete certain hormones, such as leptin, in patterns that are orderly, whereas other hormones, such as adiponectin, are secreted in patterns that appear to be random. The cross-approximate entropy statistic revealed pattern synchrony among ghrelin-leptin, ghrelin-adiponectin, and leptin-adiponectin hormone time series in the lean and obese subjects. Plasma ghrelin concentrations showed a nocturnal rise that exceeded the meal-associated increases in lean subjects, and this newly identified nocturnal rise was blunted in the obese. We suggest that the blunting of the nocturnal rise of ghrelin is a biological feature of human obesity.

Self Modeling with Flexible, Random Time Transformations

Methods for modeling sets of complex curves where the curves must be aligned in time (or in another continuous predictor) fall into the general class of functional data analysis and include self-modeling regression and time-warping procedures. Self-modeling regression (SEMOR), also known as a shape invariant model (SIM), assumes the curves have a common shape, modeled nonparametrically, and curve-specific differences in amplitude and timing, traditionally modeled by linear transformations. When curves contain multiple features that need to be aligned in time, SEMOR may be inadequate since a linear time transformation generally cannot align more than one feature. Time warping procedures focus on timing variability and on finding flexible time warps to align multiple data features. We draw on these methods to develop a SIM that models the time transformations as random, flexible, monotone functions. The model is motivated by speech movement data from the University of Wisconsin X-ray microbeam speech production project and is applied to these data to test the effect of different speaking conditions on the shape and relative timing of movement profiles.

Shape‐Invariant Modeling of Circadian Rhythms with Random Effects and Smoothing Spline ANOVA Decompositions

Medical studies often collect physiological and/or psychological measurements over time from multiple subjects, to study dynamics such as circadian rhythms. Under the assumption that the expected response functions of all subjects are the same after shift and scale transformations, shape-invariant models have been applied to analyze this kind of data. The shift and scale parameters provide efficient and interpretable data summaries, while the common shape function is usually modeled nonparametrically, to provide flexibility. However, due to the deterministic nature of the shift and scale parameters, potential correlations within a subject are ignored. Furthermore, the shape of the common function may depend on other factors, such as disease. In this article, we propose shape-invariant mixed effects models. A second-stage model with fixed and random effects is used to model individual shift and scale parameters. A second-stage smoothing spline ANOVA model is used to study potential covariate effects on the common shape function. We apply our methods to a real data set to investigate disease effects on circadian rhythms of cortisol, a hormone that is affected by stress. We find that patients with Cushing's syndrome lost circadian rhythms and their 24-hour means were elevated to very high levels. Patients with major depression had the same circadian shape and phases as normal subjects. However, their 24-hour mean levels were elevated and amplitudes were dampened for some patients.