Mixed Model Representation Research Articles

An ANOVA-based test for random effects in a nonlinear mixed model using a Monte Carlo permutation procedure

Nonlinear mixed models are essential for data analysis due to their versatility and adaptability to many research objectives and data formats. Testing misspecification is crucial to selecting a working nonlinear model. A useful and simpler model is often chosen using variance component tests. Nonlinear models are rarely studied in the literature, unlike linear models, which have numerous successful test development attempts. The latter test requires linearity between the answer and the explanatory factors. Thus, we use an existing linearization technique to build a linear model using pseudo-responses. If random effects were needed in the original nonlinear mixed model, the pseudo-responses vector inherits a mixed linear model representation. Simulation experiments and an application to a real-world bioassay data set evaluate this permutation test.

A flexible estimation of sectoral portfolio exposure to climate transition risks in the European stock market

I investigate the exposure of sectoral equity portfolios to climate transition risks by augmenting a three-factor asset pricing model with a green-minus-brown (GMB) factor as a proxy. I estimate the relationship between risk factors and excess returns within an additive mixed model representation, which flexibly captures possible changes in investors’ subjective beliefs as reflected in the determinants of asset pricing. Empirical evidence is provided based on European sectoral portfolios covering the 2016–2021 period. Compared to classic linear models, the results show an improvement in model goodness-of-fit when flexibly estimating the relationship between risk factors and excess returns. I confirm previous studies that exposure to climate transition risks particularly affects high-energy-intensity sectors. I also find heterogeneity in exposure between firms within each sectoral portfolio in terms of the sign and/or magnitude of estimates. Moreover, some firms still have no statistically significant exposure to climate transition risks.

Joint penalized spline modeling of multivariate longitudinal data, with application to HIV-1 RNA load levels and CD4 cell counts.

Motivated by the need to jointly model the longitudinal trajectories of HIV viral load levels and CD4 counts during the primary infection stage, we propose a joint penalized spline modeling approach that can be used to model the repeated measurements from multiple biomarkers of various types (eg, continuous, binary) simultaneously. This approach allows for flexible trajectories for each marker, accounts for potentially time-varying correlation between markers, and is robust to misspecification of knots. Despite its advantages, the application of multivariate penalized spline models, especially when biomarkers may be of different data types, has been limited in part due to its seemingly complexity in implementation. To overcome this, we describe a procedure that transforms the multivariate setting to the univariate one, and then makes use of the generalized linear mixed effect model representation of a penalized spline model to facilitate its implementation with standard statistical software. We performed simulation studies to evaluate the validity and efficiency through joint modeling of correlated biomarkers measured longitudinally compared to the univariate modeling approach. We applied this modeling approach to longitudinal HIV-1 RNA load and CD4 count data from Southern African cohorts to estimate features of the joint distributions such as the correlation and the proportion of subjects with high viral load levels and high CD4 cell counts overtime.

Tensor Cubic Smoothing Splines in Designed Experiments Requiring Residual Modelling

Modelling response surfaces using tensor cubic smoothing splines is presented for three designed experiments. The aim is to show how the analyses can be carried out using the asreml software in the R environment, and details of the analyses including the code to do so are presented in a tutorial style. The experiments were all run over time and involve an explanatory quantitative treatment variable; one experiment is a field trial which has a spatial component and involves an additional treatment. Thus, in addition to the response surface for the time by explanatory variable, modelling of temporal and, for the third experiment, of temporal and spatial effects at the residual level is required. A linear mixed model is used for analysis, and a mixed model representation of the tensor cubic smoothing spline is described and seamlessly incorporated in the full linear mixed model. The analyses show the flexibility and capacity of asreml for complex modelling. Supplementary materials accompanying this paper appear online.

Semiparametric analysis of correlated and interval-censored event-history data.

