Semiparametric Mixed-effects Models Research Articles

Flexible Bayesian semiparametric mixed-effects model for skewed longitudinal data

BackgroundIn clinical trials and epidemiological research, mixed-effects models are commonly used to examine population-level and subject-specific trajectories of biomarkers over time. Despite their increasing popularity and application, the specification of these models necessitates a great deal of care when analysing longitudinal data with non-linear patterns and asymmetry. Parametric (linear) mixed-effect models may not capture these complexities flexibly and adequately. Additionally, assuming a Gaussian distribution for random effects and/or model errors may be overly restrictive, as it lacks robustness against deviations from symmetry.MethodsThis paper presents a semiparametric mixed-effects model with flexible distributions for complex longitudinal data in the Bayesian paradigm. The non-linear time effect on the longitudinal response was modelled using a spline approach. The multivariate skew-t distribution, which is a more flexible distribution, is utilized to relax the normality assumptions associated with both random-effects and model errors.ResultsTo assess the effectiveness of the proposed methods in various model settings, simulation studies were conducted. We then applied these models on chronic kidney disease (CKD) data and assessed the relationship between covariates and estimated glomerular filtration rate (eGFR). First, we compared the proposed semiparametric partially linear mixed-effect (SPPLM) model with the fully parametric one (FPLM), and the results indicated that the SPPLM model outperformed the FPLM model. We then further compared four different SPPLM models, each assuming different distributions for the random effects and model errors. The model with a skew-t distribution exhibited a superior fit to the CKD data compared to the Gaussian model. The findings from the application revealed that hypertension, diabetes, and follow-up time had a substantial association with kidney function, specifically leading to a decrease in GFR estimates.ConclusionsThe application and simulation studies have demonstrated that our work has made a significant contribution towards a more robust and adaptable methodology for modeling intricate longitudinal data. We achieved this by proposing a semiparametric Bayesian modeling approach with a spline smoothing function and a skew-t distribution.

Semiparametric multinomial mixed-effects models: A university students profiling tool

Many applicative studies deal with multinomial responses and hierarchical data. Performing clustering at the highest level of grouping, in multilevel multinomial regression, is also often of interest. In this study we analyse Politecnico di Milano data with the aim of profiling students, modelling their probabilities of belonging to different categories and considering their nested structure within engineering degree programmes. In particular, we are interested in clustering degree programmes standing on their effects on different types of student career. To this end, we propose an EM algorithm for implementing semiparametric mixed-effects models dealing with a multinomial response. The novel semiparametric approach assumes the random effects to follow a multivariate discrete distribution with an a priori unknown number of support points, that is, allowed to differ across response categories. The advantage of this modelling is twofold: the discrete distribution on random effects allows, first, to express the marginal density as a weighted sum, avoiding numerical problems in the integration step, typical of the parametric approach, and, second, to identify a latent structure at the highest level of the hierarchy where groups are clustered into subpopulations.

Semiparametric mixed-effects model for analysis of non-invasive longitudinal hemodynamic responses during bone graft healing.

When dealing with longitudinal data, linear mixed-effects models (LMMs) are often used by researchers. However, LMMs are not always the most adequate models, especially if we expect a nonlinear relationship between the outcome and a continuous covariate. To allow for more flexibility, we propose the use of a semiparametric mixed-effects model to evaluate the overall treatment effect on the hemodynamic responses during bone graft healing and build a prediction model for the healing process. The model relies on a closed-form expectation-maximization algorithm, where the unknown nonlinear function is estimated using a Lasso-type procedure. Using this model, we were able to estimate the effect of time for individual mice in each group in a nonparametric fashion and the effect of the treatment while accounting for correlation between observations due to the repeated measurements. The treatment effect was found to be statistically significant, with the autograft group having higher total hemoglobin concentration than the allograft group.

Modelling trends of CD4 counts for patients on antiretroviral therapy (ART): a comprehensive health care clinic in Nairobi, Kenya

