Advertisement

A Dependent Bayesian Nonparametric Model for Test Equating

  • Jorge GonzálezEmail author
  • Andrés F. Barrientos
  • Fernando A. Quintana
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 89)

Abstract

Equating methods utilize functions to transform scores on two or more versions of a test, so that they can be compared and used interchangeably. In common practice, traditional methods of equating use parametric models where, apart from the test scores themselves, no additional information is used for the estimation of the equating transformation. We propose a flexible Bayesian nonparametric model for test equating which allows the use of covariates in the estimation of the score distribution functions that lead to the equating transformation. A major feature of this approach is that the complete shape of the score distribution may change as a function of the covariates. As a consequence, the form of the equating transformation can change according to covariate values. We discuss applications of the proposed model to real and simulated data. We conclude that our method has good performance compared to alternative approaches.

Keywords

Score Distribution Dirichlet Process Test Equate Dirichlet Process Mixture High Probability Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The first author acknowledges partial support of Fondecyt 11110044 and Anillo SOC1107 grants. The second author was partially funded by Fondecyt 3130400 grant. The third author was partially funded by Fondecyt grant 1100010.

References

  1. Barrientos AF, Jara A, Quintana F (2012) Fully nonparametric regression for bounded data using bernstein polynomials. Technical report, Department of Statistics, Pontificia Universidad Católica de ChileGoogle Scholar
  2. Caron F, Davy M, Doucet A, Duflos E, Vanheeghe P (2006) Bayesian inference for dynamic models with Dirichlet process mixtures. In: International conference on information fusion, Florence, 10–13 July 2006Google Scholar
  3. De Iorio M, Müller P, Rosner GL, MacEachern SN (2004) An ANOVA model for dependent random measures. J Am Stat Assoc 99:205–215CrossRefzbMATHGoogle Scholar
  4. De Iorio M, Johnson WO, Müller P, Rosner GL (2009) Bayesian nonparametric non-proportional hazards survival modelling. Biometrics 65:762–771CrossRefzbMATHMathSciNetGoogle Scholar
  5. De la Cruz R, Quintana FA, Müller P (2007) Semiparametric Bayesian classification with longitudinal markers. Appl Stat 56(2):119–137zbMATHGoogle Scholar
  6. Dey D, Mueller P, Sinha D (1998) Practical nonparametric and semiparametric Bayesian statistics. New York: SpringerzbMATHGoogle Scholar
  7. Dorans N, Pommerich M, Holland P (2007) Linking and aligning scores and scales. New York: Springer.CrossRefzbMATHGoogle Scholar
  8. Dunson DB, Herring AH (2006) Semiparametric Bayesian latent trajectory models. Technical report, ISDS Discussion Paper 16, Duke UniversityGoogle Scholar
  9. Dunson DB, Park JH (2008) Kernel stick-breaking processes. Biometrika 95:307–323CrossRefzbMATHMathSciNetGoogle Scholar
  10. Ferguson T (1973) A bayesian analysis of some nonparametric problems. Ann Stat 1:209–230CrossRefzbMATHMathSciNetGoogle Scholar
  11. Ferguson TS (1974) Prior distribution on the spaces of probability measures. Ann Stat 2:615–629CrossRefzbMATHMathSciNetGoogle Scholar
  12. Ferguson TS (1983) Bayesian density estimation by mixtures of normal distributions. In: Siegmund D, Rustage J, Rizvi GG (eds) Recent advances in statistics: papers in honor of Herman Chernoff on his sixtieth birthday, Bibliohound, Carlsbad, pp 287–302CrossRefGoogle Scholar
  13. Gelfand AE, Kottas A, MacEachern SN (2005) Bayesian nonparametric spatial modeling with Dirichlet process mixing. J Am Stat Assoc 100:1021–1035CrossRefzbMATHMathSciNetGoogle Scholar
  14. Gelman A, Carlin J, Stern H, Rubin D (2003) Bayesian data analysis, 2nd edn. Chapman and Hall, LondonGoogle Scholar
  15. Ghosh J, Ramamoorthi R (2003) Bayesian nonparametrics. New York: SpringerzbMATHGoogle Scholar
  16. Ghosal S, Van der Vaart AW (2007) Posterior convergence rates of Dirichlet mixtures at smooth densities. Ann Stat 35:697–723CrossRefzbMATHGoogle Scholar
  17. Ghosal S, Ghosh JK, Ramamoorthi RV (1999) Posterior consistency of Dirichlet mixtures in density estimation. Ann Stat 27:143–158CrossRefzbMATHMathSciNetGoogle Scholar
  18. González J (2014) SNSequate: Standard and Nonstandard Statistical Models and Methods for Test Equating. J Stat Softw 59(7):1–30Google Scholar
  19. González J, von Davier M (2013) Statistical models and inference for the true equating transformation in the context of local equating. J Educ Meas 50(3):315–320CrossRefGoogle Scholar