We propose a semiparametric multi-state frailty model to analyze clustered event-history data subject to interval censoring. The proposed model is motivated by an attempt to study the life course of dental caries at the tooth level, taking into account the multiplicity of caries states and the intra-oral clustering of observations made at periodic time points. Of particular interest is the study of the intra-oral distribution of processes leading to carious lesions, and whether this distribution varies with gender. The model assumes, in view of the covariate profile, a proportionality of the transition intensities conditional on subject-level frailties, coupled with a linear spline approximation of the log baseline intensities. The model estimation is conducted using a penalized likelihood where the smoothing parameters are estimated as reciprocal variance components under a mixed-model representation. A Bayesian method is proposed to predict tooth-level caries transition probabilities, which can be used for tailoring tooth-level caries treatment and prevention plans. Intensive simulation studies indicate that the model fitting and prediction perform reasonably well under realistic sample sizes. The practical utility of the methods is illustrated using data from a longitudinal study on oral health among children from low-income families residing in the city of Detroit, Michigan.

Fast symmetric additive covariance smoothing

A fast bivariate smoothing approach for symmetric surfaces is proposed that has a wide range of applications. It is shown how it can be applied to estimate the covariance function in longitudinal data as well as multiple additive covariances in functional data with complex correlation structures. The proposed symmetric smoother can handle (possibly noisy) data sampled on a common, dense grid as well as irregularly or sparsely sampled data. Estimation is based on bivariate penalized spline smoothing using a mixed model representation and the symmetry is used to reduce computation time compared to the usual non-symmetric smoothers. The application of the approach in functional principal component analysis for very general functional linear mixed models is outlined and its practical value is demonstrated in two applications. The approach is evaluated in extensive simulations. Documented open source software is provided that implements the fast symmetric bivariate smoother building on established algorithms for additivemodels.

Dividend initiators, winners during 2008 financial crisis

Purpose– The purpose of this paper is to describe and compare the mean response for selected financial variables in three dividend paying groups before and after the financial crisis of 2008. Dividend initiators are expected to be rewarded by investors over traditional dividend paying firms.Design/methodology/approach– Quarterly CRSP data from 2000 to 2012 are used to define dividend paying groups. Highly unbalanced financial data on dividend paying firms are analyzed in two truncated sample periods defined before and after the financial crisis. Fitted models describing differences in dividend paying groups are based on the linear mixed model representation of penalized splines with random effects to account for repeated measures over time.Findings– Results are presented on the important differences in selected financial variables before and after the financial crisis by dividend paying pattern group (traditional, initiators, residual/catering). Special emphasis is given to the analysis of market/book value ratio. Results demonstrate dividend initiators are rewarded over traditional dividend firms by investors. Firms with an intermittent paying pattern have no advantage. All dividend paying firms grow during the 2008 financial crisis. Traditional dividend payers have larger size than other dividend payers. The size effect explains results for several of the financial variables studied.Research limitations/implications– Future work can include an industry effect on the three dividend paying groups.Practical implications– Investors appear to prefer certainty as to when they receive a dividend over uncertainty, especially in times of economic downturn and economic recovery.Social implications– Enhanced awareness that different payment patterns exist and are rewarded differently over time on both the corporate issuer and investor sides.Originality/value– This study adds to body of knowledge of practical dividend payment patterns around a financial crisis. It also provides added support for dividend initiators.

Longitudinal Functional Models with Structured Penalties.

This article addresses estimation in regression models for longitudinally-collected functional covariates (time-varying predictor curves) with a longitudinal scaler outcome. The framework consists of estimating a time-varying coefficient function that is modeled as a linear combination of time-invariant functions with time-varying coefficients. The model uses extrinsic information to inform the structure of the penalty, while the estimation procedure exploits the equivalence between penalized least squares estimation and a linear mixed model representation. The process is empirically evaluated with several simulations and it is applied to analyze the neurocognitive impairment of HIV patients and its association with longitudinally-collected magnetic resonance spectroscopy (MRS) curves.

