Variance Component Parameters Research Articles

Modified BIC Criterion for Model Selection in Linear Mixed Models

Linear mixed-effects models are widely used in applications to analyze clustered, hierarchical, and longitudinal data. Model selection in linear mixed models is more challenging than that of linear models as the parameter vector in a linear mixed model includes both fixed effects and variance component parameters. When selecting the variance components of the random effects, the variance of the random effects must be non-negative and the parameters may lie on the boundary of the parameter space. Therefore, classical model selection methods cannot be directly used to handle this situation. In this article, we propose a modified BIC for model selection with linear mixed-effects models that can solve the case when the variance components are on the boundary of the parameter space. Through the simulation results, we found that the modified BIC performed better than the regular BIC in most cases for linear mixed models. The modified BIC was also applied to a real dataset to choose the most-appropriate model.

Multilevel multinomial logit regression model with random effects: application to flash EuroBarometer euro survey data

A multilevel model for nominal data in the framework of generalized linear mixed models (GLMM) is developed to account for the inherent dependencies among observations within clusters. Motivated by a dataset from the Flash EuroBarometer survey (FEBS), the random region and country effects are incorporated into the linear predictor of a GLMM to accommodate the nested clusterings. The fixed effects are estimated by maximizing the penalized likelihood function, whereas the random variance component parameters are predicted via the restricted maximum likelihood (REML) estimation method. The model is employed to analyse the FEBS data. A Monte Carlo simulation study is conducted to evaluate the performance of estimators.

Fast heritability estimation based on MINQUE and batch training.

Heritability, the proportion of phenotypic variance explained by genome-wide single nucleotide polymorphisms (SNPs) in unrelated individuals, is an important measure of the genetic contribution to human diseases and plays a critical role in studying the genetic architecture of human diseases. Linear mixed model (LMM) has been widely used for SNP heritability estimation, where variance component parameters are commonly estimated by using a restricted maximum likelihood (REML) method. REML is an iterative optimization algorithm, which is computationally intensive when applied to large-scale datasets (e.g. UK Biobank). To facilitate the heritability analysis of large-scale genetic datasets, we develop a fast approach, minimum norm quadratic unbiased estimator (MINQUE) with batch training, to estimate variance components from LMM (LMM.MNQ.BCH). In LMM.MNQ.BCH, the parameters are estimated by MINQUE, which has a closed-form solution for fast computation and has no convergence issue. Batch training has also been adopted in LMM.MNQ.BCH to accelerate the computation for large-scale genetic datasets. Through simulations and real data analysis, we demonstrate that LMM.MNQ.BCH is much faster than two existing approaches, GCTA and BOLT-REML.

Bag of little bootstraps for massive and distributed longitudinal data.

Linear mixed models are widely used for analyzing longitudinal datasets, and the inference for variance component parameters relies on the bootstrap method. However, health systems and technology companies routinely generate massive longitudinal datasets that make the traditional bootstrap method infeasible. To solve this problem, we extend the highly scalable bag of little bootstraps method for independent data to longitudinal data and develop a highly efficient Julia package MixedModelsBLB.jl. Simulation experiments and real data analysis demonstrate the favorable statistical performance and computational advantages of our method compared to the traditional bootstrap method. For the statistical inference of variance components, it achieves 200 times speedup on the scale of 1 million subjects (20 million total observations), and is the only currently available tool that can handle more than 10 million subjects (200 million total observations) using desktop computers.

Bayesian multi-way balanced nested MANOVA models with random effects and a large number of the main factor levels

This article considers the balanced nested multi-way multivariate analysis of variance (MANOVA) models with random effects and a large number of main factor levels under certain prior assumptions. Two different parametrizations for the MANOVA models with random effects and the corresponding explicit asymptotics are established. The asymptotic approximations are then compared with those obtained from the classical large-sample approximation and Markov chain Monte Carlo method via a balanced nested two-way MANOVA model with random effects. Simulation results demonstrate that our approach is superior to the classical approximation method on estimating the posterior standard deviations of variance component parameters.

Multiple imputation methods for handling incomplete longitudinal and clustered data where the target analysis is a linear mixed effects model.

