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**Bayesian modeling and prior sensitivity analysis for zero-one augmented beta regression models with an application to psychometric data.**
*(English)*
Zbl 1445.62171

Summary: The interest on the analysis of the zero-one augmented beta regression (ZOABR) model has been increasing over the last few years. In this work, we developed a Bayesian inference for the ZOABR model, providing some contributions, namely: we explored the use of Jeffreys-rule and independence Jeffreys prior for some of the parameters, performing a sensitivity study of prior choice, comparing the Bayesian estimates with the maximum likelihood ones and measuring the accuracy of the estimates under several scenarios of interest. The results indicate, in a general way, that: the Bayesian approach, under the Jeffreys-rule prior, was as accurate as the ML one. Also, different from other approaches, we use the predictive distribution of the response to implement Bayesian residuals. To further illustrate the advantages of our approach, we conduct an analysis of a real psychometric data set including a Bayesian residual analysis, where it is shown that misleading inference can be obtained when the data is transformed. That is, when the zeros and ones are transformed to suitable values and the usual beta regression model is considered, instead of the ZOABR model. Finally, future developments are discussed.

### MSC:

62J05 | Linear regression; mixed models |

62F15 | Bayesian inference |

62P15 | Applications of statistics to psychology |

### Keywords:

augmented beta regression; Bayesian inference; Jeffreys prior; MCMC methods; residual analysis
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\textit{D. C. Nogarotto} et al., Braz. J. Probab. Stat. 34, No. 2, 304--322 (2020; Zbl 1445.62171)

### References:

[1] | Bayes, C. L. and Valdivieso, L. (2016). A beta inflated mean regression model for fractional response variables. Journal of Applied Statistics 43, 1814-1830. |

[2] | Bazan, J. L., Romeo, J. R. and Rodrigues, J. (2014). Bayesian skew-probit regression for binary response data. Brazilian Journal of Probability and Statistics 28, 467-482. · Zbl 1304.62049 |

[3] | Branscum, A. J., Johnson, W. O. and Thurmond, M. C. (2007). Bayesian beta regression: Application to household data and genetic distance between foot-and-mouth disease viruses. Australian & New Zealand Journal of Statistics 49, 287-301. · Zbl 1136.62323 |

[4] | Buckley, J. (2003). Estimation of models with beta-distributed dependent variables: A replication and extension of Paolinos study. Political Analysis 11, 204-205. |

[5] | Carlstrom, L., Woodward, J. and Palmer, C. (2000). Evaluating the simplified conjoint expected risk model: Comparing the use of objective and subjective information. Risk Analysis 20, 385-392. |

[6] | Cepeda-Cuervo, E., Jaimes, D., Marín, M. and Rojas, J. (2016). Bayesian beta regression with Bayesian beta reg R-package. Computational Statistics 31, 165-187. · Zbl 1342.65023 |

[7] | Cribari-Neto, F. and Zeiles, A. (2010). Beta regression in R. Journal of Statistical Software 34, 1-24. |

[8] | Dey, D. K., Ghosh, S. K. and Mallick, B. N. (2000). Generalized Linear Models: A Bayesian Perspective. · Zbl 1006.00009 |

[9] | Espinheira, P., Ferrari, S. L. P. and Cribari-Neto, F. (2008). On beta regression residuals. Journal of Applied Statistics 35, 407-419. · Zbl 1147.62315 |

[10] | Ferrari, S. L. P. and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics 31, 799-815. · Zbl 1121.62367 |

[11] | Figueroa-Zuñiga, J. I., Arellano-Valle, R. B. and Ferrari, S. L. P. (2013). Mixed beta regression: A Bayesian perspective. Computational Statistics & Data Analysis 61, 137-147. · Zbl 1348.62194 |

[12] | Galvis, D. M., Bandyopadhyay, D. and Lachos, V. H. (2014). Augmented mixed beta regression models for periodontal proportion data. Statistics in Medicine 33, 3759-3771. |

[13] | Gamerman, D. and Lopes, H. (2006). Stochastic Simulation for Bayesian Inference, 2nd ed. New York: Chapman & Hall/CRC. · Zbl 1137.62011 |

[14] | Gelfand, A. E. and Sahu, S. K. (1999). Identifiability, improper priors, and Gibbs sampling for generalized linear models. Journal of the American Statistical Association 94, 247-253. · Zbl 1072.62611 |

[15] | Homan, M. D. and Gelman, A. (2014). The no-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research 15, 1593-1623. · Zbl 1319.60150 |

