Bayesian and likelihood-based methods in multilevel modeling 1 A comparison of Bayesian and likelihood-based methods for fitting multilevel models

By William J. Browne. 2006

589 citations recorded by the source corpus.

We use simulation studies, whose design is realistic for educational andmedicalresearch(aswellasotherfleldsofinquiry),tocompareBayesianand likelihood-basedmethodsforflttingvariance-components(VC)andrandom-efiects logistic regression (RELR) models. The likelihood (and approximate likelihood) approachesweexaminearebasedonthemethodsmostwidelyusedincurrentap- plied multilevel (hierarchical) analyses: maximum likelihood (ML) and restricted ML(REML)forGaussianoutcomes,andmarginalandpenalizedquasi-likelihood (MQL and PQL) for Bernoulli outcomes. Our Bayesian methods use Markov chain Monte Carlo (MCMC) estimation, with adaptive hybrid Metropolis-Gibbs sampling for RELR models, and several difiuse prior distributions (i i1 (†;†) and U(0; 1 ) priors for variance components). For evaluation criteria we consider bias of point estimates and nominal versus actual coverage of interval estimates in re- peated sampling. In two-level VC models we flnd that (a) both likelihood-based and Bayesian approaches can be made to produce approximately unbiased esti- mates, although the automatic manner in which REML accomplishes this is an advantage, but (b) both approaches had di-culty achieving nominal coverage in smallsamplesandwithsmallvaluesoftheintraclasscorrelation. Withthethree- levelRELRmodelsweexamineweflndthat(c)quasi-likelihoodmethodsforesti- mating random-efiects variances perform badly with respect to bias and coverage intheexamplewesimulated,and(d)Bayesiandifiuse-priormethodsleadtowell- calibratedpointandintervalRELRestimates. Whileitistruethatthelikelihood- based methods we study are considerably faster computationally than MCMC, (i) steady improvements in recent years in both hardware speed and e-ciency of MonteCarloalgorithmsand(ii)thelackofcalibrationoflikelihood-basedmethods insomecommonhierarchicalsettingscombinetomakeMCMC-basedBayesianflt- tingofmultilevelmodelsanattractiveapproach,evenwithratherlargedatasets. Other analytic strategies based on less approximate likelihood methods are also possible butwouldbeneflt fromfurtherstudy ofthe type summarized here.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e19c11bd45680e5cf8591ddfc83f332d8c0786a5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Statistics and Probability

Citations in the corpus

Sensitivity analysis in Bayesian generalized linear mixed models for binary data (2011, background): “Sensitivity to prior assumptions for the random effect precision was discussed in Browne and Draper (2006) and discussion articles following it (Gelman 2006; Kass and Natarajan 2006; Lambert 2006).”
A default conjugate prior for variance components in generalized linear mixed models (comment on article by Browne and Draper) (2006, background): “” In the case of a scalar random effect, Browne and Draper (2005) investigated the frequentist behavior of posterior estimates based on a uniform prior and an inverted-gamma prior.”
Variational Inference for Generalized Linear Mixed Models Using Partially Noncentered Parametrizations (2012, background): “…(in simulated data sets not shown) that posterior inferences can be sensitive to prior assumptions on the variance components in Poisson models where many of the counts are close to zero or in binary data where the cluster size is small (see Browne and Draper ( 2006) and Roos and Held (2011)).”
MCMC methods for multi-response generalized linear mixed models (2010, background): “Although techniques that approximate these integrals (Breslow and Clayton 1993) are now popular, Markov chain Monte Carlo (MCMC) methods provide an alternative strategy for marginalizing the random effects that may be more robust (Zhao, Staudenmayer, Coull, and Wand 2006; Browne and Draper 2006).”
Mixed and Mixture Regression Models for Continuous Bounded Responses Using the Beta Distribution (2012, background): “Our conclusions regarding the strengths and weaknesses of this approach are largely in line with Browne and Draper’s (2006) comparisons with likelihood and quasi-likelihood approaches. For medium to large random effects and moderate to large samples, Bayesian MCMC estimates and interval coverage see”
Multilevel Modelling of Country Effects: A Cautionary Tale (2016, background): “See Austin (2010), Browne and Draper (2000, 2006), and Stegmueller (2013).”
Moving beyond noninformative priors: why and how to choose weakly informative priors in Bayesian analyses (2019, background): “It is highly sensitive to the choice of e. Typically, e = 0.001 (Browne and Draper 2006).”
Does pay for performance diminish intrinsic interest? (2012, methodology): “However, previous work shows that when using restricted maximum likelihood estimation (the method used by Proc Mixed in SAS), unbiased estimates of parameters can be obtained when the number of level two units is as small or smaller than that in our sample (Browne and Draper 2006).”
Multilevel Mediation With Small Samples: A Cautionary Note on the Multilevel Structural Equation Modeling Framework (2017, background): “Although not generally problematic, full maximum likelihood is an asymptotic method that is known to yield biased estimates (especially downwardly biased variance component and standard error estimates) when the number of clusters is small or modest (Browne & Draper, 2006; Maas & Hox, 2005).”
Fitting multilevel multivariate models with missing data in responses and covariates that may include interactions and non‐linear terms (2014, methodology): “…p.Ω/∝1, p.σ2u/∼Γ−1.0:001, 0:001/, p.σ2e /∼Γ−1.0:001, 0:001/: In the literature there are several commonly used diffuse priors for such models (see Browne and Draper (2006) and Gelman (2006) for discussion on this) but this is not of prime interest in this paper and the data set is…”
Moderated Mediation Analysis Using Bayesian Methods (2015, background): “Researchers have found that different Bayesian estimators (e.g., posterior mean, posterior median, posterior mode) combined with different priors could have different empirical performance when inferring a parameter with a skewed distribution (e.g., Bayes & Branco, 2007; Browne & Draper, 2006).”
Regression Analysis of Country Effects Using Multilevel Data: A Cautionary Tale (2013, methodology): “Other researchers have shown that Bayesian estimation methods using Markov chain Monte-Carlo methods also perform well, especially with a relatively small number of ‘countries’ (Austin 2010; Browne and Draper 2000, 2006; Stegmueller 2013).”
Dynamic Structural Equation Models (2018, background): “It is also known that the variance of the between-level effect when N 100 can be fine-tuned by using proper priors (see Browne & Draper, 2006).”
Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper) (2004, background): “Keywords: Bayesian inference, conditional conjugacy, folded-noncentral-t distribution, half-t distribution, hierarchical model, multilevel model, noninformative prior distribution, weakly informative prior distribution”