Testing for heteroscedasticity of exponential correlation mixed-effects linear models based on M-estimation

Homogeneity of variance is a basic assumption in longitudinal data analysis. However, the assumption is not necessarily appropriate. In this paper, Fisher scoring method is applied to get M-estimator in the exponential correlation mixed-effects linear model. The score tests for heteroscedasticity and correlation coefficient based on M-estimator are then studied. Monte Carlo method is applied to investigate the properties of test statistics. At last, the methods and properties are illustrated by an actual data example.     

Estimating mixed-effects differential equation models

Ordinary differential equations (ODEs) are popular tools for modeling complicated dynamic systems in many areas. When multiple replicates of measurements are available for the dynamic process, it is of great interest to estimate mixed-effects in the ODE model for the process. We propose a semiparametric method to estimate mixed-effects ODE models. Rather than using the ODE numeric solution directly, which requires providing initial conditions, this method estimates a spline function to approximate the dynamic process using smoothing splines. A roughness penalty term is defined using the ODEs, which measures the fidelity of the spline function to the ODEs. The smoothing parameter, which controls the trade-off between fitting the data and maintaining fidelity to the ODEs, can be specified by users or selected objectively by generalized cross validation. The spline coefficients, the ODE random effects, and the ODE fixed effects are estimated in three nested levels of optimization. Two simulation studies show that the proposed method obtains good estimates for mixed-effects ODE models. The semiparametric method is demonstrated with an application of a pharmacokinetic model in a study of HIV combination therapy.     

Estimation in nonlinear mixed-effects models using heavy-tailed distributions

Nonlinear mixed-effects models are very useful to analyze repeated measures data and are used in a variety of applications. Normal distributions for random effects and residual errors are usually assumed, but such assumptions make inferences vulnerable to the presence of outliers. In this work, we introduce an extension of a normal nonlinear mixed-effects model considering a subclass of elliptical contoured distributions for both random effects and residual errors. This elliptical subclass, the scale mixtures of normal (SMN) distributions, includes heavy-tailed multivariate distributions, such as Student-t, the contaminated normal and slash, among others, and represents an interesting alternative to outliers accommodation maintaining the elegance and simplicity of the maximum likelihood theory. We propose an exact estimation procedure to obtain the maximum likelihood estimates of the fixed-effects and variance components, using a stochastic approximation of the EM algorithm. We compare the performance of the normal and the SMN models with two real data sets.     

Nonlinear mixed-effects scalar-on-function models and variable selection

This paper is motivated by our collaborative research and the aim is to model clinical assessments of upper limb function after stroke using 3D-position and 4D-orientation movement data. We present a new nonlinear mixed-effects scalar-on-function regression model with a Gaussian process prior focusing on the variable selection from a large number of candidates including both scalar and function variables. A novel variable selection algorithm has been developed, namely functional least angle regression. As it is essential for this algorithm, we studied the representation of functional variables with different methods and the correlation between a scalar and a group of mixed scalar and functional variables. We also propose a new stopping rule for practical use. This algorithm is efficient and accurate for both variable selection and parameter estimation even when the number of functional variables is very large and the variables are correlated. And thus the prediction provided by the algorithm is accurate. Our comprehensive simulation study showed that the method is superior to other existing variable selection methods. When the algorithm was applied to the analysis of the movement data, the use of the nonlinear random-effect model and the function variables significantly improved the prediction accuracy for the clinical assessment.

     

Selection of linear mixed-effects models for clustered data

We consider model selection for linear mixed-effects models with clustered structure, where conditional Kullback–Leibler (CKL) loss is applied to measure the efficiency of the selection. We estimate the CKL loss by substituting the empirical best linear unbiased predictors (EBLUPs) into random effects with model parameters estimated by maximum likelihood. Although the BLUP approach is commonly used in predicting random effects and future observations, selecting random effects to achieve asymptotic loss efficiency concerning CKL loss is challenging and has not been well studied. In this paper, we propose addressing this difficulty using a conditional generalized information criterion (CGIC) with two tuning parameters. We further consider a challenging but practically relevant situation where the number, $m$, of clusters does not go to infinity with the sample size. Hence the random-effects variances are not consistently estimable. We show that via a novel decomposition of the CKL risk, the CGIC achieves consistency and asymptotic loss efficiency, whether $m$ is fixed or increases to infinity with the sample size. We also conduct numerical experiments to illustrate the theoretical findings.     

