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Mixed Beta Regression: A Bayesian Perspective

Jorge I. Figueroa-Zuñiga, Reinaldo B. Arellano-Valle, Silvia L. P. Ferrari

arXiv:1201.2375v4stat.CO

TL;DR

Mixed-effects models have mainly addressed normal or symmetric outcomes, leaving bounded continuous responses such as proportions as a distinct modeling setting. This paper develops Bayesian beta mixed regression with fixed and random effects for the mean and precision. Its approach is described as more flexible to implement for non-normal random effects than a classic approach, and the discussion notes that precision-model selection requires care.

  • Problem

    Mixed-effects models have mainly been studied for normal or symmetric responses, whereas continuous bounded outcomes such as proportions are common.

    Source: §1 · p. 1

  • Method

    The paper uses Bayesian beta mixed regression, modeling the response mean and optionally its precision through regression structures with fixed and random effects.

    Source: §1 · p. 2, §6 · pp. 15–16

  • Results

    The approach is described as more flexible to implement for normally, Student-t, or skew-normally distributed random effects than the classic approach discussed.

    Source: §6 · p. 17

  • Takeaways & Limitations

    The empirical applications yielded good results for model fit and diagnostic tests, while precision modeling requires careful model selection because the more plausible specification is not clear in advance.

    Source: §6 · pp. 16–17

    It is not clear in advance which precision model is more plausible, so careful model selection is necessary.

    Source: §6 · p. 17

Abstract

from arXiv · show

This paper builds on recent research that focuses on regression modeling of continuous bounded data, such as proportions measured on a continuous scale. Specifically, it deals with beta regression models with mixed effects from a Bayesian approach. We use a suitable parameterization of the beta law in terms of its mean and a precision parameter, and allow both parameters to be modeled through regression structures that may involve fixed and random effects. Specification of prior distributions is discussed, computational implementation via Gibbs sampling is provided, and illustrative examples are presented.

1. Introduction

Mixed-effects models are commonly used for normal or symmetric responses, but bounded continuous outcomes such as proportions require a suitable approach. The paper develops Bayesian mixed beta regression, with fixed and random effects and Gibbs-sampling implementation.

1. Introduction

  • The paper proposes Bayesian mixed-effects regression for continuous responses restricted to the unit interval, addressing bounded outcomes such as proportions.For responses on a known interval (a, b), it also describes transformation to (0, 1).Source: §1 · pp. 1–2
  • The beta model parameterizes the response through its mean and a positive precision parameter, with mixed-effects regression structures for these parameters.The mean uses a logit link; an extended model gives precision its own mixed-effects structure through a log link.Source: §1 · p. 2
  • The Bayesian approach addresses prior specification and model fitting via Gibbs sampling, with simulated and real-data examples for illustration.The paper also considers model selection using DIC, EAIC, and EBIC.Source: §1 · p. 2

2. Bayesian mixed beta regression

The section formulates beta mixed regression for continuous responses in (0, 1), modeling the mean with fixed and random effects and allowing precision to be common or modeled separately. It also specifies random-effect distributions and priors for the model parameters.

2. Bayesian mixed beta regression

  • The beta distribution is a natural choice for modeling continuous data restricted to (0, 1), parameterized by mean µ and precision φ, with Var(y) = µ(1 − µ)/(1 + φ).Source: §2 · p. 2
  • The mean and precision may have separate regression structures, with log links for the mean and observation-specific precision, whose predictors need not match.The precision model uses its own fixed- and random-effect design matrices.Source: §2 · pp. 3–4
  • A logit link relates the conditional mean to fixed and random effects, while the design matrices encode their respective predictors.The common-precision formulation assumes conditional independence of responses given the random effects, coefficients, and precision.Source: §2 · pp. 3–4
  • Random effects may follow multivariate normal or heavier-tailed Student-t distributions, with the latter presented as a candidate when measurements contain outliers.For large degrees of freedom, the multivariate t distribution approximates the multivariate normal.Source: §2 · p. 4
  • For t-distributed fixed and random effects, the section specifies priors for their degrees of freedom and scale matrices, including exponential and inverse-Wishart choices.For the observation-specific precision model, priors for δ and its random-effect distribution parameters follow the approach used for β and the corresponding random effects.Source: §2 · p. 5
  • The proposed prior for the common precision is φ = (aB)^2, with B following a beta distribution, and includes Gelman’s prior as a special case.The section also describes an inverse-gamma option and a uniform-square prior.Source: §2 · p. 5

3. Model fitting using Markov chain Monte Carlo sampling

The section derives posterior distributions for the common- and varying-precision models and describes Gibbs sampling to obtain Monte Carlo samples. The resulting procedure supports posterior inference on regression parameters, random effects, precision, and mean responses.

3. Model fitting using Markov chain Monte Carlo sampling

  • The posterior formulation for the common-precision model assumes prior independence of Σb, νb, φ, and β, while the model’s latent effects are conditionally independent given their parameters.The varying-precision model similarly assumes prior independence for its listed parameter groups.Source: §3 · pp. 5–6
  • For the common-precision model, Gibbs sampling iteratively draws from full conditional distributions to generate samples from the joint posterior.WinBUGS implementation supports inference on regression coefficients, random effects, precision, and mean responses, as well as hypothesis testing.Source: §3 · p. 6
  • For the varying-precision model, Gibbs sampling generates Monte Carlo samples from the joint posterior over mean- and precision-model parameters and latent effects.The sampler likewise proceeds through full conditional distributions.Source: §3 · pp. 6–7

4. Illustration via simulations

The simulations compare precision-parameter priors, random-effects distributions, and precision regression structures in mixed beta models. They favor t-distributed random effects for heavy-tailed data and identify the simulated constant-precision model as the best fit among the tested precision structures.