Multiple imputation (MI) is increasingly popular for handling multivariate missing data. Two general approaches are available in standard computer packages: MI based on the posterior distribution of incomplete variables under a multivariate (joint) model, and fully conditional specification (FCS), which imputes missing values using univariate conditional distributions for each incomplete variable given all the others, cycling iteratively through the univariate imputation models. In the context of longitudinal or clustered data, it is not clear whether these approaches result in consistent estimates of regression coefficient and variance component parameters when the analysis model of interest is a linear mixed effects model (LMM) that includes both random intercepts and slopes with either covariates or both covariates and outcome contain missing information. In the current paper, we compared the performance of seven different MI methods for handling missing values in longitudinal and clustered data in the context of fitting LMMs with both random intercepts and slopes. We study the theoretical compatibility between specific imputation models fitted under each of these approaches and the LMM, and also conduct simulation studies in both the longitudinal and clustered data settings. Simulations were motivated by analyses of the association between body mass index (BMI) and quality of life (QoL) in the Longitudinal Study of Australian Children (LSAC). Our findings showed that the relative performance of MI methods vary according to whether the incomplete covariate has fixed or random effects and whether there is missingnesss in the outcome variable. We showed that compatible imputation and analysis models resulted in consistent estimation of both regression parameters and variance components via simulation. We illustrate our findings with the analysis of LSACdata.

AIMS: Average information matrix splitting

For linear mixed models with co-variance matrices which are not linearly dependent on variance component parameters, we prove that the average of the observed information and the Fisher information can be split into two parts. The essential part enjoys a simple and computational friendly formula, while the other part which involves a lot of computations is a random zero matrix and thus is negligible.

Inference on variance components near boundary in linear mixed effect models

AbstractIn making inference on variance components in linear mixed effect models, variance component parameters may be located on some boundary of a constrained parameter space, and hence usual asymptotic theory on parameter estimation, test statistics, and information criteria may not hold. We illustrate such boundary issues on variance components, and introduce some methodologies and properties along with literature. The maximum likelihood estimator of the variance parameter vector near some boundary distributes asymptotically as a projection of a normal random vector onto the boundary. The null distribution of the likelihood ratio test statistic is complicated, and hence it has been studied both from asymptotic and numerical aspects. Moreover, a boundary issue in model selection using information criteria is also essential and is closely related to that on the likelihood ratio test. We also describe a boundary issue on testing for linearity of a regression function using the relationship between linear mixed effect models and penalized spline models.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Data Reduction, Smoothing, and Filtering Statistical Models > Model Selection Statistical and Graphical Methods of Data Analysis > Bootstrap and Resampling

Estimation of Genetic Parameters for Some Production and Reproduction Traits Using Different Animal Models in Egyptian Buffaloes

Estimates of (co)variance component and genetic parameters using multi-traits animal model, traits studied were lactation period (LP), total lactation milk yield (TMY), and days open (DO). In all lactations based on 4745 records of Egyptian buffaloes calved during the period from 1980 to 2014 at Mehallet Mousa Experimental farms of Animal Production Research Institute (APRI), Ministry of Agriculture. The six mixed model used in the analysis included the non genetic effects of month and year of calving, parity, and the random effects of additive direct genetics, maternal effect, maternal permanent environmental and residual. Means for LP, TMY and Do were 220.632 d, 1445.26 kg and 184.34 d, respectively. The estimates of heritability in six models for LP, TMY and Do were ranged from (0.026±0.56 to 0.047±0.16), (0.15±0.57 to 0.021±0.15) and (0.02±0.020 to 0.020±0.13), respectively. While Estimates of maternal heritability was very low in six models for LP, TMY and DO and ranged from 0.055±0.026 to 0.18E-09±0.79E-05: 0.081±0.029 to 0.52E-09±0.13E-04; 0.053±0.01 to 0.83E-09±0.12E-04, respectively. The estimates of genetic correlations between LP and TMY in six models ranged from (0.53 to 0.84). Estimates of genetic correlation between DO and TMY in six models ranged from (-0.15 to 0.58).While the estimates of genetic correlation between LP and DO in six models ranged from (0.30 to 0.94). Corresponding estimates of phenotypic correlation among same traits fluctuated (0.17 to76). Thus, the influence of the maternal genetic effect for traits studied was lowest thereby no relative efficiency of improvement. Vice-versa, direct heritabilities for TMY and LP were efficiency, therefore considerable rate of genetic improvement these traits under investigation. Genetic betterment for LP following up improve each TMY and DO. The estimates of heritabilities were low for DO indicated that the major part of the variation in this trait was environmental and selection may not prove effective in bringing about genetic improvement. Therefore, preferable improving management can play a major role in this trait.

