Semiparametric Mixed Models Research Articles

Extending multivariate Student's- semiparametric mixed models for longitudinal data with censored responses and heavy tails.

This article extends the semiparametric mixed model for longitudinal censored data with Gaussian errors by considering the Student's -distribution. This model allows us to consider a flexible, functional dependence of an outcome variable over the covariates using nonparametric regression. Moreover, the proposed model takes into account the correlation between observations by using random effects. Penalized likelihood equations are applied to derive the maximum likelihood estimates that appear to be robust against outlying observations with respect to the Mahalanobis distance. We estimate nonparametric functions using smoothing splines under an EM-type algorithm framework. Finally, the proposed approach's performance is evaluated through extensive simulation studies and an application to two datasets from acquired immunodeficiency syndrome clinical trials.

A semiparametric mixed-effects model for censored longitudinal data.

In longitudinal studies involving laboratory-based outcomes, repeated measurements can be censored due to assay detection limits. Linear mixed-effects (LMEs) models are a powerful tool to model the relationship between a response variable and covariates in longitudinal studies. However, the linear parametric form of linear mixed-effect models is often too restrictive to characterize the complex relationship between a response variable and covariates. More general and robust modeling tools, such as nonparametric and semiparametric regression models, have become increasingly popular in the last decade. In this article, we use semiparametric mixed models to analyze censored longitudinal data with irregularly observed repeated measures. The proposed model extends the censored linear mixed-effect model and provides more flexible modeling schemes by allowing the time effect to vary nonparametrically over time. We develop an Expectation-Maximization (EM) algorithm for maximum penalized likelihood estimation of model parameters and the nonparametric component. Further, as a byproduct of the EM algorithm, the smoothing parameter is estimated using a modified linear mixed-effects model, which is faster than alternative methods such as the restricted maximum likelihood approach. Finally, the performance of the proposed approaches is evaluated through extensive simulation studies as well as applications to data sets from acquired immune deficiency syndrome studies.

Bayesian semiparametric mixed effects models for meta-analysis of the literature data : An application to cadmium toxicity studies.

We propose Bayesian semiparametric mixed effects models with measurement error to analyze the literature data collected from multiple studies in a meta-analytic framework. We explore this methodology for risk assessment in cadmium toxicity studies, where the primary objective is to investigate dose-response relationships between urinary cadmium concentrations and -microglobulin. In the proposed model, a nonlinear association between exposure and response is described by a Gaussian process with shape restrictions, and study-specific random effects are modeled to have either normal or unknown distributions with Dirichlet process mixture priors. In addition, nonparametric Bayesian measurement error models are incorporated to flexibly account for the uncertainty resulting from the usage of a surrogate measurement of a true exposure. We apply the proposed model to analyze cadmium toxicity data imposing shape constraints along with measurement errors and study-specific random effects across varying characteristics, such as population gender, age, or ethnicity.

LASSO type penalized spline regression for binary data

BackgroundGeneralized linear mixed models (GLMMs), typically used for analyzing correlated data, can also be used for smoothing by considering the knot coefficients from a regression spline as random effects. The resulting models are called semiparametric mixed models (SPMMs). Allowing the random knot coefficients to follow a normal distribution with mean zero and a constant variance is equivalent to using a penalized spline with a ridge regression type penalty. We introduce the least absolute shrinkage and selection operator (LASSO) type penalty in the SPMM setting by considering the coefficients at the knots to follow a Laplace double exponential distribution with mean zero.MethodsWe adopt a Bayesian approach and use the Markov Chain Monte Carlo (MCMC) algorithm for model fitting. Through simulations, we compare the performance of curve fitting in a SPMM using a LASSO type penalty to that of using ridge penalty for binary data. We apply the proposed method to obtain smooth curves from data on the relationship between the amount of pack years of smoking and the risk of developing chronic obstructive pulmonary disease (COPD).ResultsThe LASSO penalty performs as well as ridge penalty for simple shapes of association and outperforms the ridge penalty when the shape of association is complex or linear.ConclusionWe demonstrated that LASSO penalty captured complex dose-response association better than the Ridge penalty in a SPMM.

