Rasch model

Contents

Rasch model The Rasch model is a probabilistic measurement model for dichotomous responses developed to transform qualitative observations into quantitative scales, originating in mid-20th-century psychometrics. It underpins modern item response theory and has influenced assessment practices in United States, United Kingdom, Germany, Sweden, and Netherlands through adoption by institutions such as Educational Testing Service, University of Chicago, University of Cambridge, Max Planck Society, and Royal Swedish Academy of Sciences.

Overview

The Rasch model posits that the probability of a correct response is a logistic function of the difference between person ability and item difficulty, a concept propagated in works associated with Georg Rasch and later cited by scholars at London School of Economics, University of Oxford, Harvard University, Stanford University, and Columbia University. Influential conferences at Psychometric Society meetings, American Educational Research Association, and European Conference on Educational Research disseminated Rasch-based methods linking to applications in testing programs like Graduate Record Examinations, SAT, Programme for International Student Assessment, International Association for the Evaluation of Educational Achievement, and OECD. Adoption by agencies such as World Health Organization and United Nations facilitated use in health measurement and surveys influenced by instruments developed at Johns Hopkins University, Karolinska Institute, McGill University, University of Toronto, and Monash University.

The core mathematical formulation defines P(X_ni=1) = exp(θ_n − β_i)/(1+exp(θ_n − β_i)), where θ_n and β_i represent latent person and item parameters, a specification reflecting logistic regression foundations similar to methods used by statisticians associated with Karl Pearson, Ronald Fisher, Jerzy Neyman, Edwin Bidwell Wilson, and David Cox. The model's sufficiency and separability properties were developed alongside theories from Andrey Kolmogorov, Andrey Markov, Harold Hotelling, John Tukey, and William Gosset; matrix algebra and likelihood derivations echo work from Alan Turing, Norbert Wiener, Claude Shannon, John von Neumann, and Paul Erdős. Identifiability constraints often set a sum or reference point, practices discussed in texts from Institute of Mathematical Statistics, Royal Statistical Society, American Statistical Association, International Statistical Institute, and Society for Industrial and Applied Mathematics.

Parameter estimation employs conditional maximum likelihood, joint maximum likelihood, marginal maximum likelihood, and Bayesian approaches championed by researchers at Bayesian Society, University of California, Berkeley, Princeton University, Yale University, Massachusetts Institute of Technology, and University of Pennsylvania. Algorithms include Newton–Raphson, Expectation–Maximization, Markov chain Monte Carlo, and Gibbs sampling; these computational strategies trace methodological roots to work at Bell Labs, IBM Research, Los Alamos National Laboratory, Carnegie Mellon University, and Lawrence Berkeley National Laboratory. Software implementations emerged from projects at R-project, Pennsylvania State University, University of Illinois Urbana-Champaign, Cornell University, and University of Amsterdam with packages maintained by groups at Open University, University College London, University of Melbourne, and University of Bergen.

Goodness-of-fit and diagnostic tools include item-fit statistics, person-fit statistics, differential item functioning analyses, and graphical residual checks propagated through collaborations among scholars at Medical Research Council, National Institutes of Health, Centers for Disease Control and Prevention, European Medicines Agency, and Food and Drug Administration. Techniques such as Andersen’s conditional likelihood test, item characteristic curve inspection, and principal components analysis of residuals have been refined in studies at Karlsruhe Institute of Technology, ETH Zurich, Technical University of Munich, University of Helsinki, and Trinity College Dublin. Cross-validation, bootstrap methods, and simulation studies were advanced by teams at Los Alamos National Laboratory, Sandia National Laboratories, Argonne National Laboratory, CERN, and NASA.

The Rasch model has been applied to educational testing, health outcomes, attitude surveys, and performance assessments in contexts managed by World Bank, International Monetary Fund, European Union, NATO, and UNICEF. Specific use-cases include literacy measurement projects at UNESCO, patient-reported outcome instruments at Mayo Clinic, Cleveland Clinic, Massachusetts General Hospital, and Mount Sinai Hospital, workforce assessments by McKinsey & Company and Deloitte, and certification programs run by Association of American Medical Colleges, Royal College of Physicians, American Board of Internal Medicine, and General Medical Council. Large-scale assessments utilizing Rasch-derived scales appear in studies from Pew Research Center, Gallup, Ipsos, RAND Corporation, and Brookings Institution.

Extensions include polytomous Rasch models, rating scale and partial credit formulations, multidimensional adaptations, and mixture models developed in collaboration with authors affiliated with University of Chicago, Princeton University, University of Michigan, Northwestern University, and Duke University. Related item response models such as the two-parameter and three-parameter logistic models were compared in comparative studies at Educational Testing Service, College Board, ACT, Inc., National Council on Measurement in Education, and International Association for Computerized Adaptive Testing. Connections to latent trait models, structural equation modeling, and generalized linear mixed models stem from theoretical work at European Mathematical Society, American Mathematical Society, Institute of Physics, National Academy of Sciences, and Royal Society.

Category:Psychometrics