Asymptotic posterior distributions for balanced nested multi-way models with a large number of main effect levels

ABSTRACTWe derive the explicit form for the asymptotic posterior distribution of the balanced nested multi-way variance components model with the assumption that the number of the main factor levels tends to infinity while the number of any specific effect factor levels remains fixed. Under the multi-way model, we also study two different parameterizations, called the standard and the centering, and the relationship between certain quadratic forms of random effects and the variance component parameters. The asymptotic results are illustrated by a three-way model and by a simulation study under a two-way case.

Genetic parameters and predictions for direct and maternal growth traits in a multibreed Angus–Brahman cattle population using genomic–polygenic and polygenic models

The objectives of this research were to compare variance components, genetic parameters, and EBV rankings for birth weight (BW) direct and maternal, weaning weight (WW) direct and maternal, and postweaning gain from 205d to 365d (PWG) direct using three genomic–polygenic and one polygenic model representing four plausible beef cattle genetic evaluation scenarios for growth traits under subtropical conditions in the US southern region. In addition, EBV trends as percentage Brahman increased from 0% to 100% were evaluated for each trait and model. The dataset included 5264 animals from a multibreed Angus–Brahman population born from 1987 to 2013. Genomic–polygenic models 1 (GP1; pedigree relationships for all animals; genomic relationships for genotyped animals), 2 (GP2; pedigree relationships for non-genotyped animals; genomic relationships for genotyped animals), and 3 (GP3; no pedigree relationships; genomic relationships for genotyped animals) used actual and imputed genotypes from 46,768 SNP markers. Variance components and genetic parameters were estimated using REML procedures. Variance component and genetic parameter estimates from GP1 were the most similar to those from the polygenic model, followed by those from GP2, and the least similar (especially for maternal traits) were those from GP3. Similarly, the highest rank correlations were those between animal EBV from the polygenic model and GP1, followed by those between animal EBV from GP1 and GP2 and between the polygenic model and GP2. Model GP3 performed poorly for maternal traits due to ignoring calf–dam relationships. These results indicated that the polygenic model and genomic–polygenic model 1 should be preferred. However, high genotyping costs still make the polygenic model preferable for commercial beef cattle operations. Brahman animals tended to have higher EBV for BW direct and WW direct, and lower EBV for PWG direct, BW maternal, and WW maternal. However, low regression coefficients for EBV on Brahman fraction ensured that high, medium, and low EBV animals from all breed compositions existed in this multibreed population.

Multilevel cumulative logistic regression model with random effects: Application to British social attitudes panel survey data

A multilevel model for ordinal data in generalized linear mixed models (GLMM) framework is developed to account for the inherent dependencies among observations within clusters. Motivated by a data set from the British Social Attitudes Panel Survey (BSAPS), the random district effects and respondent effects are incorporated into the linear predictor to accommodate the nested clusterings. The fixed (random) effects are estimated (predicted) by maximizing the penalized quasi likelihood (PQL) function, whereas the variance component parameters are obtained via the restricted maximum likelihood (REML) estimation method. The model is employed to analyze the BSAPS data. Simulation studies are conducted to assess the performance of estimators.

Information based model selection criteria for generalized linear mixed models with unknown variance component parameters

This paper derives the corrected conditional Akaike information criteria for generalized linear mixed models by analytic approximation and parametric bootstrap. The sampling variation of both fixed effects and variance component parameter estimators are accommodated in the bias correction term. Simulation shows that the proposed corrected criteria provide good approximation to the true conditional Akaike information and demonstrates promising model selection results. The use of the criteria is demonstrated in the analysis of the chronic asthmatic patients’ data.

