Advertisement

Statistics and Computing

, Volume 28, Issue 3, pp 699–711 | Cite as

Latent single-index models for ordinal data

Article
  • 198 Downloads

Abstract

We propose a latent semi-parametric model for ordinal data in which the single-index model is used to evaluate the effects of the latent covariates on the latent response. We develop a Bayesian sampling-based method with free-knot splines to analyze the proposed model. As the index may vary from minus infinity to plus infinity, the traditional spline that is defined on a finite interval cannot be applied directly to approximate the unknown link function. We consider a modified version to address this problem by first transforming the index into the unit interval via a continuously cumulative distribution function and then constructing the spline bases on the unit interval. To obtain a rapidly convergent algorithm, we make use of the partial collapse and parameter expansion and reparameterization techniques, improve the movement step of Bayesian splines with free knots so that all the knots can be relocated each time instead of only one knot, and design a generalized Gibbs step. We check the performance of the proposed model and estimation method by a simulation study and apply them to analyze a real dataset.

Keywords

Free-knot splines Generalized Gibbs sampler Ordinal data Parameter expansion and reparameterization Single-index model 

Notes

Acknowledgements

The work was supported by the Natural Science Foundations of China (Nos. 11471272 and 11661074) and the Natural Science Foundation of Fujian Province of China (No. 2013J01019). The authors are grateful to the anonymous referees, the Associate Editor, and the Editor for valuable suggestions for improving the manuscript.