Robust Regression Analysis for Clustered Interval-Censored Failure Time Data

Clustered interval-censored failure time data often occur in a wide variety of research and application fields such as cancer and AIDS studies. For such data, the failure times of interest are interval-censored and may be correlated for subjects coming from the same cluster. This paper presents a robust semiparametric transformation mixed effect models to analyze such data and use a U-statistic based on rank correlation to estimate the unknown parameters. The large sample properties of the estimator are also established. In addition, the authors illustrate the performance of the proposed estimate with extensive simulations and two real data examples.

Semiparametric Estimation of Task-Based Dynamic Functional Connectivity on the Population Level.

Dynamic functional connectivity (dFC) estimates time-dependent associations between pairs of brain region time series as typically acquired during functional MRI. dFC changes are most commonly quantified by pairwise correlation coefficients between the time series within a sliding window. Here, we applied a recently developed bootstrap-based technique (Kudela et al., 2017) to robustly estimate subject-level dFC and its confidence intervals in a task-based fMRI study (24 subjects who tasted their most frequently consumed beer and Gatorade as an appetitive control). We then combined information across subjects and scans utilizing semiparametric mixed models to obtain a group-level dFC estimate for each pair of brain regions, flavor, and the difference between flavors. The proposed approach relies on the estimated group-level dFC accounting for complex correlation structures of the fMRI data, multiple repeated observations per subject, experimental design, and subject-specific variability. It also provides condition-specific dFC and confidence intervals for the whole brain at the group level. As a summary dFC metric, we used the proportion of time when the estimated associations were either significantly positive or negative. For both flavors, our fully-data driven approach yielded regional associations that reflected known, biologically meaningful brain organization as shown in prior work, as well as closely resembled resting state networks (RSNs). Specifically, beer flavor-potentiated associations were detected between several reward-related regions, including the right ventral striatum (VST), lateral orbitofrontal cortex, and ventral anterior insular cortex (vAIC). The enhancement of right VST-vAIC association by a taste of beer independently validated the main activation-based finding (Oberlin et al., 2016). Most notably, our novel dFC methodology uncovered numerous associations undetected by the traditional static FC analysis. The data-driven, novel dFC methodology presented here can be used for a wide range of task-based fMRI designs to estimate the dFC at multiple levels—group-, individual-, and task-specific, utilizing a combination of well-established statistical methods.

A semi-parametric mixed models for longitudinally measured fasting blood sugar level of adult diabetic patients

BackgroundAt the diabetic clinic of Jimma University Specialized Hospital, health professionals provide regular follow-up to help people with diabetes live long and relatively healthy lives. Based on patient condition, they also provide interventions in the form of counselling to promote a healthy diet and physical activity and prescribing medicines. The main purpose of this study is to estimate the rate of change of fasting blood sugar (FBS) profile experienced by patients over time. The change may help to assess the effectiveness of interventions taken by the clinic to regulate FBS level, where rates of change close to zero over time may indicate the interventions are good regulating the level.MethodsIn the analysis of longitudinal data, the mean profile is often estimated by parametric linear mixed effects model. However, the individual and mean profile plots of FBS level for diabetic patients are nonlinear and imposing parametric models may be too restrictive and yield unsatisfactory results. We propose a semi-parametric mixed model, in particular using spline smoothing to efficiently analyze a longitudinal measured fasting blood sugar level of adult diabetic patients accounting for correlation between observations through random effects.ResultsThe semi-parametric mixed models had better fit than the linear mixed models for various variance structures of subject-specific random effects. The study revealed that the rate of change in FBS level in diabetic patients, due to the clinic interventions, does not continue as a steady pace but changes with time and weight of patients.ConclusionsThe proposed method can help a physician in clinical monitoring of diabetic patients and to assess the effect of intervention packages, such as healthy diet, physical activity and prescribed medicines, because individualized curve may be obtained to follow patient-specific FBS level trends.