Robust REML estimation for k‐component Poisson mixture with random effects: application to the epilepsy seizure count data and urinary tract infections data

A robust version of residual maximum likelihood estimation for Poisson log-linear mixed model is developed, and the method is extended to k-component Poisson mixture with random effects. The method not only provides the robust estimators for the fixed effects and variance component parameters but also gives the robust prediction of random effects. Simulation results show that the proposed method is effective in limiting the impact of outliers under different data contamination schemes. The method is adopted to analyze the epilepsy seizure count data and the urinary tract infections data, which are deemed to contain several potential outliers. The results show that the proposed method provides better goodness of fit to the data and demonstrate the effect of the robust tuning mechanism.

Estimation of (Co)Variance Components and Genetic Parameter of Withers Height, Chest Girth and Body Length of Bali Cattle Using Animal Model

Selection criteria for performance test of yearling bulls commonly include indicators of important traits, such as withers height (WH), chest girth (CG) and body length (BL). Estimations of (co)variance component and genetic parameter of WH, CG and BL of Bali cattle are rarely reported or have not been reported. Therefore, the objective of this study was to estimate (co)variance components and genetic parameter among the indicator traits WH, CG and BL. Records for WH (n = 140), CG (n = 133), and BL (n = 130) were obtained from the Bali Breeding Center which consisted of 23 sires and 134 dams represented in the data. (Co)variance components were estimated using animal model, and average information REML was used to obtain estimates of heritability and genetic correlations. Fixed effects included sex, age, and calving season. The random effect of the animal was included to estimate additive genetic effects. Generally heritability estimates of 0.41±0.08, 0.40±0.09, and 0.39±0.08 were obtained for WH, CG and BL on the underlying scale, respectively. The genetic correlation between WH and CG (0.05) was close to zero, as were the estimated genetic correlations between WH and BL (0.05) and CG and BL (0.05). Genetic and phenotypic trends did not show improvement due to selection. However, these results suggest that concomitant selection for increased withers height, chest girth and body length would not be antagonistic.

Genetic parameter estimates for live weight and daily live weight gain obtained for Nellore bulls in a test station using different models

The objective of this study was to estimate (co)variance components and genetic parameters for live weight (LW) and daily live weight gain (LWG) of Nellore bulls in a test station using multi-trait and random regression models. In addition, breeding values for these traits were predicted by multi-trait and random regression analyses, and the rank of animals based on breeding values was compared with the current selection criterion of the test station (own performance). A total of 4758 Nellore bulls tested in a central station of the Beef Cattle Research Center (CPPC) between 1978 and 2007, including 2211 bulls from the CPPC herd and 2547 from commercial herds, were used. During the test, four LWs were recorded at intervals of 56days (LW1d, LW56d, LW112d and LW168d). LWG was calculated as the difference between two consecutive weights for three periods: 1 to 55 (LWG1), 56 to 111 (LWG2), and 112 to 168 (LWG3) days on test. For LW and LWG, the multi-trait model included the fixed effects of contemporary group (year-month of birth), dam age class, and animal age at recording as covariate. For random regression analysis, direct additive genetic and animal permanent environmental effects were modeled using linear, quadratic and cubic polynomial functions. Residual variances for LW and LWG were modeled using a step function with 1 or 3 classes, respectively. Contemporary group (year-month of birth and month of recording) and dam age class were included as fixed effects. The (co)variance components were estimated by the Restricted Maximum Likelihood method using the WOMBAT software. According to model comparison criterion, the model including cubic and quadratic Legendre polynomials to fit genetic and animal permanent environmental effects, respectively, was the most appropriate to describe the covariance structure of LW. For LWG, the BIC value indicated that the model including quadratic and linear Legendre polynomials was the most appropriate to fit genetic and animal permanent environmental effects, respectively. The variance component and genetic parameter estimates for LW and LWG obtained by random regression and multi-trait analyses were similar. Random regression on Legendre polynomials of days on test was more appropriate than multi-trait models to describe the genetic variation of growth traits in station-tested Nellore bulls. Selection based on breeding values for LWG during the test would result in the selection of bulls different from those chosen if final weight is applied as a selection criterion.

