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 Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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