Stochastic Guttman Order

Louis Guttman conceptualized an ideal scale with the property that knowledge of only the number of items a respondent passes tells the researcher exactly which items the respondent passes. Guttman (1947, 1950) does this by constructing a "scalogram" of the scored responses, in which each column corresponds to a respondent, arranged left to right in decreasing raw score order, and each row corresponds to an item, arranged top to bottom in decreasing order of respondent success. The item on which the respondents are the most successful comes first. When Guttman's ideal is realized, the responses in the top left of the scalogram are all successes and in the bottom right are all failures. There is a distinct diagonal demarcation between success and failure, with no disordering or "inversion" of successes and failures.

In practice, of course, Guttman's ideal is unobtainable, as he well knew. Accordingly he established rules which position a "cutting line" for each respondent at that place in the string of responses which minimizes the number of inversions for that respondent. This is a kind of "least error" fitting of a deterministic ideal to uncertain data. Unfortunately Guttman's rule leads to ambiguous results. If the responses are 1010 to the items ordered in ascending order of difficulty, both 1!010 and 101!0 are "least inversions" placements of the cutting line for one inversion. This has not gone unnoticed and procedures for dealing with this problem have been proposed.

Kenny and Rubin (1977) object to the ambiguity, arbitrariness and lack of clear theoretical basis which underlie the attempts to solve this problem. They build on Guttman's concept of "reproducibility", the proportion of scalable, correctly placed responses in the data. Guttman minimized the inversions one respondent at a time, with ambiguous results. Kenny and Rubin assert that the inversions are more meaningfully reduced by considering all respondents with the same raw score together. This yields the unambiguous result that the cutting line is placed at that point where the observed number of successes would be were the data perfectly scalable. All respondents with the same raw score get the same cutting line, regardless of the number of inversions within each pattern of responses. This is the default in the scalogram programs, SAS and SPSS-X.

The default is well chosen. Ordering by raw score is identical to requiring that raw score be a sufficient statistic. But this requirement leads directly to the Rasch model as the only explanation for the data. Guttman reproducibility, constrained by unambiguity, is equivalent to the Rasch model.

Guttman L 1947. The Cornell technique for scale and intensity analysis. Educational and Psychological Measurement, 7, 274-279

Guttman L 1950. The basis for scalogram analysis. In Stouffer et al. Measurement and Prediction. The American Soldier Vol. IV. New York: Wiley

Kenny DA, Rubin DC 1977. Estimating chance reproducibility in Guttman scaling. Social Science Research, 6, 188-196.

Stochastic Guttman order. Linacre JM. … Rasch Measurement Transactions, 1992, 5:4 p.189

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
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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 Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang 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
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
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