A Bayesian Nonparametric Approach to Test Equating

A Bayesian nonparametric model is introduced for score equating. It is applicable to all major equating designs, and has advantages over previous equating models. Unlike the previous models, the Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous equating models are compared through the analysis of data sets famous in the equating literature. Also, the classical percentile-rank, linear, and mean equating models are each proven to be a special case of a Bayesian model under a highly-informative choice of prior distribution.

[1]  S. Walker.  New approaches to Bayesian consistency , 2004, math/0503672.

[2]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[3]  H. Joe.  Multivariate models and dependence concepts , 1998 .

[4]  N. Hjort,et al.  NONPARAMETRIC QUANTILE INFERENCE USING DIRICHLET PROCESSES , 2006 .

[5]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[6]  S. MacEachern,et al.  An ANOVA Model for Dependent Random Measures , 2004 .

[7]  Sonia Petrone.  Random Bernstein Polynomials , 1999 .

[8]  D. Blackwell,et al.  Ferguson Distributions Via Polya Urn Schemes , 1973 .

[9]  R. Tibshirani,et al.  An introduction to the bootstrap , 1993 .

[10]  Thomas R. Boucher,et al.  Test Equating, Scaling, and Linking: Methods and Practices , 2007 .

[11]  S. Walker,et al.  Coherent psychometric modelling with Bayesian nonparametrics. , 2009, The British journal of mathematical and statistical psychology.

[12]  Dorothy T. Thayer,et al.  Univariate and Bivariate Loglinear Models for Discrete Test Score Distributions , 1998 .

[13]  Fernando A. Quintana,et al.  Nonparametric Bayesian data analysis , 2004 .

[14]  J. Sethuraman.  A CONSTRUCTIVE DEFINITION OF DIRICHLET PRIORS , 1991 .

[15]  S. Walker,et al.  On rates of convergence for posterior distributions , 2004 .

[16]  R. Brennan,et al.  Test Equating, Scaling, and Linking: Methods and Practices , 2004 .

[17]  R. Linn.  Educational measurement, 3rd ed. , 1989 .

[18]  P. Müller,et al.  A method for combining inference across related nonparametric Bayesian models , 2004 .

[19]  W. J. Hall,et al.  Approximating Priors by Mixtures of Natural Conjugate Priors , 1983 .

[20]  Stephen G. Walker,et al.  A bivariate Dirichlet process , 2003 .

[21]  P. Diaconis,et al.  Conjugate Priors for Exponential Families , 1979 .

[22]  Dorothy T. Thayer,et al.  The Kernel Method of Test Equating , 2003 .

[23]  Purushottam W. Laud,et al.  Bayesian Nonparametric Inference for Random Distributions and Related Functions , 1999 .

[24]  R. Brennan,et al.  Test Equating, Scaling, and Linking , 2004 .

[25]  S. Walker,et al.  On rates of convergence for posterior distributions in infinite-dimensional models , 2007, 0708.1892.