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Discovery Algorithms for Hierarchical Relations

Published online by Cambridge University Press:  01 January 2025

Lewis C. Price ,
C. Mitchell Dayton  and
George B. Macready
Lewis C. Price*
Affiliation:
Macro Systems, Inc.
C. Mitchell Dayton
Affiliation:
University of Maryland
George B. Macready
Affiliation:
University of Maryland
*
Requests for reprints should be sent to Lewis C. Price, Macro Systems, Inc., 8630 Fenton Street, Suite 300, Silver Spring, Maryland 20910.
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Abstract

Two algorithms based on a latent class model are presented for discovering hierarchical relations that exist among a set of K dichotomous items. The two algorithms, stepwise forward selection and backward elimination, incorporate statistical criteria for selecting (or deleting) 0-1 response pattern vectors to form the subset of the total possible 2k vectors that uniquely describe the hierarchy. The performances of the algorithms are compared, using computer-constructed data, with those of three competing deterministic approaches based on ordering theory and the calculation of Phi/Phi-max coefficients. The discovery algorithms are also demonstrated on real data sets investigated in the literature.

Keywords

probabilistic modellatent class modelhierarchical relationslatent structure analysisdiscovery algorithms
Type
Original Paper
Information
Psychometrika , Volume 45 , Issue 4 , December 1980 , pp. 449 - 465
Copyright
Copyright © 1980 The Psychometric Society

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References

Reference Notes

Airasian, P. W., Bart, W. M. & Greaney, B. J. An ordering-theoretic analysis of a propositional logic game. Paper presented at the annual meeting of the American Educational Research Association, New Orleans, Louisiana, 1973.Google Scholar
Carroll, J. B. Personal communication, July 1971.Google Scholar
Carroll, J. B.. In Resnick, L. B. (Eds.), Hierarchies in children's learning, 1971, Pittsburgh: Learning Research and Development Center, University of Pittsburgh.Google Scholar
Cook, M. J., & Walbesser, H. H. How to meet accountability with behaviorial objectives and learning hierarchies. Unpublished manuscript, University of Maryland, 1973.Google Scholar
Lele, K., & Bart, W. M. Preliminary item analysis by ordering theory, 1971, Minneapolis, Minnesota: University of Minnesota Research and Development Center in Education of Handicapped Children.Google Scholar
Price, L. C. Computer programs for discovering hierarchical relationships among dichotomous items. Unpublished manuscript, 1979. (Available from first author upon request.)Google Scholar
Resnick, L. B., & Wang, M. C. Approaches to the validation of learning hierarchies, 1969, Pittsburgh: University of Pittsburgh, Learning Research and Development Center.Google Scholar
Walbesser, H. H. An evaluation model and it applications, Appendix B: Science—A process approach—Second Report, 1968, College Park, Maryland: AAAS Commission on Science Education.Google Scholar
Murray, J. R. Statistical models for qualitative data with classification errors. Unpublished doctoral dissertation, University of Chicago.Google Scholar
Price, L. C. Algorithms for discovering hierarchical relationships among dichotomous items. Unpublished doctoral dissertation, University of Maryland, 1978.Google Scholar
Airasian, P. W. Formative evaluation instruments: A construction and validation of tests to evaluate learning over short time periods. Unpublished doctoral dissertation, University of Chicago, 1969.Google Scholar