References

  1. Albert, J.H., Chib, S.: Bayesian analysis of binary and polychotomous response data. J. Am. Stat. Assoc. 88, 669–679 (1993)MathSciNetCrossRefMATHGoogle Scholar
  2. Antoniadis, A., Grégoire, G., McKeague, I.W.: Bayesian estimation in single-index models. Stat. Sin. 14, 1147–1164 (2004)MathSciNetMATHGoogle Scholar
  3. Barnard, J., McCulloch, R., Meng, X.L.: Modeling covariance matrices in terms of standard deviations and correlations with application to shrinkage. Stat. Sin. 10, 1281–1311 (2000)MathSciNetMATHGoogle Scholar
  4. Biller, C.: Adaptive Bayesian regression spline in semiparametric generalized linear models. J. Comput. Graph. Stat. 9, 122–140 (2000)MathSciNetGoogle Scholar
  5. Bradlow, E.T., Zaslavsky, A.M.: Hierarchical latent variable model for ordinal data from a customer satisfaction survey with ‘no answer’ responses. J. Am. Stat. Assoc. 94, 43–52 (1999)Google Scholar
  6. Carroll, R.J., Fan, J., Gijbels, I., Wand, M.P.: Generalized partially linear single-index models. J. Am. Stat. Assoc. 92, 477–489 (1997)MathSciNetCrossRefMATHGoogle Scholar
  7. Chen, M.-H., Schmeiser, B.W.: General hit-and-run Monte Carlo sampling for evaluating multidimensional integrals. Oper. Res. Lett. 19, 161–169 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. Chib, S., Greenberg, E.: Analysis of multivariate probit models. Biomertrika 85, 347–361 (1998)CrossRefMATHGoogle Scholar
  9. Chib, S., Greenberg, E.: Additive cubic spline regression with Dirichlet process mixture errors. J. Econom. 156, 322–336 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. Cowles, M.K.: Accelerating Monte Carlo Markov chain convergence for cumulative-link generalized linear models. Stat. Comput. 6, 101–111 (1996)CrossRefGoogle Scholar
  11. Denison, D.G.T., Mallick, B.K., Smith, A.F.M.: Automatic Bayesian curving fitting. J. R. Stat. Soc. Ser. B 60, 333–350 (1998)CrossRefMATHGoogle Scholar
  12. Dimatteo, I., Genovese, C.R., Kass, R.E.: Bayesian curve fitting with free-knot splines. Biometrika 88, 1055–1071 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. Friedman, J.H., Stuetzle, W.: Projection pursuit regression. J. Am. Stat. Assoc. 76, 817–823 (1981)MathSciNetCrossRefGoogle Scholar
  14. Gelman, A., Rubin, D.B.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7, 457–511 (1992)CrossRefGoogle Scholar
  15. Green, P.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732 (1995)MathSciNetCrossRefMATHGoogle Scholar
  16. Härdle, W., Stoker, T.M.: Investing smooth multiple regression by the method of average derivatives. J. Am. Stat. Assoc. 84, 986–995 (1989)MATHGoogle Scholar
  17. Hastie, T.J., Tibshirani, R.J.: Generalized Additive Models. Chapman and Hall, London (1990)MATHGoogle Scholar
  18. Hastie, T.J., Tibshirani, R.J.: Varying-coefficient models. J. R. Stat. Soc. Ser. B 55, 757–796 (1993)MathSciNetMATHGoogle Scholar
  19. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)MathSciNetCrossRefMATHGoogle Scholar
  20. Holmes, C.C., Mallick, B.K.: Bayesian regression with multivariate linear splines. J. R. Stat. Soc. Ser. B 63, 3–17 (2001)MathSciNetCrossRefMATHGoogle Scholar
  21. Holmes, C.C., Mallick, B.K.: Generalized nonlinear modeling with multivariate free-knot regression splines. J. Am. Stat. Assoc. 98, 352–368 (2003)MathSciNetCrossRefMATHGoogle Scholar
  22. Hu, Y., Gramacy, R.B., Lian, H.: Bayesian quantile regression for single-index models. Stat. Comput. 23(4), 437–454 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. Ichimura, H.: Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econom. 58, 71–120 (1993)MathSciNetCrossRefMATHGoogle Scholar
  24. ISSP (International Social Survey Program) International Social Survey Program: Work Orientations. [Computer file] (1989)Google Scholar
  25. Johnson, T.R.: Generalised linear models with ordinally-observed covariates. Br. J. Math. Stat. Psychol. 59, 275–300 (2006)CrossRefGoogle Scholar
  26. Kukuk, M.: Indirect estimation of (latent) linear models with ordinal regressors: a Monte Carlo study and some empirical illustrations. Stat. Pap. 43, 379–399 (2002)MathSciNetCrossRefMATHGoogle Scholar
  27. Lang, S., Brezger, A.: Bayesian P-splines. J. Comput. Graph. Stat. 13, 183–212 (2004)MathSciNetCrossRefMATHGoogle Scholar
  28. Li, K.C.: Sliced inverse regression for dimension reduction. J. Am. Stat. Assoc. 86, 316–342 (1991)MathSciNetCrossRefMATHGoogle Scholar
  29. Lindstrom, M.J.: Bayesian estimation of free-knot splines using reversible jump. Comput. Stat. Data Anal. 41, 255–269 (2002)MathSciNetCrossRefMATHGoogle Scholar
  30. Liu, J.S.: The collapsed Gibbs sampler in Bayesian computations with applications to a gene regulation problem. J. Am. Stat. Assoc. 89, 958–966 (1994)MathSciNetCrossRefMATHGoogle Scholar
  31. Liu, J.S., Sabatti, C.: Generalised Gibbs sampler and multigrid Monte Carlo for Bayesian computation. Biometrika 87, 353–369 (2000)MathSciNetCrossRefMATHGoogle Scholar
  32. Liu, J.S., Wu, Y.N.: Parameter expansion for data augmentation. J. Am. Stat. Assoc. 94, 1264–1274 (1999)MathSciNetCrossRefMATHGoogle Scholar
  33. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equations of state calculations by fast computing machine. J. Chem. Phys. 21, 1087–1091 (1953)CrossRefGoogle Scholar
  34. Poon, W.-Y., Wang, H.-B.: Latent variable models with ordinal categorical covariates. Stat. Comput. 22, 1135–1154 (2012)MathSciNetCrossRefMATHGoogle Scholar
  35. Poon, W.-Y., Wang, H.-B.: Bayesian analysis of generalized partially linear single-index models. Comput. Stat. Data Anal. 68, 251–261 (2013)MathSciNetCrossRefGoogle Scholar
  36. Poon, W.-Y., Wang, H.-B.: Multivariate partially linear single-index models: Bayesian analysis. J. Nonparametr. Stat. 26(4), 755–768 (2014)MathSciNetCrossRefMATHGoogle Scholar
  37. Robert, C.P.: Simulation of truncated normal variables. Stat. Comput. 5, 121–125 (1995)CrossRefGoogle Scholar
  38. Ronning, G., Kukuk, M.: Efficient estimation of ordered probit models. J. Am. Stat. Assoc. 91, 1120–1129 (1996)MathSciNetCrossRefMATHGoogle Scholar
  39. Shi, J.Q., Lee, S.Y.: Bayesian sampling-based approach for factor analysis models with continuous and polytomous data. Br. J. Math. Stat. Psychol. 51, 233–252 (1998)CrossRefGoogle Scholar
  40. Shi, J.Q., Lee, S.Y.: Latent variable models with mixed continuous and polytomous data. J. R. Stat. Soc. Ser. B 62, 77–87 (2000)MathSciNetCrossRefMATHGoogle Scholar
  41. Song, X.Y., Lu, Z.H., Feng, X.N.: Latent variable models with nonparametric interaction effects of latent variables. Stat. Med. 33, 1723–1737 (2014)MathSciNetCrossRefGoogle Scholar
  42. Stoker, T.M.: Consistent estimation of scaled coefficients. Econometrica 54, 1461–1481 (1986)MathSciNetCrossRefMATHGoogle Scholar
  43. Tierney, L.: Markov chains for exploring posterior distributions. Ann. Stat. 22, 1701–1728 (1994)MathSciNetCrossRefMATHGoogle Scholar
  44. van Dyk, D.A., Park, T.: Partially collapsed Gibbs samplers: theory and methods. J. Am. Stat. Assoc. 103, 790–796 (2008)MathSciNetCrossRefMATHGoogle Scholar
  45. Wang, H.-B.: Bayesian estimation and variable selection for single index models. Comput. Stat. Data Anal. 53, 2617–2627 (2009)MathSciNetCrossRefMATHGoogle Scholar
  46. Yu, Y., Ruppert, D.: Penalized spline estimation for partially linear single-index model. J. Am. Stat. Assoc. 97, 1042–1054 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina

Personalised recommendations