Detecting the violation of variance homogeneity in mixed models

2016 ◽  
Vol 25 (6) ◽  
pp. 2506-2520 ◽  
Author(s):  
Xicheng Fang ◽  
Jialiang Li ◽  
Weng Kee Wong ◽  
Bo Fu
Keyword(s):  
Random Effects ◽  
Mixed Models ◽  
Systematic Approach ◽  
Covariance Structure ◽  
Mixed Effects ◽  
Mixed Effects Models ◽  
Simulation Studies ◽  
Homogeneity Assumption ◽  
Variance Homogeneity

Mixed-effects models are increasingly used in many areas of applied science. Despite their popularity, there is virtually no systematic approach for examining the homogeneity of the random-effects covariance structure commonly assumed for such models. We propose two tests for evaluating the homogeneity of the covariance structure assumption across subjects: one is based on the covariance matrices computed from the fitted model and the other is based on the empirical variation computed from the estimated random effects. We used simulation studies to compare performances of the two tests for detecting violations of the homogeneity assumption in the mixed-effects models and showed that they were able to identify abnormal clusters of subjects with dissimilar random-effects covariance structures; in particular, their removal from the fitted model might change the signs and the magnitudes of important predictors in the analysis. In a case study, we applied our proposed tests to a longitudinal cohort study of rheumatoid arthritis patients and compared their abilities to ascertain whether the assumption of covariance homogeneity for subject-specific random effects holds.

scholarly journals Estimating Bayes Factors for Linear Models with Random Slopes on Continuous Predictors

2017 ◽  
Author(s):  
Mirko Thalmann ◽  
Marcel Niklaus ◽  
Klaus Oberauer
Keyword(s):  
Bayesian Statistics ◽  
Random Effects ◽  
Mixed Models ◽  
Mixed Effects ◽  
Bayes Factors ◽  
Mixed Effects Models ◽  
Type I ◽  
Frequentist Statistics ◽  
Random Slopes ◽  
Continuous Predictors

Using mixed-effects models and Bayesian statistics has been advocated by statisticians in recent years. Mixed-effects models allow researchers to adequately account for the structure in the data. Bayesian statistics – in contrast to frequentist statistics – can state the evidence in favor of or against an effect of interest. For frequentist statistical methods, it is known that mixed models can lead to serious over-estimation of evidence in favor of an effect (i.e., inflated Type-I error rate) when models fail to include individual differences in the effect sizes of predictors ("random slopes") that are actually present in the data. Here, we show through simulation that the same problem exists for Bayesian mixed models. Yet, at present there is no easy-to-use application that allows for the estimation of Bayes Factors for mixed models with random slopes on continuous predictors. Here, we close this gap by introducing a new R package called BayesRS. We tested its functionality in four simulation studies. They show that BayesRS offers a reliable and valid tool to compute Bayes Factors. BayesRS also allows users to account for correlations between random effects. In a fifth simulation study we show, however, that doing so leads to slight underestimation of the evidence in favor of an actually present effect. We only recommend modeling correlations between random effects when they are of primary interest and when sample size is large enough. BayesRS is available under https://cran.r-project.org/web/packages/BayesRS/, R code for all simulations is available under https://osf.io/nse5x/?view_only=b9a7caccd26a4764a084de3b8d459388

scholarly journals Benefits of Bayesian Model Selection and Averaging for Mixed-Effects Modeling

2021 ◽  
Author(s):  
Daniel W. Heck ◽  
Florence Bockting
Keyword(s):  
Model Selection ◽  
Random Effects ◽  
Mixed Models ◽  
Bayesian Model ◽  
Fixed Effects ◽  
Bayes Factor ◽  
Mixed Effects ◽  
Bayes Factors ◽  
Mixed Effects Models ◽  
Within Subjects