[16] | Kieschnick, R. and McCullough, B. D. (2003). Regression analysis of variates observed on \((0,1)\): Percentages, proportions, and fractions. Statistical Modelling 3, 193-213. · Zbl 1070.62056 |

[17] | Kim, S., Chen, M.-H. and Dey, D. K. (2008). Flexible generalized t-link models for binary response data. Biometrika 95, 93-106. · Zbl 1437.62513 |

[18] | Lemonte, A. J. and Bazán, J. L. (2016). New class of Johnson SB distributions and its associated regression model for rates and proportions. Biometrical Journal 58, 727-746. · Zbl 1386.62004 |

[19] | Liu, F. and Kong, Y. (2015). zoib: An R package for Bayesian inference for beta regression and zero/one inflated beta regression. The R Journal 7, 34-51. |

[20] | López, F. O. (2013). A Bayesian approach to parameter estimation in simplex regression model: A comparison with beta regression. Revista Colombiana de Estadística 36, 1-21. · Zbl 06257233 |

[21] | Maddala, G. S. (1983). Limited-Dependent and Qualitative Variables in Economics. New York: Cambridge University Press. · Zbl 0527.62098 |

[22] | Nogarotto, D. C. (2013). Bayesian inference EM beta and inflated beta regression model. Master’s dissertation (in Portuguese), IMECC-Unicamp. |

[23] | Nogarotto, D. C., Azevedo, C. L. N. and Bazán, J. L. (2020). Supplement to “Bayesian modeling and prior sensitivity analysis for zero-one augmented beta regression models with an application to psychometric data.” https://doi.org/10.1214/18-BJPS423SUPP |

[24] | Oliveira, M. S. (2004). A beta regression model: Theory and application. Master’s dissertation (in Portuguese), IME-USP. |

[25] | Ospina, R. (2008). Inflated beta regression modelo. Doctoral’s thesis (in Portuguese), IME-USP. |

[26] | Ospina, R. and Ferrari, S. L. P. (2010). Inflated beta distributions. Statistical Papers 51, 111-126. · Zbl 1247.62043 |

[27] | Ospina, R. and Ferrari, S. L. P. (2012). A general class of zero-or-one inflated beta regression models. Computational Statistics & Data Analysis 56, 1609-1623. · Zbl 1243.62099 |

[28] | Paolino, P. (2001). Maximum likelihood estimation of models with beta-distributed dependent variables. Political Analysis 9, 325-346. |

[29] | Papke, L. and Wooldridge, J. (1996). Econometric methods for fractional response variables with an application to 401(K) plan participation rates. Journal of Applied Econometrics 11, 619-632. |

[30] | Parker, A. J., Bandyopadhyay, D. and Slate, E. H. (2014). A spatial augmented beta regression model for periodontal proportion data. Statistical Modelling 14, 503-521. |

[31] | Paulino, C. D., Turkman, M. A. A. and Murteira, B. (2003). Bayesian Statistic (in Portuguese). Lisboa: Fundação Calouste Gulbenkian. |

[32] | Pereira, T. L. and Cribari-Neto, F. (2014). Detecting model misspecification in inflated beta regressions. Communications in Statistics Simulation and Computation 43, 631-656. · Zbl 1291.62126 |

[33] | R Development Core Team (2015). R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing. Available at http://www.R-project.org. |

[34] | Ramalho, J. J. S. and Silva, J. V. (2009). A two-part fractional regression model for the financial leverage decisions of micro, small, medium and large firms. Quantitative Finance 9, 621-636. |

[35] | Rue, H. and Martino, S. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. Journal of the Royal Statistical Society, Series B 71, 319-392. · Zbl 1248.62156 |

[36] | Smithson, M. and Verkuilen, J. (2006). A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychological Methods 11, 54-71. |

[37] | Souza, T. C., Pereira, T. L., Cribari-Neto, F. and Lima, M. C. V. (2016). Testing inference in inflated beta regressions under model misspecification. Communications in Statistics Simulation and Computation 45, 625-642. · Zbl 1341.62205 |

[38] | Stavrunova, O. and Yerokhin, O. (2012). Two-part fractional regression model for the demand for risky assets. Applied Economics 44, 21-26. |

[39] | Swearingen, C. J., Castro, M. S. M. and Bursac, Z. (2012). Inflated beta regression: Zero, one, and everything in between. In SAS Global Forum 2012. Available at http://support.sas.com/resources/papers/proceedings12/325-2012.pdf. |

[40] | Wieczorek, J. |

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