[Paper Review] Bayesian analysis for a class of beta mixed models

Wagner Hugo Bonat, Paulo Justiniano Ribeiro+1 more|arXiv (Cornell University)|Jan 13, 2014

Statistical Methods and Bayesian Inference35 references9 citations

TL;DR

This paper proposes a Bayesian inference approach for beta mixed models using Integrated Nested Laplace Approximation (INLA), demonstrating its computational efficiency and accuracy in modeling bounded continuous responses such as proportions and rates. Results show INLA produces results comparable to MCMC and likelihood methods while significantly reducing computation time, making it ideal for extensive model comparison and sensitivity analysis in hierarchical data structures.

ABSTRACT

Generalized linear mixed models (GLMM) encompass large class of statistical models, with a vast range of applications areas. GLMM extends the linear mixed models allowing for different types of response variable. Three most common data types are continuous, counts and binary and standard distributions for these types of response variables are Gaussian, Poisson and Binomial, respectively. Despite that flexibility, there are situations where the response variable is continuous, but bounded, such as rates, percentages, indexes and proportions. In such situations the usual GLMM's are not adequate because bounds are ignored and the beta distribution can be used. Likelihood and Bayesian inference for beta mixed models are not straightforward demanding a computational overhead. Recently, a new algorithm for Bayesian inference called INLA (Integrated Nested Laplace Approximation) was proposed.INLA allows computation of many Bayesian GLMMs in a reasonable amount time allowing extensive comparison among models. We explore Bayesian inference for beta mixed models by INLA. We discuss the choice of prior distributions, sensitivity analysis and model selection measures through a real data set. The results obtained from INLA are compared with those obtained by an MCMC algorithm and likelihood analysis. We analyze data from an study on a life quality index of industry workers collected according to a hierarchical sampling scheme. Results show that the INLA approach is suitable and faster to fit the proposed beta mixed models producing results similar to alternative algorithms and with easier handling of modeling alternatives. Sensitivity analysis, measures of goodness of fit and model choice are discussed.

Motivation & Objective

• To develop and evaluate a Bayesian framework for beta mixed models suitable for bounded continuous responses such as proportions, rates, and indexes.
• To compare the performance of INLA with traditional MCMC and likelihood-based inference methods in fitting beta mixed models.
• To assess the sensitivity of posterior distributions to prior distributions on the dispersion parameter (φ) and random effects precision (τ).
• To evaluate model selection criteria—LML, DIC, and CPO—for identifying the best-fitting model in a real-world hierarchical dataset.
• To provide practical guidance on prior specification and model diagnostics in beta mixed models using real data from a life quality index study.

Proposed method

• Utilizes Integrated Nested Laplace Approximation (INLA) for fast Bayesian inference in beta mixed models, avoiding the computational burden of MCMC.
• Applies a hierarchical modeling structure with Gaussian random effects to account for clustering in the data, such as workers nested within companies and states.
• Models the mean of the beta distribution using a linear predictor with a logit link function, and allows the precision parameter (φ) to vary via covariates.
• Employs weakly informative gamma priors for the dispersion parameter (φ) and precision of random effects (τ), with sensitivity analysis using Hellinger divergence.
• Uses model comparison criteria including Log-Marginal Likelihood (LML), Deviance Information Criterion (DIC), and Conditional Predictive Ordinates (CPO) for model selection.
• Validates results against MCMC and likelihood-based inference, ensuring consistency across methods.

Experimental results

Research questions

• RQ1 Can INLA provide accurate and computationally efficient Bayesian inference for beta mixed models compared to MCMC and likelihood methods?
• RQ2 How sensitive are posterior estimates of the dispersion parameter (φ) and random effects precision (τ) to the choice of prior distributions?
• RQ3 Which model selection criteria—LML, DIC, or CPO—best identify the optimal beta mixed model in a hierarchical data structure?
• RQ4 To what extent do different prior distributions for φ and τ affect the posterior distributions and model conclusions?
• RQ5 How do covariates such as company size and average income influence the life quality index in a hierarchical industrial worker dataset?

Key findings

• INLA produces posterior estimates for beta mixed models that are highly consistent with those obtained via MCMC and likelihood inference, validating its accuracy.
• The Hellinger distance between prior and posterior distributions for φ decreased from 0.6 to 0.1628, indicating low sensitivity to prior choice, while for τ it decreased from 0.6 to 0.2827, showing moderate sensitivity.
• Despite a 52.05% change in the posterior mean of τ under extreme priors, the regression coefficients remained stable, with changes of less than 0.5% across models.
• The final model selected via LML, DIC, and CPO was consistent across all three criteria, indicating robust model choice.
• Company size and average income were found to be significant predictors of life quality index, with random intercepts capturing state-level variation.
• The computational time for INLA was substantially lower than MCMC, making it feasible to explore multiple model specifications efficiently.