Journal of Statistical Theory and Practice

, Volume 6, Issue 3, pp 501–509 | Cite as

Dimensional Reduction for Latent Scores Modeling Using Recursive Integration

  • S. KiatsupaibulEmail author
  • A. J. Hayter


This article addresses the problem of estimating the cutpoints that are used to model ordinal categorical data from continuous latent variables. A Bayesian approach is taken, and the cutpoint estimates are obtained as the expectation of their posterior distribution. Ostensibly this involves a high-dimensional integral evaluation with the dimension equivalent to the number of cutpoints. However, it shows how recursive integration techniques can be used to reduce the calculation to a series of two-dimensional integral evaluations, resulting in a practical estimation algorithm whose computational complexity is only a linear function of the number of cutpoints. Observed covariates can also be incorporated into the model, and the efficient estimation of the cutpoints assists in the estimation of other parameters. The new methodology is illustrated with an application to credit risk rating modeling of Thai corporations, and the computational advantages over other standard approaches are demonstrated.

AMS Subject Classification

62F15 62P05 


Bayesian modeling Covariates credit risk ratings Cutpoints Latent variables Numerical integration Posterior expectation Recursive intregration 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Agresti, A. 2002. Categorical data analysis. Hoboken, NJ, Wiley.CrossRefGoogle Scholar
  2. Albert, J. H., and S. Chib. 1993. Bayesian analysis of binary and polychotomous response data. J. Am. Stat. Assoc., 88, 669–679.MathSciNetCrossRefGoogle Scholar
  3. Altman, E. I., and H. A. Rijken. 2004. How rating agencies achieve rating stability. J. Banking Finance, 28, 2679–2714.CrossRefGoogle Scholar
  4. Anderson, J. A., and P. R. Philips. 1981. Regression, discrimination, and measurement models for ordered categorical variables. Appl. Stat., 30, 22–31.MathSciNetCrossRefGoogle Scholar
  5. Asmussen, S., and P. W. Glynn. 2010. Stochastic simulation: Algorithm and analysis. New York, Springer.zbMATHGoogle Scholar
  6. Basel Committee on Banking Supervision. 2005. Convergence of capital measurement and capital standards: A revised framework. Basel, Switzerland, Bank for International Settlements.Google Scholar
  7. Baumert, S., A. Ghate, S. Kiatsupaibul, Y. Shen, R. L. Smith, and Z. B. Zabinsky. 2009. Discrete hit-and-run for sampling points from arbitrary distributions over subsets of integer hyperrectangles. Operations Res., 57(3), 727–739.MathSciNetCrossRefGoogle Scholar
  8. Chatterjee, A., G. Horgan, and C. Theobald. 2008. Exposure assessment for pesticide intake from multiple food products: A Bayesian latent-variable approach. Risk Anal., 28(6), 1727–1736.CrossRefGoogle Scholar
  9. Cowles, M. K. 1996. Accelerating Monte Carlo Markov chain convergence for cumulative link generalized linear models. Stat. Comput., 6, 101–111.CrossRefGoogle Scholar
  10. Dunson, D. B. 2007. Bayesian methods for latent trait modelling of longitudinal data. Stat. Methods Med. Res., 16, 399–415.MathSciNetCrossRefGoogle Scholar
  11. Everitt, B. 1984. An introduction to latent variable models. London, Chapman and Hall.CrossRefGoogle Scholar
  12. Hayter, A. J. 2006. Recursive integration methodologies with statistical applications. J. Stat. Plan. Inference, 136, 2284–2296.MathSciNetCrossRefGoogle Scholar
  13. Henderson, S. G. 2006. Mathematics for simulation. In Handbooks in operations research and management science, Volume 13: Simulation, ed. S. G. Henderson and B. L. Nelson. p. 18–53, Amsterdam, North-Holland.Google Scholar
  14. Hoshino, T. 2008. A Bayesian propensity score adjustment for latent variable modeling and MCMC algorithm. Comput. Stat. Data Anal., 52, 1413–1429.MathSciNetCrossRefGoogle Scholar
  15. Jacquier, E., M. Johannes, and N. Polson. 2007. MCMC maximum likelihood for latent state models. J. Econometrics, 137, 615–640.MathSciNetCrossRefGoogle Scholar
  16. Kiatsupaibul, S., Z. B. Zabinsky, and R. L. Smith. 2011. An analysis of a variation of hit-and-run for uniform sampling from general regions. ACM Trans. Modeling Comput. Simulation, 21(3), article 16.Google Scholar
  17. Leon-Novelo, L. G., X. Zhou, B. Nebiyou-Bekele, and P. Muller. 2010. Assessing toxicities in a clinical trial: Bayesian inference for ordinal data nested within categories. Biometrics, 66(3), 966–974.MathSciNetCrossRefGoogle Scholar
  18. Lovász, L. 1999. Hit-and-run mixes fast. Math. Programming, 86(3), 443–461.MathSciNetCrossRefGoogle Scholar
  19. Lovász, L., and S. S. Vempala. 2006. Fast algorithms for logconcave functions: Sampling, rounding, integration and optimization. Proc. 47th IEEE Symposium on Foundations of Computer Science (FOCS '06), 57–68.Google Scholar
  20. McCullagh, P. 1980. Regression models for ordinal data. J. R. Stat. Soc. Ser. B, 42(2), 109–142.MathSciNetzbMATHGoogle Scholar
  21. Skrondal, A., and S. Rabe-Hesketh. 2004. Generalized latent variable modeling: Multilevel, longitudinal, and structural equation models. Boca Raton, FL, Chapman and Hall.CrossRefGoogle Scholar
  22. Slaughter, J. C., A. H. Herring. and J. M. Thorp. 2009. A Bayesian latent variable mixture model for longitudinal fetal growth. Biometrics, 64(2), 1233–1242.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

  1. 1.Faculty of Commerce and AccountancyChulalongkorn UniversityBangkokThailand
  2. 2.Department of Business Information and AnalyticsUniversity of DenverDenverUSA

Personalised recommendations