Bayesian Semiparametric Mixed Effects Markov Models With Application to Vocalization Syntax

ABSTRACTStudying the neurological, genetic, and evolutionary basis of human vocal communication mechanisms using animal vocalization models is an important field of neuroscience. The datasets typically comprise structured sequences of syllables or “songs” produced by animals from different genotypes under different social contexts. It has been difficult to come up with sophisticated statistical methods that appropriately model animal vocal communication syntax. We address this need by developing a novel Bayesian semiparametric framework for inference in such datasets. Our approach is built on a novel class of mixed effects Markov transition models for the songs that accommodate exogenous influences of genotype and context as well as animal-specific heterogeneity. Crucial advantages of the proposed approach include its ability to provide insights into key scientific queries related to global and local influences of the exogenous predictors on the transition dynamics via automated tests of hypotheses. The methodology is illustrated using simulation experiments and the aforementioned motivating application in neuroscience. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

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.

Glycemic Excursions in Type 1 Diabetes in Pregnancy: A Semiparametric Statistical Approach to Identify Sensitive Time Points during Gestation.

Aim. To examine the gestational glycemic profile and identify specific times during pregnancy that variability in glucose levels, measured by change in velocity and acceleration/deceleration of blood glucose fluctuations, is associated with delivery of a large-for-gestational-age (LGA) baby, in women with type 1 diabetes. Methods. Retrospective analysis of capillary blood glucose levels measured multiple times daily throughout gestation in women with type 1 diabetes was performed using semiparametric mixed models. Results. Velocity and acceleration/deceleration in glucose levels varied across gestation regardless of delivery outcome. Compared to women delivering LGA babies, those delivering babies appropriate for gestational age exhibited significantly smaller rates of change and less variation in glucose levels between 180 days of gestation and birth. Conclusions. Use of innovative statistical methods enabled detection of gestational intervals in which blood glucose fluctuation parameters might influence the likelihood of delivering LGA baby in mothers with type 1 diabetes. Understanding dynamics and being able to visualize gestational changes in blood glucose are a potentially useful tool to assist care providers in determining the optimal timing to initiate continuous glucose monitoring.

Semiparametric Mixed Models for Nested Repeated Measures Applied to Ambulatory Blood Pressure Monitoring Data.

Semiparametric mixed models are increasingly popular for statistical analysis of medical device studies in which long sequences of repeated measurements are recorded. Monitoring these sequences at different periods over time on the same individual, such as before and after an intervention, results in nested repeated measures (NRM). Covariance models to account for NRM and simultaneously address mean profile estimation with penalized splines via semiparametric regression are considered with application to a prospective study of 24-hour ambulatory blood pressure and the impact of surgical intervention on obstructive sleep apnea.

Effect of Smoothing in Generalized Linear Mixed Models on the Estimation of Covariance Parameters for Longitudinal Data.

Besides being mainly used for analyzing clustered or longitudinal data, generalized linear mixed models can also be used for smoothing via restricting changes in the fit at the knots in regression splines. The resulting models are usually called semiparametric mixed models (SPMMs). We investigate the effect of smoothing using SPMMs on the correlation and variance parameter estimates for serially correlated longitudinal normal, Poisson and binary data. Through simulations, we compare the performance of SPMMs to other simpler methods for estimating the nonlinear association such as fractional polynomials, and using a parametric nonlinear function. Simulation results suggest that, in general, the SPMMs recover the true curves very well and yield reasonable estimates of the correlation and variance parameters. However, for binary outcomes, SPMMs produce biased estimates of the variance parameters for high serially correlated data. We apply these methods to a dataset investigating the association between CD4 cell count and time since seroconversion for HIV infected men enrolled in the Multicenter AIDS Cohort Study.

Locally epistatic genomic relationship matrices for genomic association and prediction.

In plant and animal breeding studies a distinction is made between the genetic value (additive plus epistatic genetic effects) and the breeding value (additive genetic effects) of an individual since it is expected that some of the epistatic genetic effects will be lost due to recombination. In this article, we argue that the breeder can take advantage of the epistatic marker effects in regions of low recombination. The models introduced here aim to estimate local epistatic line heritability by using genetic map information and combining local additive and epistatic effects. To this end, we have used semiparametric mixed models with multiple local genomic relationship matrices with hierarchical designs. Elastic-net postprocessing was used to introduce sparsity. Our models produce good predictive performance along with useful explanatory information.

