Anomaly, Paradox and Progress

"The aim of science is to maximize the scope of solved empirical problems, while minimizing the scope of anomalous and conceptual problems" (Laudan, 1977, p. 66).

When comparing different research traditions, such as the "test score" and "scaling" traditions, progress should be evaluated in terms of the adequacy of solutions offered for both empirical (data dominated) anomalies and conceptual (theory dominated) paradoxes in educational and psychological measurement. An important question in the study of progress is: How do scientists react to anomalies and paradoxes?

Kuhn (1970) defines an anomaly as a violation of the "paradigm-induced expectations that govern normal science" (pp. 52-53). Anomalies are detected through empirical analyses and have formed the basis for most discoveries in the natural sciences. For Kuhn, the discovery of anomalies provides the impetus for paradigm change within a field of study. Anomalies are empirical difficulties that reflect differences between the observed and theoretically expected data.

A paradox is a statement that seems to be contradictory or absurd, but may in fact be true. Both anomalies and paradoxes appear only within the framework of specific theories. Crossing item characteristic curves (ICCs) are viewed as paradoxical within the framework of Rasch measurement. If ICCs cross for two items, then the item difficulty order reverses above and below the crossing point. When ICCs cross, sample- invariant item calibration cannot be achieved. Crossing ICCs are not viewed as paradoxical within the framework of the two- or three-parameter IRT models.

How does scientific progress occur? According to positivists, such as Popper, theories are abandoned when anomalies occur. Post-positivist scholars (Kuhn, Lakatos, Laudan) question this. Laudan, Laudan and Donovan (1988) have proposed seven theses regarding how scientists react to anomalies:

When a theory encounters an anomaly or a paradox, then scientists (1) believe that this reflects adversely on their skills rather than on the inadequacies of the theory.
(2) leave the anomaly/paradox unresolved.
(3) refuse to change their assumptions.
(4) ignore the anomaly/paradox as long as the theory continues to anticipate novel phenomena successfully.
(5) believe that the anomaly/paradox becomes grounds for rejecting the theory only if it persistently resists solution.
(6) introduce hypotheses which are not testable in order to save the theory.
(7) believe that the anomaly/paradox becomes acute only if a rival theory explains it.

Evidence from the natural sciences reported by Laudan et al. suggests that when problems in a theory are encountered by scientists, these problems are not ignored, but the theory is not immediately abandoned. Typically, the scientists who use the theory seek a way of explaining and dealing with the problem that is not ad hoc. If they cannot address the problem, then the theory is likely to be abandoned; this is even more likely if an alternative theory is available that can explain the problem. Laudan et al. do not distinguish between anomalies and paradoxes, and scientists probably react in the same way to both.

What roles have anomalies and paradoxes played in progress in measurement theory? How have measurement practitioners and theorists reacted to conceptual problems? In the next two columns, I will explore the seven theses using, as two case studies, the "attenuation paradox" and the crossing of item characteristic curves.

Kuhn, T. 1970. The structure of scientific revolutions. 2nd Ed. Chicago: University of Chicago Press.

Laudan, L. 1977. Progress and its problems: Towards a theory of scientific growth. Berkeley, CA: University of California Press.

Laudan, R., Laudan, L., & Donovan, A. 1988. Testing theories of scientific change. In A. Donovan, L. Laudan, & R. Laudan (Eds.), Scrutinizing science: Empirical studies of scientific change (pp. 3-44). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Anomaly, Paradox and Progress, G Engelhard Jr. … Rasch Measurement Transactions, 1992, 6:2 p. 212

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

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