Prior sensitivity

  • Across the tested precision priors, DIC, EAIC, and EBIC were similar, with model 1d showing a slightly better fit.Model 1d used the proposed beta-based prior for φ.Source: Table 1 · p. 8, §4 · p. 8

Estimation and diagnostics

  • Model 1d’s posterior estimates were similar to the true parameter values, and changing regression-coefficient or random-effect dispersion priors did not appreciably alter inferences.The multivariate convergence diagnostic was 1.01, and other reported diagnostics supported convergence and stationary, uncorrelated chains.Source: §4 · pp. 8–9, Table 2 · p. 9

Random-effects distributions

  • For heavy-tailed random effects (νb = 5, 10), the multivariate t fit outperformed the normal fit in posterior estimation and information criteria.The advantage in DIC, EAIC, and EBIC was clearer when νb was small.Source: Table 3 · p. 10, §4 · p. 10, Table 4 · p. 11

Precision regression structures

  • Among five precision structures, model 2a—the data-generating constant-precision model—fit better than the alternatives, while models 2a, 2c, and 2d were similarly good.Models without random effects for precision achieved the best fits overall.Source: Table 5 · p. 11, §4 · p. 11
  • For model 2a, the estimates were similar to the true parameter values, and the reported convergence diagnostic was mprf = 1.00 < 1.2.Estimates used the last 90,000 of 100,000 Monte Carlo iterations.Source: §4 · pp. 11–12, Table 6 · p. 12

5. A real data application

The Prater data application compares mixed beta regression models with constant and observation-varying precision. The variable-precision model fits best by the reported information criteria, and the estimated covariate EP is associated with both mean and precision.

5. A real data application

  • Under model 1, the proposed precision priors yield similar DIC, EAIC, and EBIC values.The smallest EAIC and EBIC occur with beta prior ϵ = 0.1, while the smallest DIC occurs with ϵ = 0.5.Source: §5 · p. 13, Table 7 · p. 13
  • EP positively affects both the mean and precision of the crude-oil-to-gasoline proportion.The response records the proportion of crude oil converted into gasoline; the dataset has 32 observations from 10 crude batches.Source: §5 · pp. 12, 14
  • Model 2.5, with variable precision, is favored over constant-precision model 1.4 by DIC, EAIC, and EBIC.The comparison indicates that both the location and precision submodels are affected by random additive effects and EP; zero lies in the credibility interval for association between the random effects.Source: §5 · p. 13, Table 8 · p. 14
  • Diagnostics support the reported estimates for models 1.4 and 2.5, with multivariate Gelman–Rubin values of 1.09 and 1.06, respectively.The authors also report suitable diagnostic plots and no posterior-density multimodality for model 1.4; similar evidence is reported for model 2.5.Source: §5 · pp. 14–15, Table 9 · p. 14, Table 10 · p. 15

6. Discussion

The discussion presents Bayesian mixed beta regression with either fixed precision or a regression model for precision. Its applications had good model fit and diagnostic performance, while the precision structure requires careful selection.

6. Discussion

  • The paper develops Bayesian mixed beta regression with either common precision or a separate regression structure for precision.The mean uses a logit link in the described model, and the approach can accommodate other link choices.Source: §6 · pp. 15–16
  • Alternative beta priors for common precision gave good model fit and diagnostic-test performance.The paper also reports good fit and diagnostic results in its empirical applications, while noting that precision-model selection requires care because the plausible model is not clear in advance.Source: §6 · pp. 16–17
  • For mixed-effects beta regression, the Bayesian approach can accommodate normal, Student-t, skew-normal, or other random-effect distributions using WinBUGS.The discussion contrasts this flexibility with the reported difficulty of implementing non-normal random effects using the classic adaptive Gaussian quadrature approach.Source: §6 · p. 17
  • The approach also supports easy implementation of missing-data imputation, a setting described as more complicated for the classic approach.The discussion cites Carrigan et al. (2007) in connection with missing-data imputation.Source: §6 · pp. 17–18

Appendix: BUGS codes for the mixed beta regression

The appendix provides BUGS code for fitting mixed beta regression in simulated-data examples. The code specifies beta responses and regression structures for the mean and, in the variable-precision version, precision.

Appendix: BUGS codes for the mixed beta regression

  • The BUGS code models each response as beta-distributed, with its shape parameters formed from the mean and precision.The constant-precision code uses a shared φ, while the variable-precision code uses φ[i,j].Source: Appendix: BUGS codes for the mixed… · pp. 18–19
  • The mean submodel uses a logit link combining fixed covariate effects and random effects.The code expresses the fixed and random contributions as inner products with x and z, respectively.Source: Appendix: BUGS codes for the mixed… · p. 18
  • The appendix code includes alternative specifications for the constant-precision prior, including inverse-gamma, uniform, and transformed-beta forms.The displayed code also specifies priors for regression coefficients and random effects.Source: Appendix: BUGS codes for the mixed… · pp. 18–19
  • For variable precision, the code models log(φ[i,j]) with fixed and random covariate effects.The precision submodel uses x and z predictors with coefficients δ and γ.Source: Appendix: BUGS codes for the mixed… · p. 19
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