Hearing function in patients living with HIV/AIDS.

During the earlier years of the HIV/AIDS epidemic, initial reports described sensorineural hearing loss in up to 49% of individuals with HIV/AIDS. During those years, patients commonly progressed to advanced stages of HIV disease and frequently had neurological complications. However, the abnormalities on pure-tone audiometry and brainstem-evoked responses outlined in small studies were not always consistently correlated with advanced stages of HIV/AIDS. Moreover, these studies could not exclude the confounding effect of concurrent opportunistic infections and syphilis. Additional reports also have indicated that some antiretroviral medications may be ototoxic; thus, it has been difficult to make conclusions regarding the cause of changes in hearing function in HIV-infected patients. More recently, accelerated aging has been suggested as a potential explanation for the disproportionate increase in complications of aging described in many HIV-infected patients; hence, accelerated aging-associated hearing loss may also be playing a role in these patients. We conducted a large cross-sectional analysis of hearing function in over 300 patients with HIV-1 infection and in 137 HIV-uninfected controls. HIV-infected participants and HIV-uninfected controls underwent a 2-hr battery of hearing tests including the Hearing Handicap Inventory, standard audiometric pure-tone air and bone conduction testing, tympanometric testing, and speech reception and discrimination testing. Three-way analysis of variance (ANOVA) and logistic regression analysis of 278 eligible HIV-infected subjects stratified by disease stage in early HIV disease (n = 127) and late HIV disease (n = 148) and 120 eligible HIV-uninfected controls revealed no statistically significant differences among the three study groups in either overall 4-frequency pure-tone average (4-PTA) or hearing loss prevalence in either ear. Three-way ANOVA showed significant differences in word recognition scores in the right ear among groups, a significant group effect on tympanogram static admittance in both ears and a significant group effect on tympanic gradient in the right ear. There was significantly larger admittance and gradient in controls as compared to the HIV-infected group at late stage of disease. Hearing loss in the HIV-infected groups was associated with increased age and was similar to that described in the literature for the general population. Three-way ANOVA analysis also indicated significantly greater pure-tone thresholds (worse hearing) at low frequencies in HIV patients in the late stage of disease compared with HIV-uninfected controls. This difference was also found by semi-parametric mixed effects models. Despite reports of "premature" or "accelerated" aging in HIV-infected subjects, we found no evidence of hearing loss occurring at an earlier age in HIV-infected patients compared to HIV-uninfected controls. Similar to what is described in the general population, the probability of hearing loss increased with age in the HIV-infected subjects and was more common in patients over 60 years of age. Interestingly, HIV-infected subjects had worse hearing at lower frequencies and have significant differences in tympanometry compared to HIV-uninfected controls; these findings deserve further study.

Semi-parametric mixed effects models for longitudinal data with applications in business and economics

Longitudinal data is becoming increasingly common in business, social sciences, and biological sciences due to the advantages it offers over cross-section data in modeling and incorporating heterogeneity among subjects and in being able to make causal inferences from observational data. Parametric models and methods are widely used for analyzing longitudinal data for continuous, discrete, and count data occurring in these disciplines. Some popular models are Gaussian, Logit, and Poisson fixed and random effects models. These models are unreliable in situations in which the link function is nonlinear and the form of nonlinearity is not known with certainty. This paper employs a semi-parametric extension of fixed and random effects models called generalized additive mixed models (GAMMs) to analyze several longitudinal data sets. These semi-parametric models are flexible and robust extensions of generalized linear models. Following Wood [19], the GAMMs are represented using penalized regression splines and estimated by penalized regression methods treating the penalized component of each smooth as a random effect term and the unpenalized component as a fixed effect term. The degree of smoothness for the unknown functions in the linear predictor part of the GAMM is estimated as the variance parameter of the term. Applications of GAMMs studied include analysis of anti-social behavior, decision to use a professional tax preparer, and analysis of patent data on manufacturing firms. For each application, several GAMMs are compared with their parametric counterparts. Keywords: Generalized Additive Mixed Models (GAMMS), Generalized Linear Mixed Models (GLMMS), Logit Models, Poisson Regression Models, Penalized Regression Splines.