References

Airasian, P. W. Validating sequential instructional hierarchies, Educational Technology, 1971, December, 54–56.Google Scholar
Airasian, P. W., Madaus, G. F., & Woods, E. M. Scaling attitude items: A comparison of scalogram analysis and ordering theory. Educational and Psychological Measurement, 1975, 35, 809–819.CrossRefGoogle Scholar
Bart, W. M. Some results of ordering theory for Guttman scaling. Educational and Psychological Measurement, 1976, 36, 141–148.CrossRefGoogle Scholar
Bart, W. M., & Airasian, P. W. Determination of the ordering among seven Piagetian tasks by an ordering-theoretic method. Journal of Educational Psychology, 1974, 66, 277–284.CrossRefGoogle Scholar
Bart, W. M., & Krus, D. J. An ordering-theoretic method to determine hierarchies among items. Educational and Psychological Measurement, 1973, 33, 291–300.CrossRefGoogle Scholar
Carroll, J. B. The nature of the data, or how to choose a correlation coefficient. Psychometrika, 1961, 26, 347–372.CrossRefGoogle Scholar
Cohen, J. A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 1960, 10, 37–46.CrossRefGoogle Scholar
Dayton, C. M., & Macready, G. B. A probabilistic model for validation of behavioral hierarchies. Psychometrika, 1976, 41, 189–204.CrossRefGoogle Scholar
Dayton, C. M., & Macready, G. B. A scaling model with response errors and intrinsically unscalable respondents. Psychometrika, 1980, 45, 343–356.CrossRefGoogle Scholar
Flavell, J. H. The developmental psychology of Jean Piaget, 1963, New York: Van Nostrand.CrossRefGoogle Scholar
Gagné, R. M. The acquisition of knowledge. Psychological Review, 1962, 69, 355–365.CrossRefGoogle Scholar
Gagné, R. M., & Paradise, N. E. Abilities and learning sets in knowledge acquisition. Psychological Monographs, 1961, 75, (Whole No. 518).CrossRefGoogle Scholar
Goodman, L. A. Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 1974, 61, 215–231.CrossRefGoogle Scholar
Goodman, L. A. A new model for scaling response patterns: An application of the quasi-independent concept. Journal of the American Statistical Association, 1975, 70(352), 755–768.CrossRefGoogle Scholar
Guttman, L. A. A basis for scaling qualitative data. American Sociological Review, 1944, 9, 139–150.CrossRefGoogle Scholar
Guttman, L. A. A basis for scalogram analysis. In Stouffer, S. A., Guttman, L., Suchman, E. A., Lazarsfeld, P. F., Star, S. A. & Clausen, J. A. (Eds.), Studies in social psychology in World War II, Volume 4, Measurement and prediction, 1950, Princeton, New Jersey: Princeton University Press.Google Scholar
Guttman, L. A. On Festinger's evaluation of scale analysis. Psychological Bulletin, 1974, 44, 451–495.CrossRefGoogle Scholar
Lazarsfeld, P. F., & Henry, N. W. Latent structure analysis, 1968, Boston: Houghton-Mifflin.Google Scholar
Lingoes, J. C. Multiple scalogram analysis: A set-theoretic model for analyzing dichotomous items. Educational and Psychological Measurement, 1963, 23, 501–524.CrossRefGoogle Scholar
Macready, G. B. The structure of domain hierarchies found within a domain referenced testing system. Educational and Psychological Measurement, 1975, 35, 583–598.CrossRefGoogle Scholar
Macready, G. B., & Dayton, C. M. The use of probabilistic models in the assessment of mastery. Journal of Educational Statistics, 1977, 2, 99–120.CrossRefGoogle Scholar
Macready, G. B., & Dayton, C. M. A two-stage conditional estimation procedure for unrestricted latent class models. Journal of Educational Statistics, 1980, 45, in press.Google Scholar
Proctor, C. H. A probabilistic formulation and statistical analysis for Guttman scaling. Psychometrika, 1970, 35, 73–78.CrossRefGoogle Scholar
Rao, C. R. Linear statistical inference and its applications, 1965, New York: Wiley.Google Scholar
Rimoldi, H. J. A., & Grib, T. F. Pattern analysis. The British Journal of Statistical Psychology, 1960, 13, 137–149.CrossRefGoogle Scholar
White, R. T., & Clark, R. M. A test of inclusion which allows for errors of measurement. Psychometrika, 1973, 38, 77–86.CrossRefGoogle Scholar