Screening for prostate cancer using multivariate mixed-effects models

Using several variables known to be related to prostate cancer, a multivariate classification method is developed to predict the onset of clinical prostate cancer. A multivariate mixed-effects model is used to describe longitudinal changes in prostate specific antigen (PSA), a free testosterone index (FTI), and body mass index (BMI) before any clinical evidence of prostate cancer. The patterns of change in these three variables are allowed to vary depending on whether the subject develops prostate cancer or not and the severity of the prostate cancer at diagnosis. An application of Bayes' theorem provides posterior probabilities that we use to predict whether an individual will develop prostate cancer and, if so, whether it is a high-risk or a low-risk cancer. The classification rule is applied sequentially one multivariate observation at a time until the subject is classified as a cancer case or until the last observation has been used. We perform the analyses using each of the three variables individually, combined together in pairs, and all three variables together in one analysis. We compare the classification results among the various analyses and a simulation study demonstrates how the sensitivity of prediction changes with respect to the number and type of variables used in the prediction process.     

Semiparametric mixed-effects models for clustered doubly censored data

The Cox proportional frailty model with a random effect has been proposed for the analysis of right-censored data which consist of a large number of small clusters of correlated failure time observations. For right-censored data, Cai et al. [3] proposed a class of semiparametric mixed-effects models which provides useful alternatives to the Cox model. We demonstrate that the approach of Cai et al. [3] can be used to analyze clustered doubly censored data when both left- and right-censoring variables are always observed. The asymptotic properties of the proposed estimator are derived. A simulation study is conducted to investigate the performance of the proposed estimator.     

Bayesian composite quantile regression for linear mixed-effects models

Longitudinal data are commonly modeled with the normal mixed-effects models. Most modeling methods are based on traditional mean regression, which results in non robust estimation when suffering extreme values or outliers. Median regression is also not a best choice to estimation especially for non normal errors. Compared to conventional modeling methods, composite quantile regression can provide robust estimation results even for non normal errors. In this paper, based on a so-called pseudo composite asymmetric Laplace distribution (PCALD), we develop a Bayesian treatment to composite quantile regression for mixed-effects models. Furthermore, with the location-scale mixture representation of the PCALD, we establish a Bayesian hierarchical model and achieve the posterior inference of all unknown parameters and latent variables using Markov Chain Monte Carlo (MCMC) method. Finally, this newly developed procedure is illustrated by some Monte Carlo simulations and a case analysis of HIV/AIDS clinical data set.     

LASSO-type estimators for semiparametric nonlinear mixed-effects models estimation

Parametric nonlinear mixed effects models (NLMEs) are now widely used in biometrical studies, especially in pharmacokinetics research and HIV dynamics models, due to, among other aspects, the computational advances achieved during the last years. However, this kind of models may not be flexible enough for complex longitudinal data analysis. Semiparametric NLMEs (SNMMs) have been proposed as an extension of NLMEs. These models are a good compromise and retain nice features of both parametric and nonparametric models resulting in more flexible models than standard parametric NLMEs. However, SNMMs are complex models for which estimation still remains a challenge. Previous estimation procedures are based on a combination of log-likelihood approximation methods for parametric estimation and smoothing splines techniques for nonparametric estimation. In this work, we propose new estimation strategies in SNMMs. On the one hand, we use the Stochastic Approximation version of EM algorithm (SAEM) to obtain exact ML and REML estimates of the fixed effects and variance components. On the other hand, we propose a LASSO-type method to estimate the unknown nonlinear function. We derive oracle inequalities for this nonparametric estimator. We combine the two approaches in a general estimation procedure that we illustrate with simulations and through the analysis of a real data set of price evolution in on-line auctions.     