Kernel smoothers and bootstrapping for semiparametric mixed effects models

While today linear mixed effects models are frequently used tools in different fields of statistics, in particular for studying data with clusters, longitudinal or multi-level structure, the nonparametric formulation of mixed effects models is still quite recent. In this paper we discuss and compare different nonparametric estimation methods. In this context we introduce a computationally inexpensive bootstrap method, which is used to estimate local mean squared errors, to construct confidence intervals and to find locally optimal smoothing parameters. The theoretical considerations are accompanied by the provision of algorithms and simulation studies of the finite sample behavior of the methods. We show that our confidence intervals have good coverage probabilities, and that our bandwidth selection method succeeds to minimize the mean squared error for the nonparametric function locally.

Penalized likelihood estimation for semiparametric mixed models, with application to alcohol treatment research

In this article, we implement a practical computational method for various semiparametric mixed effects models, estimating nonlinear functions by penalized splines. We approximate the integration of the penalized likelihood with respect to random effects with the use of adaptive Gaussian quadrature, which we can conveniently implement in SAS procedure NLMIXED. We carry out the selection of smoothing parameters through approximated generalized cross-validation scores. Our method has two advantages: (1) the estimation is more accurate than the current available quasi-likelihood method for sparse data, for example, binary data; and (2) it can be used in fitting more sophisticated models. We show the performance of our approach in simulation studies with longitudinal outcomes from three settings: binary, normal data after Box-Cox transformation, and count data with log-Gamma random effects. We also develop an estimation method for a longitudinal two-part nonparametric random effects model and apply it to analyze repeated measures of semicontinuous daily drinking records in a randomized controlled trial of topiramate.

Influence diagnostics for elliptical semiparametric mixed models

In this paper we extend semiparametric mixed linear models with normal errors to elliptical errors in order to permit distributions with heavier and lighter tails than the normal ones. Penalized li...

Flexible estimation of covariance function by penalized spline with application to longitudinal family data.

Longitudinal data are routinely collected in biomedical research studies. A natural model describing longitudinal data decomposes an individual's outcome as the sum of a population mean function and random subject-specific deviations. When parametric assumptions are too restrictive, methods modeling the population mean function and the random subject-specific functions nonparametrically are in demand. In some applications, it is desirable to estimate a covariance function of random subject-specific deviations. In this work, flexible yet computationally efficient methods are developed for a general class of semiparametric mixed effects models, where the functional forms of the population mean and the subject-specific curves are unspecified. We estimate nonparametric components of the model by penalized spline (P-spline, Biometrics 2001; 57:253-259), and reparameterize the random curve covariance function by a modified Cholesky decomposition (Biometrics 2002; 58:121-128) which allows for unconstrained estimation of a positive-semidefinite matrix. To provide smooth estimates, we penalize roughness of fitted curves and derive closed-form solutions in the maximization step of an EM algorithm. In addition, we present models and methods for longitudinal family data where subjects in a family are correlated and we decompose the covariance function into a subject-level source and observation-level source. We apply these methods to the multi-level Framingham Heart Study data to estimate age-specific heritability of systolic blood pressure nonparametrically.

On Variance Components in Semiparametric Mixed Models for Longitudinal Data

. First, to test the existence of random effects in semiparametric mixed models (SMMs) under only moment conditions on random effects and errors, we propose a very simple and easily implemented non-parametric test based on a difference between two estimators of the error variance. One test is consistent only under the null and the other can be so under both the null and alternatives. Instead of erroneously solving the non-standard two-sided testing problem, as in most papers in the literature, we solve it correctly and prove that the asymptotic distribution of our test statistic is standard normal. This avoids Monte Carlo approximations to obtain p-values, as is needed for many existing methods, and the test can detect local alternatives approaching the null at rates up to root n. Second, as the higher moments of the error are necessarily estimated because the standardizing constant involves these quantities, we propose a general method to conveniently estimate any moments of the error. Finally, a simulation study and a real data analysis are conducted to investigate the properties of our procedures.