Bayes factors allow researchers to test the effects of experimental manipulations in within-subjects designs using mixed-effects models. van Doorn et al. (2021) showed that such hypothesis tests can be performed by comparing different pairs of models which vary in the specification of the fixed- and random-effect structure for the within-subjects factor. To discuss the question of which of these model comparisons is most appropriate, van Doorn et al. used a case study to compare the corresponding Bayes factors. We argue that researchers should not only focus on pairwise comparisons of two nested models but rather use the Bayes factor for performing model selection among a larger set of mixed models that represent different auxiliary assumptions. In a standard one-factorial, repeated-measures design, the comparison should include four mixed-effects models: fixed-effects H0, fixed-effects H1, random-effects H0, and random-effects H1. Thereby, the Bayes factor enables testing both the average effect of condition and the heterogeneity of effect sizes across individuals. Bayesian model averaging provides an inclusion Bayes factor which quantifies the evidence for or against the presence of an effect of condition while taking model-selection uncertainty about the heterogeneity of individual effects into account. We present a simulation study showing that model selection among a larger set of mixed models performs well in recovering the true, data-generating model.

scholarly journals Basic Features of the Analysis of Germination Data with Generalized Linear Mixed Models

Data ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 6 ◽  
Author(s):  
Alberto Gianinetti
Keyword(s):  
Boundary Conditions ◽  
Longitudinal Studies ◽  
Random Effects ◽  
Mixed Models ◽  
Generalized Linear Mixed Models ◽  
Linear Mixed Models ◽  
Covariance Structure ◽  
Conditional Models ◽  
Error Terms ◽  
Germination Indices

Germination data are discrete and binomial. Although analysis of variance (ANOVA) has long been used for the statistical analysis of these data, generalized linear mixed models (GzLMMs) provide a more consistent theoretical framework. GzLMMs are suitable for final germination percentages (FGP) as well as longitudinal studies of germination time-courses. Germination indices (i.e., single-value parameters summarizing the results of a germination assay by combining the level and rapidity of germination) and other data with a Gaussian error distribution can be analyzed too. There are, however, different kinds of GzLMMs: Conditional (i.e., random effects are modeled as deviations from the general intercept with a specific covariance structure), marginal (i.e., random effects are modeled solely as a variance/covariance structure of the error terms), and quasi-marginal (some random effects are modeled as deviations from the intercept and some are modeled as a covariance structure of the error terms) models can be applied to the same data. It is shown that: (a) For germination data, conditional, marginal, and quasi-marginal GzLMMs tend to converge to a similar inference; (b) conditional models are the first choice for FGP; (c) marginal or quasi-marginal models are more suited for longitudinal studies, although conditional models lead to a congruent inference; (d) in general, common random factors are better dealt with as random intercepts, whereas serial correlation is easier to model in terms of the covariance structure of the error terms; (e) germination indices are not binomial and can be easier to analyze with a marginal model; (f) in boundary conditions (when some means approach 0% or 100%), conditional models with an integral approximation of true likelihood are more appropriate; in non-boundary conditions, (g) germination data can be fitted with default pseudo-likelihood estimation techniques, on the basis of the SAS-based code templates provided here; (h) GzLMMs are remarkably good for the analysis of germination data except if some means are 0% or 100%. In this case, alternative statistical approaches may be used, such as survival analysis or linear mixed models (LMMs) with transformed data, unless an ad hoc data adjustment in estimates of limit means is considered, either experimentally or computationally. This review is intended as a basic tutorial for the application of GzLMMs, and is, therefore, of interest primarily to researchers in the agricultural sciences.

scholarly journals Fixed and Random Effects Selection in Mixed Effects Models

Biometrics ◽  
2010 ◽  
Vol 67 (2) ◽  
pp. 495-503 ◽  
Author(s):  
Joseph G. Ibrahim ◽  
Hongtu Zhu ◽  
Ramon I. Garcia ◽  
Ruixin Guo
Keyword(s):  
Random Effects ◽  
Mixed Effects ◽  
Mixed Effects Models ◽  
Fixed And Random Effects
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