Conditional Akaike information criterion for generalized linear mixed models

In this study, a model identification instrument to determine the variance component structure for generalized linear mixed models ( glmm s) is developed based on the conditional Akaike information ( c ai ). In particular, an asymptotically unbiased estimator of the c ai (denoted as c aicc ) is derived as the model selection criterion which takes the estimation uncertainty in the variance component parameters into consideration. The relationship between bias correction and generalized degree of freedom for glmm s is also explored. Simulation results show that the estimator performs well. The proposed criterion demonstrates a high proportion of correct model identification for glmm s. Two sets of real data (epilepsy seizure count data and polio incidence data) are used to illustrate the proposed model identification method.

Inferences in generalized linear longitudinal mixed models

AbstractWithout realizing the fact that the time‐dependent covariates corresponding to the repeated discrete responses under a generalized linear longitudinal model (GLLM) cause non‐stationary (time dependent) correlations for the repeated responses, many existing studies use a stationary (either “working” or true) correlation structure to develop certain estimating equations for the regression effects involved in the model. By constructing suitable non‐stationary correlation structures both for longitudinal count and binary data, this article first demonstrates that the stationary correlations based estimation approaches may yield inefficient regression estimates. For efficient estimation, the article then suggests a true non‐stationary correlation structure based generalized quasi‐likelihood (GQL) estimation approach, where non‐stationary correlation structure is identified by exploiting the estimated lag correlations of the responses. A generalization of the GLLM to the familial‐longitudinal set up both for count and binary data is also discussed, where the data exhibit familial as well as non‐stationary longitudinal correlations, the familial correlations among the responses of the family members are being generated through a random common family effect. The GQL estimating equations are provided for the estimation of the regression and the variance component parameters of this generalized linear longitudinal mixed model (GLLMM), whereas the longitudinal correlations are estimated by solving suitable moment estimating equations. The Canadian Journal of Statistics 38: 174–196; 2010 © 2010 Statistical Society of Canada

Empirical Bayes Estimation of Small Area Means under a Nested Error Linear Regression Model with Measurement Errors in the Covariates

Abstract. Previously, small area estimation under a nested error linear regression model was studied with area level covariates subject to measurement error. However, the information on observed covariates was not used in finding the Bayes predictor of a small area mean. In this paper, we first derive the fully efficient Bayes predictor by utilizing all the available data. We then estimate the regression and variance component parameters in the model to get an empirical Bayes (EB) predictor and show that the EB predictor is asymptotically optimal. In addition, we employ the jackknife method to obtain an estimator of mean squared prediction error (MSPE) of the EB predictor. Finally, we report the results of a simulation study on the performance of our EB predictor and associated jackknife MSPE estimators. Our results show that the proposed EB predictor can lead to significant gain in efficiency over the previously proposed EB predictor.

A computer graphical user interface for survival mixture modelling of recurrent infections

Recurrent infections data are commonly encountered in medical research, where the recurrent events are characterised by an acute phase followed by a stable phase after the index episode. Two-component survival mixture models, in both proportional hazards and accelerated failure time settings, are presented as a flexible method of analysing such data. To account for the inherent dependency of the recurrent observations, random effects are incorporated within the conditional hazard function, in the manner of generalised linear mixed models. Assuming a Weibull or log-logistic baseline hazard in both mixture components of the survival mixture model, an EM algorithm is developed for the residual maximum quasi-likelihood estimation of fixed effect and variance component parameters. The methodology is implemented as a graphical user interface coded using Microsoft visual C ++ . Application to model recurrent urinary tract infections for elderly women is illustrated, where significant individual variations are evident at both acute and stable phases. The survival mixture methodology developed enable practitioners to identify pertinent risk factors affecting the recurrent times and to draw valid conclusions inferred from these correlated and heterogeneous survival data.