An evaluation of ridge estimator in linear mixed models: an example from kidney failure data

This paper is concerned with the ridge estimation of fixed and random effects in the context of Henderson's mixed model equations in the linear mixed model. For this purpose, a penalized likelihood method is proposed. A linear combination of ridge estimator for fixed and random effects is compared to a linear combination of best linear unbiased estimator for fixed and random effects under the mean-square error (MSE) matrix criterion. Additionally, for choosing the biasing parameter, a method of MSE under the ridge estimator is given. A real data analysis is provided to illustrate the theoretical results and a simulation study is conducted to characterize the performance of ridge and best linear unbiased estimators approach in the linear mixed model.     

A modified nested-error regression model for small area estimation

A nested-error regression model having both fixed and random effects is introduced to estimate linear parameters of small areas. The model is applicable to data having a proportion of domains where the variable of interest cannot be described by a standard linear mixed model. Algorithms and formulas to fit the model, to calculate EBLUP and to estimate mean-squared errors are given. A Monte Carlo simulation experiment is presented to illustrate the gain of precision obtained by using the proposed model and to obtain some practical conclusions. A motivating application to Spanish Labour Force Survey data is also given.     

Modelling and estimating heavy-tailed non-homogeneous correlated queues: Pareto-inverse gamma HGLM with covariates

Evidence of communication traffic complexity reveals correlation in a within-queue and heterogeneity among queues. We show how a random-effect model can be used to accommodate these kinds of phenomena. We apply a Pareto distribution for arrival (service) time of individual queue for given arrival (service) rate. For modelling potential correlation in arrival (service) times within a queue and heterogeneity of the arrival (service) rates among queues, we use an inverse gamma distribution. This modelling approach is then applied to the cache access log data processed through an Internet server. We believe that our approach is potentially useful in the area of network resource management.     

Sparse alternatives to ridge regression: a random effects approach

In a calibration of near-infrared (NIR) instrument, we regress some chemical compositions of interest as a function of their NIR spectra. In this process, we have two immediate challenges: first, the number of variables exceeds the number of observations and, second, the multicollinearity between variables are extremely high. To deal with the challenges, prediction models that produce sparse solutions have recently been proposed. The term ‘sparse’ means that some model parameters are zero estimated and the other parameters are estimated naturally away from zero. In effect, a variable selection is embedded in the model to potentially achieve a better prediction. Many studies have investigated sparse solutions for latent variable models, such as partial least squares and principal component regression, and for direct regression models such as ridge regression (RR). However, in the latter, it mainly involves an L₁ norm penalty to the objective function such as lasso regression. In this study, we investigate new sparse alternative models for RR within a random effects model framework, where we consider Cauchy and mixture-of-normals distributions on the random effects. The results indicate that the mixture-of-normals model produces a sparse solution with good prediction and better interpretation. We illustrate the methods using NIR spectra datasets from milk and corn specimens.     

Discrepancy in regression estimates between log-normal and gamma: some case studies

In regression models with multiplicative error, estimation is often based on either the log-normal or the gamma model. It is well known that the gamma model with constant coefficient of variation and the log-normal model with constant variance give almost the same analysis. This article focuses on the discrepancies of the regression estimates between the two models based on real examples. It identifies that even though the variance or the coefficient of variation remains constant, but regression estimates may be different between the two models. It also identifies that for the same positive data set, the variance is constant under the log-normal model but non-constant under the gamma model. For this data set, the regression estimates are completely different between the two models. In the process, it explains the causes of discrepancies between the two models.     

A power comparison of three tests for design effects in a random effects covariance model

A preliminary testing procedure for design ettecta in a ran-dom effects covariance model is Compared with the usual procedure to see if the power of the latter can be improved. A procedure which ignores the random covariate effects is included for comparison and for study of misspecification effects. Methodology is based on Roebruck's (1982) results for regular linear models.     

A generalized,multi-stage adjusted,latent class linear mixed model for testing genetic association

ABSTRACT

In this study, a Generalized, Multi-Stage Adjusted, Latent Class Linear Mixed Model is proposed for modeling the heterogeneous distributed phenotype and genetic information across the whole genome in the presence of both serial and familial correlations. Genome data were analyzed by applying the proposed model to Genetic Analysis Workshop (GAW) data, and the model results were compared to the results of standard models. Moreover, the potential of the model is discussed compared to simulated data. As a result of model comparisons, the information criteria and the genomic control parameter were found to be smaller. The results of a power analysis show that the proposed model is more powerful.     