Nonlinear mixed-effects models with scale mixture of skew-normal distributions

Aiming to avoid the sensitivity in the parameters estimation due to atypical observations or skewness, we develop asymmetric nonlinear regression models with mixed-effects, which provide alternatives to the use of normal distribution and other symmetric distributions. Nonlinear models with mixed-effects are explored in several areas of knowledge, especially when data are correlated, such as longitudinal data, repeated measures and multilevel data, in particular, for their flexibility in dealing with measures of areas such as economics and pharmacokinetics. The random components of the present model are assumed to follow distributions that belong to scale mixtures of skew-normal (SMSN) distribution family, that encompasses distributions with light and heavy tails, such as skew-normal, skew-Student-t, skew-contaminated normal and skew-slash, as well as symmetrical versions of these distributions. For the parameters estimation we obtain a numerical solution via the EM algorithm and its extensions, and the Newton-Raphson algorithm. An application with pharmacokinetic data shows the superiority of the proposed models, for which the skew-contaminated normal distribution has shown to be the most adequate distribution. A brief simulation study points to good properties of the parameter vector estimators obtained by the maximum likelihood method.     

Bayesian variable selection in a finite mixture of linear mixed-effects models

Mixture of linear mixed-effects models has received considerable attention in longitudinal studies, including medical research, social science and economics. The inferential question of interest is often the identification of critical factors that affect the responses. We consider a Bayesian approach to select the important fixed and random effects in the finite mixture of linear mixed-effects models. To accomplish our goal, latent variables are introduced to facilitate the identification of influential fixed and random components and to classify the membership of observations in the longitudinal data. A spike-and-slab prior for the regression coefficients is adopted to sidestep the potential complications of highly collinear covariates and to handle large p and small n issues in the variable selection problems. Here we employ Markov chain Monte Carlo (MCMC) sampling techniques for posterior inferences and explore the performance of the proposed method in simulation studies, followed by an actual psychiatric data analysis concerning depressive disorder.     

Conditions for consistent estimation in mixed-effects models for binary matched-pairs data

Parametric mixed-effects logistic models can provide effective analysis of binary matched-pairs data. Responses are assumed to follow a logistic model within pairs, with an intercept which varies across pairs according to a specified family of probability distributions G. In this paper we give necessary and sufficient conditions for consistent covariate effect estimation and present a geometric view of estimation which shows that when the assumed family of mixture distributions is rich enough, estimates of the effect of the binary covariate are typically consistent. The geometric view also shows that under the conditions for consistent estimation, the mixed-model estimator is identical to the familar conditional-likelihood estimator for matched pairs. We illustrate the findings with some examples.     

Efficient estimation of variance components in nonparametric mixed-effects models with large samples

Linear mixed-effects (LME) regression models are a popular approach for analyzing correlated data. Nonparametric extensions of the LME regression model have been proposed, but the heavy computational cost makes these extensions impractical for analyzing large samples. In particular, simultaneous estimation of the variance components and smoothing parameters poses a computational challenge when working with large samples. To overcome this computational burden, we propose a two-stage estimation procedure for fitting nonparametric mixed-effects regression models. Our results reveal that, compared to currently popular approaches, our two-stage approach produces more accurate estimates that can be computed in a fraction of the time.     

Adaptive fitting of linear mixed-effects models with correlated random effects

Linear mixed-effects model has been widely used in longitudinal data analyses. In practice, the fitting algorithm can fail to converge due to boundary issues of the estimated random-effects covariance matrix G, that is, being near-singular, non-positive definite, or both. Current available algorithms are not computationally optimal because the condition number of matrix G is unnecessarily increased when the random-effects correlation estimate is not zero. We propose an adaptive fitting (AF) algorithm using an optimal linear transformation of the random-effects design matrix. It is a data-driven adaptive procedure, aiming at reducing subsequent random-effects correlation estimates down to zero in the optimal transformed estimation space. Simulations show that AF significantly improves the convergent properties, especially under small sample size, relative large noise and high correlation settings. One real data for insulin-like growth factor protein is used to illustrate the application of this algorithm implemented with software package R (nlme).     

