Rasch Models (Gerhard Fischer, Handbook of Statistics, Vol. 26, 2007)

Gerhard Fischer, a leading Rasch theoretician since the 1960s, and a pillar of the Rasch community, has retired. His article in the Handbook of Statistics summarizes his legacy and his perspective on Rasch measurement. The article is 71 pages long. It is a thorough algebraic exposition of major aspects of the Rasch dichotomous model (RM), supported by numerical examples. Here are the section headings:

Rasch Models

1. Some history of the Rasch Model (2 pages). This recounts the work of Georg Rasch, focusing on the Poisson model, and ending with the RM.
2. Some basic concepts and properties of the RM (6 pages). Local independence, likelihood functions, the raw score as a sufficient statistic.
3. Characterizations and scale properties of the RM (10 pages). Mathematical comparisons with other IRT models based on item response functions. Specific objectivity.
4. Item parameter estimation (15 pages), including
4.1 Joint maximum likelihood estimation (2 pages)
4.2 Conditional maximum likelihood estimation (4 pages)
4.3 Marginal maximum likelihood estimation (5)
4.4 An approximate estimation method (1 page). This minimizes a variance-weighted sum of squares.
[Absent from this list is the pairwise estimation method (Rasch, 1980, pp. 171-2, Choppin, RUMM2020, etc.)]
5. Person parameter estimation. (2 pages). With known item difficulties, Maximum Likelihood Estimation and Warm Likelihood estimation.
6. Testing of fit (15 pages)
6.1 Conditional likelihood ratio tests for comparing person groups. (3 pages). Item response functions (IRFs), model and empirical.
6.2 Pearson-type tests. (3 pages). Glas-Verhelst tests of deviations between observed and expected frequencies.
[Absent: mention of Wright's INFIT and OUTFIT statistics.]
6.3 Wald-type tests. (1 page)
6.4 Lagrange multiplier tests (1 page)
6.5 Exact tests and approximate Monte Carlo tests (5 pages). Fit tests for person response strings when the item difficulties are known.
7. The linear logistic test model (7 pages)
7.1 Testing the fit of an LLTM (1 page)
7.2 Differential item functioning (DIF) [using LLTM] (3 pages)
8. Longitudinal linear logistic models (LLTM). (6 pages). Using LLTM across time-points.
8.1 a unidimensional LLTM of change (1 page)
8.2 A multidimensional LLTM of change (3 pages). Multiple dimensions modeled as parallel unidimensional LLTMs.
8.3 The special case of two time points: The LLRA (1 page). Linear Logistic Test Model With Relaxed Assumptions
9. Some remarks on applications and extension of the RM. (2 pages)
9.1 Dichotomous generalizations (1 Page). Mentioned are multifactorial RM, FACETS model, Mixed RM, One Parameter Logistic Model (OPLM), dynamic RMs.
9.2 Polytomous generalizations (1 Page). Mentioned are Rating Scale Model (RSM). Partial Credit Model (PCM), IRT models and multidimensional IRT models, and a Rasch model for continuous data.
References (7 pages). Gerhard Fischer has 20 references as first author, Erling Andersen 9, Georg Rasch 7, no one else more than 4 references.

Gerhard Fischer's Insights:

"G. Rasch generally showed a preference for heuristic graphical methods over significance tests." p. 549

"Given these results, an applied research worker might be inclined to conclude that "the RM fits the data". Statisticians are usually more reserved: they know that models never fit; models can at best fit to some degree of approximation. If the sample is large and the test powerful enough, the model will always be refuted." p. 552.

Comment: other Rasch philosophers would have worded this: "the data fit the RM", and "If the sample is large enough and the (statistical) test powerful enough, empirical data will always be shown to be defective."

But most revealing of Fischer's measurement philosophy is this critique of the modern use of the Rasch model from section 9:

"Applying the RM has recently become quite popular not only in psychology and education, but also in many other scientific domains. It is tempting to use the RM whenever a 'scale' consists of dichotomous observations ('items') and the raw score suggests itself as a useful data reduction or 'measure'. More often than not, such enterprises are futile, however, because the strict limitations of the RM are violated: unidimensionality, no guessing, parallel IRFs (or SO), no DIF with respect to gender, age, education, etc. Unidimensionality of the items requires, on the substantive level, that the items are of very homogeneous content; this often conflicts with psychologists' diagnostic aims. The requirement of no guessing strictly speaking excludes the popular multiple choice item format; in particular, it is hopeless to fit a RM to personality or attitude questionnaires with two (`yes' vs. `no') answer categories, because answers determined by the latent trait to be measured are indistinguishable from random responses. Absence of DIF is also a very hard criterion: experience with many intelligence and achievement tests shows that all verbal items - or items having a substantial verbal component - are prone to DIF with respect to the subpopulations mentioned above. Even elementary arithmetic or science problems often show considerable DIF with respect to gender, depending on their content. Therefore, the RM can by no means be considered as an omnibus method for the analysis and scoring of all sorts of tests. Rather, it should be viewed as a guideline for the construction or improvement of tests, as an ideal to which a test should be gradually approximated, so that measurement can profit from the unique properties of the RM."

Comment: Fischer's conclusion that the RM is "an ideal to which a test should be gradually approximated" is one with which surely all Rasch practitioners agree. However, Fischer's intermediate finding, "More often than not, such enterprises are futile", is strongly contradicted by 40 years of the practical application of Rasch methodology by numerous analysts to messy ordinal data originating from many different sources. Rasch measures, and the insights obtained from Rasch analysis of the data, have generally proved to be informative.

Rasch Models (Gerhard Fischer, Handbook of Statistics, Vol. 26, 2007), G.H. Fischer ... Rasch Measurement Transactions, 2010, 24:3 p. 1296-7

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

Forum

Rasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

The URL of this page is www.rasch.org/rmt/rmt243e.htm

Website: www.rasch.org/rmt/contents.htm