Linear mixed models with skew-elliptical distributions: A Bayesian approach (original) (raw)
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The International Journal of Biostatistics, 2000
In recent years, various mixed-effects models have been suggested for estimating viral decay rates in HIV dynamic models for complex longitudinal data. Among those models are linear mixed-effects (LME), nonlinear mixed-effects (NLME), and semiparametric nonlinear mixedeffects (SNLME) models. However, a critical question is whether these models produce coherent estimates of viral decay rates, and if not, which model is appropriate and should be used in practice. In addition, one often assumes that a model random error is normally distributed, but the normality assumption may be unrealistic, particularly if the data exhibit skewness. Moreover, some covariates such as CD4 cell count may be often measured with substantial errors. This paper addresses these issues simultaneously by jointly modeling the response variable with skewness and a covariate process with measurement errors using a Bayesian approach to investigate how estimated parameters are changed or different under these three models. A real data set from an AIDS clinical trial study was used to illustrate the proposed models and methods. It was found that there was a significant incongruity in the estimated decay rates in viral loads based on the three mixed-effects models, suggesting that the decay rates estimated by using Bayesian LME or NLME joint models should be interpreted differently from those estimated by using Bayesian SNLME joint models. The findings also suggest that the Bayesian SNLME joint model is preferred to other models because an arbitrary data truncation is not necessary; and it is also shown that the models with a skew-normal distribution and/or measurement errors in covariate may achieve reliable results when the data exhibit skewness.
Bayesian inference in nonlinear mixed-effects models using normal independent distributions
Computational Statistics & Data Analysis
Nonlinear mixed-effects (NLME) models are popular in many longitudinal studies, including those on human immunodeficiency virus (HIV) viral dynamics, pharmacokinetic analyses, and studies of growth and decay analysis. Generally, the normality of the random effects is a common assumption in NLME models but it can sometimes be unrealistic, suppressing important features of amongsubjects variation. In this context, the use of normal/independent distributions arises as a tool for robust modeling of NLME models. These distributions fall in a class of symmetric heavy-tailed distributions that includes the normal distribution, the generalized Student-t, Student-t, slash and the contaminated normal distributions as special cases, providing an appealing robust alternative to the routine use of normal distributions in these types of models. The aim of this paper is the estimation of NLME models considering normal/independent distributions for the error term and random effects, under the Bayesian paradigm. A Bayesian case deletion influence diagnostic based on the q-divergence measure and model selections criteria is also developed. These analyses are computationally possible due to an important result that approximates the likelihood function of a NLME model with normal/independent distributions for a simple normal/independent distribution with specified parameters.
2020
A standard assumption is that the random effects of Generalized Linear Mixed Effects Models (GLMMs) follow the normal distribution. However, this assumption has been found to be quite unrealistic and sometimes too restrictive as revealed in many real-life situations. A common case of departures from normality includes the presence of outliers leading to heavy-tailed distributed random effects. This work, therefore, aims to develop a robust GLMM framework by replacing the normality assumption on the random effects by the distributions belonging to the Normal-Independent (NI) class. The resulting models are called the Normal-Independent GLMM (NI-GLMM). The four special cases of the NI class considered in these models’ formulations include the normal, Student-t, Slash and contaminated normal distributions. A full Bayesian technique was adopted for estimation and inference. A real-life data set on cotton bolls was used to demonstrate the performance of the proposed NI-GLMM methodology.
Inference and Local Influence Assessment in a Multifactor Skew-Normal Linear Mixed Model
Mathematics
This work considers a multifactor linear mixed model under heteroscedasticity in random-effect factors and the skew-normal errors for modeling the correlated datasets. We implement an expectation–maximization (EM) algorithm to achieve the maximum likelihood estimates using conditional distributions of the skew-normal distribution. The EM algorithm is also implemented to extend the local influence approach under three model perturbation schemes in this model. Furthermore, a Monte Carlo simulation is conducted to evaluate the efficiency of the estimators. Finally, a real data set is used to make an illustrative comparison among the following four scenarios: normal/skew-normal errors and heteroscedasticity/homoscedasticity in random-effect factors. The empirical studies show our methodology can improve the estimates when the model errors follow from a skew-normal distribution. In addition, the local influence analysis indicates that our model can decrease the effects of anomalous obser...
Estimation in nonlinear mixed-effects models using heavy-tailed distributions
Statistics and Computing
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.
Bayesian Model Uncertainty in Mixed Effects Models
Lecture Notes in Statistics, 2008
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A linear mixed model for analyzing longitudinal skew-normal responses with random dropout
Journal of the Korean Statistical Society
In this paper, a linear mixed effects model is used to fit skewed longitudinal data in the presence of dropout. Two distributional assumptions are considered to produce background for heavy tailed models. One is the linear mixed model with skew-normal random effects and normal errors and the other one is the linear mixed model with skew-normal errors and normal random effects. An ECM algorithm is developed to obtain the parameter estimates. Also an empirical Bayes approach is used for estimating random effects. A simulation study is implemented to investigate the performance of the presented algorithm. Results of an application are also reported where standard errors of estimates are calculated using the Bootstrap approach.
Flexible Bayesian semiparametric mixed-effects model for skewed longitudinal data
BMC Medical research methodology, 2024
Background In 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. Methods This 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. Results To 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. Conclusions The 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.
Biometrics & Biostatistics International Journal
Joint modeling of longitudinal and time to event data have been widely used for analyzing medical data, where longitudinal measurements is gathered with a time to event or survival data. In most of these studies, distributional assumption for modeling longitudinal response is normal, which leads to vulnerable inference in the presence of outliers in longitudinal measurements and violation of this assumption. Violation of the normality assumption can also make the statistical inference vague. Powerful distributions for robust analyzing and relaxing normality assumption, are skew-elliptical distributions, which include univariate and multivariate versions of the student's t, the Laplace and normal distributions. In this paper, a linear mixed effects model with skew-elliptical distribution for both random effects and residuals and a Cox's model for time to event data are used for joint modeling. This strategy allows for the skewness and the heavy tails of random effect distributions and thus makes inferences robust to the violation. For estimation, a Bayesian parametric approach using Markov chain Monte Carlo is adopted. The method is illustrated in a real Intensive Care Unit (ICU) data set and the best model is selected using some Bayesian criteria for model selection.