BackgroundIn resource-limited settings, changes in CD4 counts constitute an important component in patient monitoring and evaluation of treatment response as these patients do not have access to routine viral load testing. In this study, we quantified trends on CD4 counts in patients on highly active antiretroviral therapy (HAART) in a comprehensive health care clinic in Kenya between 2011 and 2017. We evaluated the rate of change in CD4 cell count in response to antiretroviral treatment. We further assessed factors that influenced time to treatment change focusing on baseline characteristics of the patients and different initial drug regimens used. This was a retrospective study involving 432 naïve HIV patients that had at least two CD4 count measurements for the period. The relationship between CD4 cell count and time was modeled using a semi parametric mixed effects model while the Cox proportional hazards model was used to assess factors associated with the first regimen change.ResultsMajority of the patients were females and the average CD4 count at start of treatment was 362.1 cell/mm^3. The CD4 count measurements increased nonlinearly over time and these trends were similar regardless of the treatment regimen administered to the patients. The change of logarithm CD4 cell count rises fast for in the first 450 days of antiretroviral initiation. The average time to first regimen change was 2142 days. Tenoforvir (TDF) based regimens had a lower drug substitution(aHR 0.2682, 95% CI:0.08263- 0.8706) compared to Zidovudine(AZT).ConclusionThe backbone used was found to be associated with regimen changes among the patients with fewer switches being observed, with the use of TDF when compared to AZT. There was however no significant difference between TDF and AZT in terms of the rate of change in logarithm CD4 count over time.

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.

Regression analysis of intelligent education based on linear mixed effect model

In order to explore the regression effect of the intelligent education model, based on the linear mixed effect model, this paper constructs a regression analysis model based on the linear mixed effect model and studies the estimation algorithm of the linear mixed effect model under massive data. Moreover, this paper combines the three-step estimation method with the divide-and-conquer algorithm to propose a three-stage estimation algorithm. In addition, this paper studies the estimation algorithm of the semi-parametric mixed effect model under massive data, and combines the local linear least square method with the divide and conquer algorithm to obtain a new estimation algorithm. At the same time, this article extends the estimation algorithm to the semi-parametric mixed effect model under massive data to obtain a new estimation algorithm, so that the semi-parametric mixed model is also applicable under massive data. Finally, this paper constructs an intelligent education regression analysis model with data mining function based on actual needs, and designs experiments to verify the performance of the model. The research results show that the model constructed in this paper has a certain effect.

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

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

Multivariate t semiparametric mixed-effects model for longitudinal data with multiple characteristics

Semiparametric mixed-effects models (SMM) have received increasing attention in recent years because of the greater flexibility in analysing longitudinal trajectories. However, the normality assumption of SMM may be unrealistic when outliers occur in the data. This paper presents a semiparametric extension of the multivariate t linear mixed-effects model (MtLMM), called the multivariate t semiparametric mixed model (MtSMM). To be specific, the MtSMM incorporates a parametric linear function related to the fixed covariate effects and random effects which have a joint multivariate t distribution together with an arbitrary nonparametric smooth function to capture the unexpected patterns. A computationally analytical EM-based algorithm is developed for carrying out maximum likelihood estimation of the MtSMM. Simulation studies and a real example concerning the analysis of PBCseq data are used to investigate the empirical behaviour of the proposed methodology.

Random forests for high-dimensional longitudinal data.

Random forests are one of the state-of-the-art supervised machine learning methods and achieve good performance in high-dimensional settings where p, the number of predictors, is much larger than n, the number of observations. Repeated measurements provide, in general, additional information, hence they are worth accounted especially when analyzing high-dimensional data. Tree-based methods have already been adapted to clustered and longitudinal data by using a semi-parametric mixed effects model, in which the non-parametric part is estimated using regression trees or random forests. We propose a general approach of random forests for high-dimensional longitudinal data. It includes a flexible stochastic model which allows the covariance structure to vary over time. Furthermore, we introduce a new method which takes intra-individual covariance into consideration to build random forests. Through simulation experiments, we then study the behavior of different estimation methods, especially in the context of high-dimensional data. Finally, the proposed method has been applied to an HIV vaccine trial including 17 HIV-infected patients with 10 repeated measurements of 20,000 gene transcripts and blood concentration of human immunodeficiency virus RNA. The approach selected 21 gene transcripts for which the association with HIV viral load was fully relevant and consistent with results observed during primary infection.

Kernel estimation in semiparametric mixed effect longitudinal modeling

HIV damages the immune system by targeting the CD4 cells. Hence CD4 count data modeling is important in the analysis of HIV infection. This paper considers a semiparametric mixed effects model for the analysis of CD4 longitudinal data. The model is a natural extension to the linear mixed and semiparametric models that uses parametric linear model to present the covariate effects and an arbitrary nonparametric smooth function to model the time effect and account for the within subject correlation using random effects. We approximate the nonparametric function by the profile kernel method, and make use of the weighted least squares to estimate the regression coefficients. Under some regularity conditions, the asymptotic normality of the resulting estimator is established and the performance is compared with the backfitting method. Although, two estimators share the same asymptotic variance matrix for independent data, it is shown that, backfitting often produces larger bias and variance than the profile-kernel method, asymptotically. Consequently, the use of backfitting method is no longer advised in semiparametric mixed effect longitudinal model. For practical implementation and also improve efficiency, the estimation of the covariance function is accomplished using an iterative algorithm. Performance of the proposed methods are compared through a simulation study and the analysis of CD4 data.