Statistical and Computational Challenges in Whole Genome Prediction and Genome-Wide Association Analyses for Plant and Animal Breeding

Whole genome prediction (WGP) modeling and genome-wide association (GWA) analyses are big data issues in agricultural quantitative genetics. Both areas require meaningful input from the statistical scholarly community in order to further improve the accuracy of prediction of genetic merit and inference on putative causal variants as well as improving the computational efficiency of existing methods and algorithms. These concerns have become increasingly critical as new sequencing technologies will only exacerbate current model dimensionality problems. We focus primarily on mixed model and hierarchical Bayesian analyses which have been most commonly pursued by animal and plant breeders for WGP thus far. We draw attention to our observation that many such previous analyses have not carefully inferred upon hyperparameters defined at the top levels of the Bayesian model hierarchy, but simply arbitrarily specify their values. We also reassess previous discussions on WGP model dimensionality, believing that useful data augmentation schemes utilized in various Markov Chain Monte Carlo (MCMC) schemes have led to a general misunderstanding that heavy-tailed or variable selection-based WGP models may be highly parameterized relative to more standard mixed model representations. Computational efficiency is addressed with respect to MCMC and competitive, albeit approximate, alternatives. Furthermore, GWA analyses are reassessed, encouraging a greater reliance on shrinkage-based inferences based on critically chosen priors, instead of potentially nonreproducible fixed effects P value-based inference.

Bayesian nonlinear model selection for gene regulatory networks.

Gene regulatory networks represent the regulatory relationships between genes and their products and are important for exploring and defining the underlying biological processes of cellular systems. We develop a novel framework to recover the structure of nonlinear gene regulatory networks using semiparametric spline-based directed acyclic graphical models. Our use of splines allows the model to have both flexibility in capturing nonlinear dependencies as well as control of overfitting via shrinkage, using mixed model representations of penalized splines. We propose a novel discrete mixture prior on the smoothing parameter of the splines that allows for simultaneous selection of both linear and nonlinear functional relationships as well as inducing sparsity in the edge selection. Using simulation studies, we demonstrate the superior performance of our methods in comparison with several existing approaches in terms of network reconstruction and functional selection. We apply our methods to a gene expression dataset in glioblastoma multiforme, which reveals several interesting and biologically relevant nonlinear relationships.

Structured Additive Regression Models: AnRInterface toBayesX

Structured additive regression (STAR) models provide a flexible framework for modeling possible nonlinear effects of covariates: They contain the well established frameworks of generalized linear models and generalized additive models as special cases but also allow a wider class of effects, e.g., for geographical or spatio-temporal data, allowing for specification of complex and realistic models. BayesX is standalone software package providing software for fitting general class of STAR models. Based on a comprehensive open-source regression toolbox written in C++, BayesX uses Bayesian inference for estimating STAR models based on Markov chain Monte Carlo simulation techniques, a mixed model representation of STAR models, or stepwise regression techniques combining penalized least squares estimation with model selection. BayesX not only covers models for responses from univariate exponential families, but also models from less-standard regression situations such as models for multi-categorical responses with either ordered or unordered categories, continuous time survival data, or continuous time multi-state models. This paper presents a new fully interactive R interface to BayesX: the R package R2BayesX. With the new package, STAR models can be conveniently specified using R’s formula language (with some extended terms), fitted using the BayesX binary, represented in R with objects of suitable classes, and finally printed/summarized/plotted. This makes BayesX much more accessible to users familiar with R and adds extensive graphics capabilities for visualizing fitted STAR models. Furthermore, R2BayesX complements the already impressive capabilities for semiparametric regression in R by a comprehensive toolbox comprising in particular more complex response types and alternative inferential procedures such as simulation-based Bayesian inference.

