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Guttman Algebras and a Model Checking Procedure for Guttman Scales

  • Ivo Düntsch
  • Günther Gediga
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 17)

Abstract

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.

Keywords

Item-response structure Guttman scales Concept lattice Discrete duality Model checking 

Notes

Acknowledgements

We should like to express our gratitude to the anonymous referees for careful reading and useful suggestions.

References

  1. Birkhoff, G. (1948). Lattice Theory. Providence: AMS Colloquium Publications, AMS.Google Scholar
  2. Bogardus, E. S. (1925). Measuring social distances. Journal of Applied Sociology, 9, 299–308.Google Scholar
  3. Demri, S. & Orłowska, E. (2002). Incomplete Information: Structure, Inference, Complexity. Monographs in Theoretical Computer Science. An EATCS series. Berlin: Springer.CrossRefGoogle Scholar
  4. Doignon, J.-P. & Falmagne, J.-C. (1985). Spaces for the assessment of knowledge. International Journal of Man-Machine Studies, 23(2), 175–196.CrossRefGoogle Scholar
  5. Ducamp, A. & Falmagne, J.-C. (1969). Composite measurement. Journal of Mathematical Psychology, 6, 359–390.CrossRefGoogle Scholar
  6. Düntsch, I., Gediga, G., & Orłowska, E. (2001). Relational attribute systems. International Journal of Human Computer Studies, 55(3), 293–309.CrossRefGoogle Scholar
  7. Düntsch, I. & Orłowska, E. (2008). A discrete duality between the apartness algebras and apartness frames. Journal of Applied Non-classical Logics, 18(2– 3), 209–223.CrossRefGoogle Scholar
  8. 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
  9. Düntsch, I., Orłowska, E., & Wang, H. (2001). Algebras of approximating regions. Fundamenta Informaticae, 46, 71–82.Google Scholar
  10. Efron, B. (1981). Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods. Biometrika, 68(3), 589–599.CrossRefGoogle Scholar
  11. Efron, B. & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. New York: Chapman & Hall.CrossRefGoogle Scholar
  12. Gediga, G. & Düntsch, I. (2001). Rough approximation quality revisited. Artificial Intelligence, 132(2), 219–234.CrossRefGoogle Scholar
  13. Gediga, G. & Düntsch, I. (2002). Skill set analysis in knowledge structures. British Journal of Mathematical and Statistical Psychology, 55(2), 361–384.CrossRefGoogle Scholar
  14. Gothwal, V. K., Wright, T. A., Lamoureux, E. L., & Pesudovs, K. (2009). Guttman scale analysis of the distance vision scale. Investigative Ophthalmology & Visual Science, 50(9), 4496–4501.CrossRefGoogle Scholar
  15. Guttman, L. (1944). A basis for scaling qualitative data. American Sociological Review, 9, 139–150.CrossRefGoogle Scholar
  16. 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
  17. Lazarsfeld, P. F. (1968). Latent Structure Analysis. Boston: Houghton Mifflin.Google Scholar
  18. 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
  19. Orłowska, E. & Rewitzky, I. (2008). Context algebras, context frames and their discrete duality. In J. Peters, A. Skowron, & H. Rybiński (Eds.), Transactions on Rough Sets IX (Vol. 5390, pp. 212–229). Lecture Notes in Computer Science. Berlin: Springer.CrossRefGoogle Scholar
  20. 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
  21. Schooler, C. (1968). A note of extreme caution on the use of Guttman scales. American Journal of Sociology, 74(3), 296–301.CrossRefGoogle Scholar
  22. Wille, R. (1982). Restructuring lattice theory: An approach based on hierarchies of concepts. In I. Rival (Ed.), Ordered Sets (pp. 445–470). NATO Advanced Studies Institute. Dordrecht: Reidel.CrossRefGoogle Scholar
  23. Wright, B. D. (1977). Solving measurement problems with the Rasch model. Journal of Educational Measurement, 14(2), 97–116.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Brock UniversitySt. CatherinesCanada
  2. 2.Institut für Evaluation und MarktanalysenJeggenGermany

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