Analysis of generalized semiparametric mixed varying-coefficients models for longitudinal data.

The generalized semiparametric mixed varying-coefficient effects model for longitudinal data can accommodate a variety of link functions and flexibly model different types of covariate effects, including time-constant, time-varying, and covariate-varying effects. The time-varying effects are unspecified functions of time and the covariate-varying effects are nonparametric functions of a possibly time-dependent exposure variable. A semiparametric estimation procedure is developed that uses local linear smoothing and profile weighted least squares, which requires smoothing in the two different and yet connected domains of time and the time-dependent exposure variable. The asymptotic properties of the estimators of both nonparametric and parametric effects are investigated. In addition, hypothesis testing procedures are developed to examine the covariate effects. The finite-sample properties of the proposed estimators and testing procedures are examined through simulations, indicating satisfactory performances. The proposed methods are applied to analyze the ACTG 244 clinical trial to investigate the effects of antiretroviral treatment switching in HIV-infected patients before and after developing the T215Y antiretroviral drug resistance mutation.

Semiparametric Mixed Effects Models for Unsupervised Classification of Italian Schools

Summary The main purpose of the paper is to improve research on school effectiveness by applying a new strategy for uncovering subpopulations of schools that differ in terms of distribution of student outcomes. We propose a semiparametric mixed effects model with an expectation–maximization algorithm to estimate its parameters and we apply it to the Italian Institute for the Educational Evaluation of Instruction and Training data of 2013–2014 as a tool for the identification of latent subpopulations of schools. The semiparametric assumption provides the random effects of the mixed effects model to be distributed according to a discrete distribution with an (a priori) unknown number of support points. This modelling induces an automatic clustering of schools (the higher level of hierarchy), where schools within the same cluster share the same random effects. The latent subpopulations of schools identified may then be exploited through the use of multinomial models that include school level features. The novelties introduced by this paper are twofold: first, the semiparametric expectation–maximization algorithm is an innovative method that could be used in many classification problems; second, its application to education data represents a new approach to study school effectiveness.

A Joint Modeling Approach for Longitudinal Data with Informative Observation Times and a Terminal Event

In this article, we propose a new joint modeling approach for the analysis of longitudinal data with informative observation times and a dependent terminal event. We specify a semiparametric mixed effects model for the longitudinal process, a proportional rate frailty model for the observation process, and a proportional hazards frailty model for the terminal event. The association among the three related processes is modeled via two latent variables. Estimating equation approaches are developed for parameter estimation, and the asymptotic properties of the proposed estimators are established. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a medical cost study of chronic heart failure patients is illustrated.

Robust probabilistic classification applicable to irregularly sampled functional data

A robust probabilistic classifier for functional data is developed to predict class membership based on functional input measurements and to provide a reliable probability estimate for class membership. The method combines a Bayes classifier and semi-parametric mixed effects model with robust tuning parameter to make the method robust to outlying curves, and to improve the accuracy of the risk or uncertainty estimates, which is crucial in medical diagnostic applications. The approach applies to functional data with varying ranges and irregular sampling without making parametric assumptions on the within-curve covariance. Simulation studies evaluate the proposed method and competitors in terms of sensitivity to heavy tailed functional distributions and outlying curves. Classification performance is evaluated by both error rate and logloss, the latter of which imposes heavier penalties on highly confident errors than on less confident errors. Run-time experiments on the R implementation indicate that the proposed method scales well computationally. Illustrative applications include data from quantitative ultrasound analysis and phoneme recognition.

A semiparametric method for estimating the progression of cognitive decline in dementia

ABSTRACTAlthough many patients develop cognitive decline, their trajectories of cognitive decline are diverse and incompletely understood. Accurate prediction of the cognitive decline process is critical for early treatment and management of dementia. We used the Clinical Dementia Rating Score Sum of Boxes (CDR SUM) score as a cognitive decline proxy, and investigated factors that are potentially associated with cognitive decline, including duration, age at onset, sex, education, health-related symptoms, and neuropsychiatric symptoms. We analyzed data from an established 10-year longitudinal patient registry of patients diagnosed with non-normal cognition. We compared a multi-level polynomial regression model and two semiparametric mixed-effects models, and applied Nakagawa and Schielzeth's R2GLMM and correlation coefficient as model selection criteria. The semiparametric method was selected to describe and predict the cognitive decline trajectory. Neuropsychiatric symptoms were indicators of a higher CDR SUM score. History of stroke, presence of disinhibition, and nighttime behavior disturbances were also associated with higher CDR SUM score. Older age of onset (> 86 years), educational level higher than high school education (≥ 12 education years), and the presence of irritation were indicators of slower cognitive decline. The semiparametric model can assist in estimating cognitive decline in terms of CDR SUM score, given individual characteristics.