Penalized function-on-function regression

A general framework for smooth regression of a functional response on one or multiple functional predictors is proposed. Using the mixed model representation of penalized regression expands the scope of function-on-function regression to many realistic scenarios. In particular, the approach can accommodate a densely or sparsely sampled functional response as well as multiple functional predictors that are observed on the same or different domains than the functional response, on a dense or sparse grid, and with or without noise. It also allows for seamless integration of continuous or categorical covariates and provides approximate confidence intervals as a by-product of the mixed model inference. The proposed methods are accompanied by easy to use and robust software implemented in the pffr function of the R package refund. Methodological developments are general, but were inspired by and applied to a diffusion tensor imaging brain tractography dataset.

ESTIMATION AND INFERENCE FOR VARYING-COEFFICIENT MODELS WITH NONSTATIONARY REGRESSORS USING PENALIZED SPLINES

This paper considers estimation and inference for varying-coefficient models with nonstationary regressors. We propose a nonparametric estimation method using penalized splines, which achieves the same optimal convergence rate as kernel-based methods, but enjoys computation advantages. Utilizing the mixed model representation of penalized splines, we develop a likelihood ratio test statistic for checking the stability of the regression coefficients. We derive both the exact and the asymptotic null distributions of this test statistic. We also demonstrate its optimality by examining its local power performance. These theoretical findings are well supported by simulation studies.

A Bayesian multinomial modeling of spatial pattern of co-morbidity of malaria and non-malarial febrile illness among young children in Nigeria.

Children in developing countries continue to suffer mortality and morbidity from a number of illnesses, among which are malaria and non-malarial febrile illnesses, which epidemiologically overlap. We examined the spatial pattern and risk factors of co-morbidity of malaria and non-malarial febrile illness among children aged 6-59 months in Nigeria. Using data from the 2010 Nigeria Malaria Indicator Survey, we considered the co-morbidity of malaria and non-malarial febrile illness among the children as multicategorical and selected a mixed multinomial logit model capable of incorporating covariates of different types. Inference was Bayesian, based on multicategorical linear mixed-model representation. We found that the risk of co-morbidity of malaria and non-malarial febrile illness increases as a child advances in age while the risk of non-malarial fever reduces after about 32 months of age. Area of residence (urban or rural), wealth index and type of roofing material used in the dwelling are other important risk factors for the co-morbidity found in this study. Further, children from four of Nigeria's 37 states are at high risk of malaria. Disease preventive measures need to be intensified, with more focus on rural areas and the poor. Campaigns for use of insecticide-treated bed nets need be more aggressive in all Nigerian states.

Additive mixed models with approximate Dirichlet process mixtures: the EM approach

We consider additive mixed models for longitudinal data with a nonlinear time trend. As random effects distribution an approximate Dirichlet process mixture is proposed that is based on the truncated version of the stick breaking presentation of the Dirichlet process and provides a Gaussian mixture with a data driven choice of the number of mixture components. The main advantage of the specification is its ability to identify clusters of subjects with a similar random effects structure. For the estimation of the trend curve the mixed model representation of penalized splines is used. An Expectation-Maximization algorithm is given that solves the estimation problem and that exhibits advantages over Markov chain Monte Carlo approaches, which are typically used when modeling with Dirichlet processes. The method is evaluated in a simulation study and applied to theophylline data and to body mass index profiles of children.

Fast smoothing parameter separation in multidimensional generalized P-splines: the SAP algorithm

A new computational algorithm for estimating the smoothing parameters of a multidimensional penalized spline generalized linear model with anisotropic penalty is presented. This new proposal is based on the mixed model representation of a multidimensional P-spline, in which the smoothing parameter for each covariate is expressed in terms of variance components. On the basis of penalized quasi-likelihood methods, closed-form expressions for the estimates of the variance components are obtained. This formulation leads to an efficient implementation that considerably reduces the computational burden. The proposed algorithm can be seen as a generalization of the algorithm by Schall (1991)--for variance components estimation--to deal with non-standard structures of the covariance matrix of the random effects. The practical performance of the proposed algorithm is evaluated by means of simulations, and comparisons with alternative methods are made on the basis of the mean square error criterion and the computing time. Finally, we illustrate our proposal with the analysis of two real datasets: a two dimensional example of historical records of monthly precipitation data in USA and a three dimensional one of mortality data from respiratory disease according to the age at death, the year of death and the month of death.