Influence analysis of additive mixed-effects nonlinear regression models via EM algorithm

This paper presents a unified method for influence analysis to deal with random effects appeared in additive nonlinear regression models for repeated measurement data. The basic idea is to apply the Q-function, the conditional expectation of the complete-data log-likelihood function obtained from EM algorithm, instead of the observed-data log-likelihood function as used in standard influence analysis. Diagnostic measures are derived based on the case-deletion approach and the local influence approach. Two real examples and a simulation study are examined to illustrate our methodology.     

A flexible approach for multivariate mixed-effects models with non-ignorable missing values

We propose a flexible model approach for the distribution of random effects when both response variables and covariates have non-ignorable missing values in a longitudinal study. A Bayesian approach is developed with a choice of nonparametric prior for the distribution of random effects. We apply the proposed method to a real data example from a national long-term survey by Statistics Canada. We also design simulation studies to further check the performance of the proposed approach. The result of simulation studies indicates that the proposed approach outperforms the conventional approach with normality assumption when the heterogeneity in random effects distribution is salient.     

Some asymptotic results for semiparametric nonlinear mixed-effects models with incomplete data

In modeling complex longitudinal data, semiparametric nonlinear mixed-effects (SNLME) models are very flexible and useful. Covariates are often introduced in the models to partially explain the inter-individual variations. In practice, data are often incomplete in the sense that there are often measurement errors and missing data in longitudinal studies. The likelihood method is a standard approach for inference for these models but it can be computationally very challenging, so computationally efficient approximate methods are quite valuable. However, the performance of these approximate methods is often based on limited simulation studies, and theoretical results are unavailable for many approximate methods. In this article, we consider a computationally efficient approximate method for a class of SNLME models with incomplete data and investigate its theoretical properties. We show that the estimates based on the approximate method are consistent and asymptotically normally distributed.     

Empirical likelihood for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data

In this paper, the empirical likelihood inferences for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data are investigated. We construct the empirical log-likelihood ratio function for the fixed-effects parameters and the mean parameters of random-effects. The empirical log-likelihood ratio at the true parameters is proven to be asymptotically $\chi ^2_{q+r}$ , where $q$ and $r$ are dimensions of the fixed and random effects respectively, and the corresponding confidence regions for them are then constructed. We also obtain the maximum empirical likelihood estimator of the parameters of interest, and prove it is the asymptotically normal under some suitable conditions. A simulation study and a real data application are undertaken to assess the finite sample performance of the proposed method.     

Unbalanced cluster sizes and rates of convergence in mixed-effects models for clustered data

ABSTRACT

Convergence problems often arise when complex linear mixed-effects models are fitted. Previous simulation studies (see, e.g. [Buyse M, Molenberghs G, Burzykowski T, Renard D, Geys H. The validation of surrogate endpoints in meta-analyses of randomized experiments. Biostatistics. 2000;1:49–67, Renard D, Geys H, Molenberghs G, Burzykowski T, Buyse M. Validation of surrogate endpoints in multiple randomized clinical trials with discrete outcomes. Biom J. 2002;44:921–935]) have shown that model convergence rates were higher (i) when the number of available clusters in the data increased, and (ii) when the size of the between-cluster variability increased (relative to the size of the residual variability). The aim of the present simulation study is to further extend these findings by examining the effect of an additional factor that is hypothesized to affect model convergence, i.e. imbalance in cluster size. The results showed that divergence rates were substantially higher for data sets with unbalanced cluster sizes – in particular when the model at hand had a complex hierarchical structure. Furthermore, the use of multiple imputation to restore ‘balance’ in unbalanced data sets reduces model convergence problems.