A Simulation-based Goodness-of-fit Test for Random Effects in Generalized Linear Mixed Models

Abstract.  The goodness-of-fit of the distribution of random effects in a generalized linear mixed model is assessed using a conditional simulation of the random effects conditional on the observations. Provided that the specified joint model for random effects and observations is correct, the marginal distribution of the simulated random effects coincides with the assumed random effects distribution. In practice, the specified model depends on some unknown parameter which is replaced by an estimate. We obtain a correction for this by deriving the asymptotic distribution of the empirical distribution function obtained from the conditional sample of the random effects. The approach is illustrated by simulation studies and data examples.     

Analyzing multivariate longitudinal binary data: A generalized estimating equations approach

The authors consider regression analysis for binary data collected repeatedly over time on members of numerous small clusters of individuals sharing a common random effect that induces dependence among them. They propose a mixed model that can accommodate both these structural and longitudinal dependencies. They estimate the parameters of the model consistently and efficiently using generalized estimating equations. They show through simulations that their approach yields significant gains in mean squared error when estimating the random effects variance and the longitudinal correlations, while providing estimates of the fixed effects that are just as precise as under a generalized penalized quasi‐likelihood approach. Their method is illustrated using smoking prevention data.     

A certain locally most powerful test for one-way random effects model: full distribution and power considerations

We investigate the properties of the locally most powerful nonparametric criterion against logistic alternatives developed by Govindarajulu (1975) for testing one-way random effects modcls. We deduce the appropriate computational forms for the test criterion T and tabulate the critical values of T for α = .01, .05 and 0.10, and various sample sizes. Certain features of the computational methods are discussed. In the tables we retain only those sample sizes beyond which the asymptotic theory is meaningful. We also study the power comparison of the test for two populations with the classical F-test under a range of normal alternatives.     

A multilevel model with autoregressive components for the analysis of tribal art prices

In this paper, we introduce a multilevel model specification with time-series components for the analysis of prices of artworks sold at auctions. Since auction data do not constitute a panel or a time series but are composed of repeated cross-sections, they require a specification with items at the first level nested in time-points. Our approach combines the flexibility of mixed effect models together with the predicting performance of time series as it allows to model the time dynamics directly. Model estimation is obtained by means of maximum likelihood through the expectation–maximization algorithm. The model is motivated by the analysis of the first database ethnic artworks sold in the most important auctions worldwide. The results show that the proposed specification improves considerably over classical proposals both in terms of fit and prediction.     

Functional Coefficient Regression Models for Non-linear Time Series: A Polynomial Spline Approach

Abstract.  We propose a global smoothing method based on polynomial splines for the estimation of functional coefficient regression models for non-linear time series. Consistency and rate of convergence results are given to support the proposed estimation method. Methods for automatic selection of the threshold variable and significant variables (or lags) are discussed. The estimated model is used to produce multi-step-ahead forecasts, including interval forecasts and density forecasts. The methodology is illustrated by simulations and two real data examples.     

Generalized Estimation of the BLUP in Mixed-Effects Models: A Comparison with ML and REML

The Best Linear Unbiased Predictor (BLUP) in mixed models is a function of the variance components and they are estimated using maximum likelihood (ML) or restricted ML methods. Nonconvergence of BLUP would occur due to a drawback of the standard likelihood-based approaches. In such situations, ML and REML either do not provide any BLUPs or all become equal. To overcome this drawback, we provide a generalized estimate (GE) of BLUP that does not suffer from the problem of negative or zero variance components, and compare its performance against the ML and REML estimates of BLUP. Simulated and published data are used to compare BLUP.     

A NONPARAMETRIC MIXED‐EFFECTS MODEL FOR CANCER MORTALITY

There are several ways to handle within‐subject correlations with a longitudinal discrete outcome, such as mortality. The most frequently used models are either marginal or random‐effects types. This paper deals with a random‐effects‐based approach. We propose a nonparametric regression model having time‐varying mixed effects for longitudinal cancer mortality data. The time‐varying mixed effects in the proposed model are estimated by combining kernel‐smoothing techniques and a growth‐curve model. As an illustration based on real data, we apply the proposed method to a set of prefecture‐specific data on mortality from large‐bowel cancer in Japan.     