Estimation and Inference for Varying-Coefficient Models With Nonstationary Regressors Using Penalized Splines

This paper considers estimation and inference for varying-coefficient models with nonstationary regressors. We propose a nonparametric estimation method using penalized splines, which achieves the same optimal convergence rate as kernel-based methods, but enjoys computation advantages. Utilizing the mixed model representation of penalized splines, we develop a likelihood ratio test statistic for checking the stability of the regression coefficients. We derive both the exact and the asymptotic null distributions of this test statistic. We also demonstrate its optimality by examining its local power performance. These theoretical findings are well supported by simulation studies.

Efficient two-dimensional smoothing with [formula omitted]-spline ANOVA mixed models and nested bases

Low-rank smoothing techniques have gained much popularity in non-standard regression modeling. In particular, penalized splines and tensor product smooths are used as flexible tools to study non-parametric relationships among several covariates. The use of standard statistical software facilitates their use for several types of problems and applications. However, when interaction terms are considered in the modeling, and multiple smoothing parameters need to be estimated standard software does not work well when datasets are large or higher-order interactions are included or need to be tested. In this paper, a general approach for constructing and estimating bivariate smooth models for additive and interaction terms using penalized splines is proposed. The formulation is based on the mixed model representation of the smooth-ANOVA model by Lee and Durbán (in press), and several nested models in terms of random effects components are proposed. Each component has a clear interpretation in terms of function shape and model identifiability constraints. The term PS-ANOVA is coined for this type of models. The estimation method is relatively straightforward based on the algorithm by Schall (1991) for generalized linear mixed models. Further, a simplification of the smooth interaction term is used by constructing lower-rank basis (nested basis). Finally, some simulation studies and real data examples are presented to evaluate the new model and the estimation method.

On Testing an Unspecified Function Through a Linear Mixed Effects Model with Multiple Variance Components

We examine a generalized F-test of a nonparametric function through penalized splines and a linear mixed effects model representation. With a mixed effects model representation of penalized splines, we imbed the test of an unspecified function into a test of some fixed effects and a variance component in a linear mixed effects model with nuisance variance components under the null. The procedure can be used to test a nonparametric function or varying-coefficient with clustered data, compare two spline functions, test the significance of an unspecified function in an additive model with multiple components, and test a row or a column effect in a two-way analysis of variance model. Through a spectral decomposition of the residual sum of squares, we provide a fast algorithm for computing the null distribution of the test, which significantly improves the computational efficiency over bootstrap. The spectral representation reveals a connection between the likelihood ratio test (LRT) in a multiple variance components model and a single component model. We examine our methods through simulations, where we show that the power of the generalized F-test may be higher than the LRT, depending on the hypothesis of interest and the true model under the alternative. We apply these methods to compute the genome-wide critical value and p-value of a genetic association test in a genome-wide association study (GWAS), where the usual bootstrap is computationally intensive (up to 10(8) simulations) and asymptotic approximation may be unreliable and conservative.

Corrected Confidence Bands for Functional Data Using Principal Components

Functional principal components (FPC) analysis is widely used to decompose and express functional observations. Curve estimates implicitly condition on basis functions and other quantities derived from FPC decompositions; however these objects are unknown in practice. In this article, we propose a method for obtaining correct curve estimates by accounting for uncertainty in FPC decompositions. Additionally, pointwise and simultaneous confidence intervals that account for both model- and decomposition-based variability are constructed. Standard mixed model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition. Iterated expectation and variance formulas combine model-based conditional estimates across the distribution of decompositions. A bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities. Our method compares favorably to competing approaches in simulation studies that include both densely and sparsely observed functions. We apply our method to sparse observations of CD4 cell counts and to dense white-matter tract profiles. Code for the analyses and simulations is publicly available, and our method is implemented in the R package refund on CRAN.