Brazilian Journal of Probability and Statistics

Bayesian skew-probit regression for binary response data

Jorge L. Bazán, José S. Romeo, and Josemar Rodrigues

Full-text: Open access

Abstract

Since many authors have emphasized the need of asymmetric link functions to fit binary regression models, we propose in this work two new skew-probit link functions for the binary response variables. These link functions will be named power probit and reciprocal power probit due to the relation between them including the probit link as a special case. Also, the probit regressions are special cases of the models considered in this work. A Bayesian inference approach using MCMC is developed for real data suggesting that the link functions proposed here are more appropriate than other link functions used in the literature. In addition, simulation study show that the use of probit model will lead to biased estimate of the regression coefficient.

Article information

Source
Braz. J. Probab. Stat., Volume 28, Number 4 (2014), 467-482.

Dates
First available in Project Euclid: 30 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1406741875

Digital Object Identifier
doi:10.1214/13-BJPS218

Mathematical Reviews number (MathSciNet)
MR3263060

Zentralblatt MATH identifier
1301.05082

Keywords
Skew-probit links binary regression Bayesian estimation power normal distribution reciprocal power normal distribution

Citation

Bazán, Jorge L.; Romeo, José S.; Rodrigues, Josemar. Bayesian skew-probit regression for binary response data. Braz. J. Probab. Stat. 28 (2014), no. 4, 467--482. doi:10.1214/13-BJPS218. https://projecteuclid.org/euclid.bjps/1406741875


Export citation

References

  • Achcar, J., Coelho-Barros, E. A. and Cordeiro, G. M. (2013). Beta generalized distributions and related exponentiated models: A Bayesian approach. Brazilian Journal of Probability and Statistics 27, 1–19.
  • Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polytomous response data. Journal of the American Statistical Association 88, 669–679.
  • Albert, J. H. and Chib, S. (1995). Bayesian residual analysis for binary response regression models. Biometrika 82, 747–759.
  • Bazán, J. L., Bolfarine, H. and Branco, M. D. (2010). A framework for skew-probit links in Binary regression. Communications in Statistics—Theory and Methods 39, 678–697.
  • Bazán, J. L., Branco, M. D. and Bolfarine, H. (2006). A skew item response model. Bayesian Analysis 1, 861–892.
  • Bedrick, E. J., Christensen, R. and Johnson, W. (1996). A new perspective on priors for generalized linear models. Journal of the American Statistical Association 91, 1450–1460.
  • Bolfarine, H. and Bazán, J. L. (2010). Bayesian estimation of the logistic positive exponent IRT model. Journal of Educational and Behavioral Statistics 35, 693–713.
  • Brooks, S. P. (2002). Discussion on “Bayesian measures of model complexity and fit” by D. J. Spiegelhalter, N. G. Best, B. P. Carlin and A. van der Linde. Journal of the Royal Statistical Society, Ser. B 64, 616–618.
  • Carlin, B. P. and Louis, T. A. (2000). Bayes and Empirical Bayes Methods for Data Analysis, 2nd ed. Boca Raton, FL: Chapman & Hall.
  • Chen, M. H., Dey, D. K. and Shao, Q.-M. (1999). A new skewed link model for dichotomous quantal response data. Journal of the American Statistical Association 94, 1172–1186.
  • Chen, M. H., Dey, D. K. and Shao, Q.-M. (2001). Bayesian analysis of binary data using skewed logit models. Calcutta Statistical Association Bulletin 51, 201–202.
  • Collet, D. (2003). Modelling Binary Data, 2nd ed. Boca Raton, FL: Chapman & Hall/CRC.
  • Cordeiro, G. and de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation 81, 883–898.
  • Cordeiro, G. M. and McCullagh P. (1991). Bias reduction in generalized linear models. Journal of the Royal Statistical Society, Ser. B 53, 629–643.
  • Czado, C. (1994). Bayesian inference of binary regression models with parametric link. Journal of Statistical Planning and Inference 41, 121–140.
  • Czado, C. and Santner, T. J. (1992). The effect of link misspecification on binary regression inference. Journal of Statistical Planning and Inference 33, 213–231.
  • Devidas, M. and George, E. O. (1999). Monotonic algorithms for maximum likelihood estimation in generalized linear models. Sankhyā, Ser. B 61, 382–396.
  • Dey, D. K., Ghosh, S. K. and Mallick, B. K. (1999). Generalized Linear Models: A Bayesian Perspective. New York: Marcel Dekker.
  • Eugene, N., Lee, C. and Famoye, F. (2002). Beta Normal distribution and its applications. Communication and Statistics—Theory and Methods 31, 497–512.
  • Finney, D. J. (1971). Probit Analysis. London: Cambridge Univ. Press.
  • Firth D. (1993). Bias reduction of maximum likelihood estimates. Biometrika 80, 27–38.
  • Guerrero, V. M. and Johnson, R. A. (1982). Use of the Box–Cox transformation with binary response models. Biometrika 69, 309–314.
  • Gupta, R. D. and Gupta, R. C. (2008). Analyzing skewed data by power normal model. Test 17, 197–210.
  • Johnson, V. and Albert, J. (1999). Ordinal Data Modeling. New York: Springer-Verlag.
  • Kundu, D. and Gupta, R. D. (2013). Power-normal distribution. Statistics 47, 110–125.
  • Longford, N. T. (1994). Logistic regression with random coefficients. Computational Statistics Data Analysis 17, 1–15.
  • Maiti, T. and Pradhan, V. (2008). A comparative study of the bias corrected estimates in logistic regression. Statistical Methods in Medical Research 17, 621–634.
  • Milicer, H. and Szczotka, F. (1966). Age at menarche in Warsaw girls in 1965. Human Biology 38, 199–203.
  • Prentice, R. L. (1976). A Generalization of the probit and logit methods for dose-response curves. Biometrika 32, 761–768.
  • SAS Institute Inc. (2009). SAS/STAT® 9.2 User’s Guide, 2nd ed. Cary, NC: SAS Institute Inc.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Ser. B 64, 583–639.
  • StataCorp (2009). Stata: Release 11. Statistical Software. College Station, TX: StataCorp LP.
  • Stukel, T. (1988). Generalized logistic models. Journal of the American Statistical Association 83, 426–431.
  • Taylor, J. and Siqueira, A. (1996). The cost of adding parameters to a model. Journal of the Royal Statistical Society 58, 593–607.
  • Wang, X. and Dey, D. (2010). Generalized extreme value regression for binary response data: An application to B2B electronic payments system adoption. The Annals of Applied Statistics 4, 2000–2023.
  • Zellner, A. and Rossi, P. E. (1984). Bayesian analysis of dichotomous quantal response models. Journal of Econometrics 25, 365–393.