A model of regression lines through a common point: estimation of the focal point in wind-blown sand phenomena

This paper discusses a model in which the regression lines will be passing through a common point. This point exists as a focal point in the wind-blown sand phenomena. The model of regression lines will be called ‘the focal point regression model’. The focal point will move according to the conditions of the experiments or the measurement site, so it must be estimated together with regression coefficients. The existence of the focal point is mathematically proved in the research field of coastal engineering, but its physical meaning and exact estimation method have not been established. Considering the experimental and/or measurement conditions, five models, that is, common or different error variance(s), passing through or not the centroid and Bayes-like approach are proposed. Moreover, the formulae of direct computation for a focal point under some conditions are given for engineering purpose. The models are applied to the wind-blown sand data, and behaviors of the models are verified by numerical experiments.     

Latent Variable Modelling: A Survey*

Abstract. Latent variable modelling has gradually become an integral part of mainstream statistics and is currently used for a multitude of applications in different subject areas. Examples of ‘traditional’ latent variable models include latent class models, item–response models, common factor models, structural equation models, mixed or random effects models and covariate measurement error models. Although latent variables have widely different interpretations in different settings, the models have a very similar mathematical structure. This has been the impetus for the formulation of general modelling frameworks which accommodate a wide range of models. Recent developments include multilevel structural equation models with both continuous and discrete latent variables, multiprocess models and nonlinear latent variable models.     

Bayesian robust transformation and variable selection: A unified approach

The authors consider the problem of simultaneous transformation and variable selection for linear regression. They propose a fully Bayesian solution to the problem, which allows averaging over all models considered including transformations of the response and predictors. The authors use the Box‐Cox family of transformations to transform the response and each predictor. To deal with the change of scale induced by the transformations, the authors propose to focus on new quantities rather than the estimated regression coefficients. These quantities, referred to as generalized regression coefficients, have a similar interpretation to the usual regression coefficients on the original scale of the data, but do not depend on the transformations. This allows probabilistic statements about the size of the effect associated with each variable, on the original scale of the data. In addition to variable and transformation selection, there is also uncertainty involved in the identification of outliers in regression. Thus, the authors also propose a more robust model to account for such outliers based on a t‐distribution with unknown degrees of freedom. Parameter estimation is carried out using an efficient Markov chain Monte Carlo algorithm, which permits moves around the space of all possible models. Using three real data sets and a simulated study, the authors show that there is considerable uncertainty about variable selection, choice of transformation, and outlier identification, and that there is advantage in dealing with all three simultaneously. The Canadian Journal of Statistics 37: 361–380; 2009 © 2009 Statistical Society of Canada     

A Markov-chain-based regression model with random effects for the analysis of 18O-labelled mass spectra

The enzymatic ¹⁸O-labelling is a useful technique for reducing the influence of the between-spectra variability on the results of mass-spectrometry experiments. A difficulty in applying the technique lies in the quantification of the corresponding peptides due to the possibility of an incomplete labelling, which may result in biased estimates of the relative peptide abundance. To address the problem, Zhu et al. [A Markov-chain-based heteroscedastic regression model for the analysis of high-resolution enzymatically ¹⁸O-labeled mass spectra, J. Proteome Res. 9(5) (2010), pp. 2669–2677] proposed a Markov-chain-based regression model with heteroscedastic residual variance, which corrects for the possible bias. In this paper, we extend the model by allowing for the estimation of the technical and/or biological variability for the mass spectra data. To this aim, we use a mixed-effects version of the model. The performance of the model is evaluated based on results of an application to real-life mass spectra data and a simulation study.     

Estimation and selection procedures in regression: An L1 approach

The authors consider the problem of estimating a regression function go involving several variables by the closest functional element of a prescribed class _G that is closest to it in the L₁ norm. They propose a new estimator ? based on independent observations and give explicit finite sample bounds for the L₁distance between ?g and go. They apply their estimation procedure to the problem of selecting the smoothing parameter in nonparametric regression.