  20. Griffin JE, Steel MFJ (2006) Order-based dependent Dirichlet processes. J Am Stat Assoc 101:179–194CrossRefzbMATHMathSciNetGoogle Scholar
  21. Hanson T, Johnson W (2002) Modeling regression error with a mixture of Polya trees. J Am Stat Assoc 97(460):1020–1033CrossRefzbMATHMathSciNetGoogle Scholar
  22. Hjort NL, Holmes C, Müller P, Walker S (2010) Bayesian nonparametrics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  23. Holland P, Rubin D (1982) Test equating. Academic, New YorkGoogle Scholar
  24. Jara A, Hanson T (2011) A class of mixtures of dependent tail-free processes. Biometrika 98: 553–566CrossRefzbMATHMathSciNetGoogle Scholar
  25. Jara A, Lesaffre E, De Iorio M, Quintana FA (2010) Bayesian semiparametric inference for multivariate doubly-interval-censored data. Ann Appl Stat 4:2126–2149CrossRefzbMATHMathSciNetGoogle Scholar
  26. Jara A, Hanson T, Quintana F, Müller P, Rosner G (2011) DPpackage: Bayesian non-and semi-parametric modelling in R. J Stat Softw 40:1–30Google Scholar
  27. Kolen M, Brennan R (2004) Test equating, scaling, and linking: methods and practices. Springer, New YorkCrossRefGoogle Scholar
  28. Lavine M (1992) Some aspects of polya tree distributions for statistical modelling. Ann Stat 20:1222–1235CrossRefzbMATHMathSciNetGoogle Scholar
  29. Lavine M (1994) More aspects of polya tree distributions for statistical modelling. Ann Stat 22:1161–1176CrossRefzbMATHMathSciNetGoogle Scholar
  30. Lijoi A, Prünster I, Walker S (2005) On consistency of non-parametric normal mixtures for Bayesian density estimation. J Am Stat Assoc 100:1292–1296CrossRefzbMATHGoogle Scholar
  31. Lo AY (1984) On a class of Bayesian nonparametric estimates I: Density estimates. Ann Stat 12:351–357CrossRefzbMATHGoogle Scholar
  32. Lorentz G (1986) Bernstein polynomials. Chelsea, New YorkzbMATHGoogle Scholar
  33. MacEachern S (1999) Dependent nonparametric processes. In: ASA proceedings of the section on Bayesian statistical science, pp 50–55Google Scholar
  34. MacEachern SN (2000) Dependent Dirichlet processes. Technical report, Department of Statistics, The Ohio State UniversityGoogle Scholar
  35. Mauldin R, Sudderth W, Williams S (1992) Polya trees and random distributions. Ann Stat 20(3):1203–1221CrossRefzbMATHMathSciNetGoogle Scholar
  36. Müller P, Mitra R (2013) Bayesian nonparametric inference–why and how. Bayesian Anal 8(2):269–302CrossRefMathSciNetGoogle Scholar
  37. Müller P, Quintana F (2004) Nonparametric bayesian data analysis. Stat Sci 19:95–110CrossRefzbMATHGoogle Scholar
  38. Müller P, Erkanli A, West M (1996) Bayesian curve fitting using multivariate normal mixtures. Biometrika 83:67–79CrossRefzbMATHMathSciNetGoogle Scholar
  39. Müller P, Quintana FA, Rosner G (2004) A method for combining inference across related nonparametric Bayesian models. J R Stat Soc Ser B 66:735–749CrossRefzbMATHGoogle Scholar
  40. Müller P, Rosner GL, De Iorio M, MacEachern S (2005) A nonparametric Bayesian model for inference in related longitudinal studies. J R Stat Soc Ser C 54:611–626CrossRefzbMATHGoogle Scholar
  41. Petrone S (1999) Random bernstein polynomials. Scand J Stat 26(3):373–393CrossRefzbMATHMathSciNetGoogle Scholar
  42. R Development Core Team (2013). R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. ISBN:3-900051-07-0Google Scholar
  43. Rodriguez A, Dunson DB, Gelfand A (2008) The nested Dirichlet process. J Am Stat Assoc 103:1131–1154CrossRefzbMATHMathSciNetGoogle Scholar
  44. Sethuraman J (1994) A constructive definition of dirichlet priors. Stat Sin 4:639–650zbMATHMathSciNetGoogle Scholar
  45. Smith BJ (2007) Boa: An r package for mcmc output convergence assessment and posterior inference. J Stat Softw 21:1–37Google Scholar
  46. Teh YW, Jordan MI, Beal MJ, Blei DM (2006) Hierarchical Dirichlet processes. J Am Stat Assoc 101:1566–1581CrossRefzbMATHMathSciNetGoogle Scholar
  47. Tokdar ST, Zhu YM, Ghosh JK (2010) Bayesian density regression with logistic Gaussian process and subspace projection. Bayesian Anal 5:1–26CrossRefMathSciNetGoogle Scholar
  48. von Davier A (2011) Statistical models for test equating, scaling, and linking. Springer, New YorkzbMATHGoogle Scholar
  49. von Davier A, Holland P, Thayer D (2004) The kernel method of test equating. Springer, New YorkzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jorge González
    • 1
    • 2
    Email author
  • Andrés F. Barrientos
    • 1
  • Fernando A. Quintana
    • 1
  1. 1.Faculty of MathematicsPontificia Universidad Católica de ChileMacul, SantiagoChile
  2. 2.Measurement Center MIDE UCPontificia Universidad Católica de ChileMacul, SantiagoChile

Personalised recommendations