A semiparametric joint model for terminal trend of quality of life and survival in palliative care research.

Palliative medicine is an interdisciplinary specialty focusing on improving quality of life (QOL) for patients with serious illness and their families. Palliative care programs are available or under development at over 80% of large US hospitals (300+ beds). Palliative care clinical trials present unique analytic challenges relative to evaluating the palliative care treatment efficacy which is to improve patients' diminishing QOL as disease progresses towards end of life (EOL). A unique feature of palliative care clinical trials is that patients will experience decreasing QOL during the trial despite potentially beneficial treatment. Often longitudinal QOL and survival data are highly correlated which, in the face of censoring, makes it challenging to properly analyze and interpret terminal QOL trend. To address these issues, we propose a novel semiparametric statistical approach to jointly model the terminal trend of QOL and survival data. There are two sub-models in our approach: a semiparametric mixed effects model for longitudinal QOL and a Cox model for survival. We use regression splines method to estimate the nonparametric curves and AIC to select knots. We assess the model performance through simulation to establish a novel modeling approach that could be used in future palliative care research trials. Application of our approach in a recently completed palliative care clinical trial is also presented.

Time-varying latent model for longitudinal data with informative observation and terminal event times

Longitudinal data often occur in follow-up studies, and in many situations, there may exist informative observation times and a dependent terminal event such as death that stops the follow-up. We propose a semiparametric mixed effect model with time-varying latent effects in the analysis of longitudinal data with informative observation times and a dependent terminal event. Estimating equation approaches are developed for parameter estimation, and asymptotic properties of the resulting estimators are established. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer study is provided.

Simple approximate MAP inference for Dirichlet processes mixtures

The Dirichlet process mixture model (DPMM) is a ubiquitous, flexible Bayesian nonparametric statistical model. However, full probabilistic inference in this model is analytically intractable, so that computationally intensive techniques such as Gibbs sampling are required. As a result, DPMM-based methods, which have considerable potential, are restricted to applications in which computational resources and time for inference is plentiful. For example, they would not be practical for digital signal processing on embedded hardware, where computational resources are at a serious premium. Here, we develop a simplified yet statistically rigorous approximate maximum a-posteriori (MAP) inference algorithm for DPMMs. This algorithm is as simple as DP-means clustering, solves the MAP problem as well as Gibbs sampling, while requiring only a fraction of the computational effort. (For freely available code that implements the MAP-DP algorithm for Gaussian mixtures see http://www.maxlittle.net/.) Unlike related small variance asymptotics (SVA), our method is non-degenerate and so inherits the “rich get richer” property of the Dirichlet process. It also retains a non-degenerate closed-form likelihood which enables out-of-sample calculations and the use of standard tools such as cross-validation. We illustrate the benefits of our algorithm on a range of examples and contrast it to variational, SVA and sampling approaches from both a computational complexity perspective as well as in terms of clustering performance. We demonstrate the wide applicabiity of our approach by presenting an approximate MAP inference method for the infinite hidden Markov model whose performance contrasts favorably with a recently proposed hybrid SVA approach. Similarly, we show how our algorithm can applied to a semiparametric mixed-effects regression model where the random effects distribution is modelled using an infinite mixture model, as used in longitudinal progression modelling in population health science. Finally, we propose directions for future research on approximate MAP inference in Bayesian nonparametrics.

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

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

Inference in semi-parametric spline mixed models for longitudinal data

In this paper, the authors investigate a robust semi-parametric mixed effects model for analyzing longitudinal data with an unspecified mean response function. The robust method, developed in the framework of the maximum likelihood, is used to bound the influence of potential outliers when estimating the model parameters. The authors also present a robust test procedure for assessing the significance of a variance component in the mixed model. An application is provided using a clinical dataset from a retinopathy of prematurity study in which longitudinal measurements were obtained from premature infants treated with supplemental oxygen. The empirical properties of the proposed estimators are also studied in simulations.