Bayesian nonparametric estimation of test equating functions with covariates

Equating is an important step in the process of collecting, analyzing, and reporting test scores in any program of assessment. Methods of equating 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 either parametric or semi-parametric models where, apart from the test scores themselves, no additional information is used to estimate the equating transformation function. 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 is proposed. A major feature of this approach is that the complete shape of the scores distribution may change as a function of the covariates. As a consequence, the form of the equating transformation can change according to covariate values. Applications of the proposed model to real and simulated data are discussed and compared to other current methods of equating.

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

[2]  Marie Wiberg,et al.  Observed Score Linear Equating with Covariates , 2011 .

[3]  Peter Müller,et al.  DPpackage: Bayesian Semi- and Nonparametric Modeling in R , 2011 .

[4]  W. Sudderth,et al.  Polya Trees and Random Distributions , 1992 .

[5]  Stephen G. Walker,et al.  Dependent mixtures of Dirichlet processes , 2011, Comput. Stat. Data Anal..

[6]  Fernando A. Quintana,et al.  Fully Nonparametric Regression for Bounded Data Using Dependent Bernstein Polynomials , 2017 .

[7]  J. E. Griffin,et al.  Order-Based Dependent Dirichlet Processes , 2006 .

[8]  Albert Y. Lo,et al.  On a Class of Bayesian Nonparametric Estimates: I. Density Estimates , 1984 .

[9]  S. MacEachern,et al.  Bayesian Nonparametric Spatial Modeling With Dirichlet Process Mixing , 2005 .

[10]  T. Ferguson.  Prior Distributions on Spaces of Probability Measures , 1974 .

[11]  Radford M. Neal.  Slice Sampling , 2003, The Annals of Statistics.

[12]  J. Ghosh,et al.  POSTERIOR CONSISTENCY OF DIRICHLET MIXTURES IN DENSITY ESTIMATION , 1999 .

[13]  David B Dunson,et al.  Nonparametric Bayesian models through probit stick-breaking processes. , 2011, Bayesian analysis.

[14]  W. Johnson,et al.  Modeling Regression Error With a Mixture of Polya Trees , 2002 .

[15]  Stephen G. Walker,et al.  A Bayesian Nonparametric Approach to Test Equating , 2009 .

[16]  P. Müller,et al.  Bayesian Nonparametrics: An invitation to Bayesian nonparametrics , 2010 .

[17]  Eswar G. Phadia,et al.  Prior Processes and Their Applications: Nonparametric Bayesian Estimation , 2013 .

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

[19]  KATHERINE B. LEHMAN.  On Support , 1984 .

[20]  Fernando A. Quintana,et al.  On the Support of MacEachern’s Dependent Dirichlet Processes and Extensions , 2012 .

[21]  Jorge González,et al.  SNSequate: Standard and Nonstandard Statistical Models and Methods for Test Equating , 2014 .

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

[23]  Stephen G. Walker,et al.  A Bayesian Nonparametric Model for Test Equating , 2009 .

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

[25]  Matthias von Davier,et al.  Statistical Models and Inference for the True Equating Transformation in the Context of Local Equating. , 2013 .

[26]  T. Ferguson.  BAYESIAN DENSITY ESTIMATION BY MIXTURES OF NORMAL DISTRIBUTIONS , 1983 .

[27]  Hong Ying Hu,et al.  Fully Nonparametric Regression Estimation Based on Empirical Mode Decomposition , 2012 .

[28]  A. V. D. Vaart,et al.  Posterior convergence rates of Dirichlet mixtures at smooth densities , 2007, 0708.1885.

[29]  Sonia Petrone,et al.  A Predictive Study of Dirichlet Process Mixture Models for Curve Fitting , 2014, Scandinavian journal of statistics, theory and applications.

[30]  S. Walker.  Invited comment on the paper "Slice Sampling" by Radford Neal , 2003 .

[31]  P. Holland,et al.  Linking and aligning scores and scales , 2007 .

[32]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[33]  Lancelot F. James,et al.  Gibbs Sampling Methods for Stick-Breaking Priors , 2001 .

[34]  Peter Müller,et al.  DPpackage: Bayesian Non- and Semi-parametric Modelling in R. , 2011, Journal of statistical software.

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

[36]  A. A. Davier,et al.  Observed-Score Equating: An Overview , 2013, Psychometrika.

[37]  R. Gnanadesikan,et al.  Probability plotting methods for the analysis of data. , 1968, Biometrika.

[38]  Brian J. Smith,et al.  boa: An R Package for MCMC Output Convergence Assessment and Posterior Inference , 2007 .

[39]  Paul W. Holland,et al.  Statistical models for test equating, scaling, and linking , 2011 .

[40]  A. Gelfand,et al.  Bayesian Model Choice: Asymptotics and Exact Calculations , 1994 .

[41]  Willem J. van der Linden.  Some Conceptual Issues in Observed-Score Equating , 2013 .

[42]  M. Lavine.  More Aspects of Polya Tree Distributions for Statistical Modelling , 1992 .

[43]  Willem J. van der Linden,et al.  Local Observed-Score Equating , 2009 .

[44]  Michael J. Kolen,et al.  The kernel method of test equating , 2006 .

[45]  David B. Dunson,et al.  Bayesian Semiparametric Joint Models for Functional Predictors , 2009, Journal of the American Statistical Association.

[46]  S. Geisser,et al.  A Predictive Approach to Model Selection , 1979 .

[47]  P. Holland,et al.  Population Invariance and the Equatability of Tests: Basic Theory and The Linear Case , 2000 .

[48]  S. Walker,et al.  On Consistency of Nonparametric Normal Mixtures for Bayesian Density Estimation , 2005 .

[49]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .