Guttman Algebras and a Model Checking Procedure for Guttman Scales
We consider Guttman scales both from an algebraic and a statistical point of view. We present a duality between a class of algebras and Guttman scalable response structures, and show that the index of reproducibility is not always a reliable indicator for the Guttman scalability of a data set. Furthermore, we present a model checking procedure, and close with an example.
KeywordsItem-response structure Guttman scales Concept lattice Discrete duality Model checking
We should like to express our gratitude to the anonymous referees for careful reading and useful suggestions.
- Birkhoff, G. (1948). Lattice Theory. Providence: AMS Colloquium Publications, AMS.Google Scholar
- Bogardus, E. S. (1925). Measuring social distances. Journal of Applied Sociology, 9, 299–308.Google Scholar
- Düntsch, I. & Orłowska, E. (2011). An algebraic approach to preference relations. In H. de Swart (Ed.), Proceedings of Relational and Algebraic Methods in Computer Science – 12th International Conference, RAMICS 2011 (Vol. 6663, pp. 141–147). Lecture Notes in Computer Science. Berlin: Springer.CrossRefGoogle Scholar
- Düntsch, I., Orłowska, E., & Wang, H. (2001). Algebras of approximating regions. Fundamenta Informaticae, 46, 71–82.Google Scholar
- Guttman, L. (1950). Measurement and prediction. In S. Stouffer, L. Guttman, E. Suchman, P. Lazarsfeld, S. Star, & J. Clausen (Eds.), Measurement and Prediction (pp. 60–90). Princeton: Princeton University Press.Google Scholar
- Lazarsfeld, P. F. (1968). Latent Structure Analysis. Boston: Houghton Mifflin.Google Scholar
- Orłowska, E., Radzikowska, A. M., & Rewitzky, I. (2015). Dualities for Structures of Applied Logics. Studies in Logic, Mathematical Logic and Foundations. London: College Publications.Google Scholar
- Rasch, G. (1961). On general laws and the meaning of measurement in psychology. In J. Neyman (Ed.), Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability (Vol. 4, pp. 321–333).Google Scholar