Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-13T13:55:59.565Z Has data issue: false hasContentIssue false
  • English
  • Français

A Latent Trait Model for Dichotomous Choice Data

Published online by Cambridge University Press:  01 January 2025

Herbert Hoijtink
Herbert Hoijtink*
Affiliation:
Department of Statistics and Measurement Theory, University of Groningen, The Netherlands
*
Requests for reprints should be send to Herbert Hoijtink, Department of Statistics and Measurement Theory, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, THE NETHERLANDS.
Get accessRights & Permissions [Opens in a new window]

Abstract

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to stimuli, with a one (zero) indicating agreement (disagreement) with the content of the stimulus. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and stimulus: the smaller the distance, the larger the probability that a person will agree with the content of the stimulus. An estimation procedure based on expectation maximization and marginal maximum likelihood is developed and the quality of the resulting parameter estimates evaluated.

Keywords

marginal maximum likelihoodexpectation maximizationnonmonotone trace linessingle-peaked preference functionslatent trait theoryunfolding

Information

Type
Original Paper
Information
Psychometrika , Volume 55 , Issue 4 , December 1990, pp. 641 - 656
Copyright
Copyright © 1990 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

I gratefully acknowledge Ivo Molenaar and Wijbrandt van Schuur for their advice and encouragement during the course of the investigation, Derk-Jan Kiewiet who constructed the program for the ML estimator for the person parameter and Anne Boomsma, Wendy Post, Tom Snijders, and David Thissen for their comments on smaller aspects of the investigation.

References

Andersen, E. B. (1970). Asymptoticl proerties of conditional maximum likelihood estimators. Journal of the Royal Statistical Society, Series B, 32, 283–301.CrossRefGoogle Scholar
Andersen, E. B., & Madsen, M. (1977). Estimating the parmeters of the latent population distribution. Psychometrika, 42, 357–374.CrossRefGoogle Scholar
Andrich, D. (1988). The application of an unfolding model of the PIRT type to the measurement of attitude. Applied Psychological Measurement, 12, 33–51.CrossRefGoogle Scholar
Bock, R. D. (1988). BILOG, Chicago: Scientific Software.Google Scholar
Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bock, R. D., & Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197.CrossRefGoogle Scholar
Cliff, N., Collins, L. M., Zatkin, J., & Gallipeau, D., McCormick, D. J. (1988). An ordinal scaling method for questionnaire and other ordinal data. Applied Psychological Measurement, 12, 83–97.CrossRefGoogle Scholar
Coombs, C. H. (1964). A theory of data, Ann Arbor: Mathesis Press.Google Scholar
Davison, M. L. (1977). On a metric unidimensional unfolding model for attitudinal and developmental data. Psychometrika, 42, 523–548.CrossRefGoogle Scholar
Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38.CrossRefGoogle Scholar
DeSarbo, W. S., & Hoffman, D. L. (1986). Simple and weighted unfolding threshold models for the spatial representation of binary choice data. Applied Psychological Measurement, 10, 247–264.CrossRefGoogle Scholar
Engelen, R. J. H. (1987). Semiparametric estimation in the Rasch model, Twente, The Netherlands: University of Twente.Google Scholar
Fischer, G. H. (1987). Einfuhrung in die theorie psychologischer Tests: Grundlagen and Anwendungen [Introduction to the theory of psychological test: Fundamentals and applications], Bern: Huber.Google Scholar
fischer, G. H. (1974). Applying the principles of specific objectivity and of generalizability to the measurement of change. Psychometrika, 52, 565–587.CrossRefGoogle Scholar
Glas, C. A. W. (1988). The derivation of some tests for the Rasch model from the multinomial distribution. Psychometrika, 52, 525–546.CrossRefGoogle Scholar
Haberman, S. J. (1977). Maximum likelihood estimation in exponential response models. The Annals of Statistics, 5, 815–841.CrossRefGoogle Scholar
Hartley, H. O. (1958). Maximum likelihood estimation from incomplete data. Biometrics, 14, 174–194.CrossRefGoogle Scholar
Hartley, H. O., & Hocking, R. R. (1971). The analysis of incomplete data. Biometrics, 27, 783–808.CrossRefGoogle Scholar
Hoijtink, H. (1990). PARELLA, measurement of latent traits by poximity items, Leiden: DSWO Press.Google Scholar
Jansen, P. G. W. (1983). Rasch analysis of attitudinal data. Doctoral dissertation, State Psychological Service, Catholic University of Nymegen, The Netherlands.Google Scholar
Kennedy, W. J. Jr., Gentle, J. E. (1980). Statistical computing, New York: Dekker.Google Scholar
Kiewiet, D. J. (1988). Een programma voor de schatting van persoons-parameters in het PPPA-model [A program for the estimation of person parameters in the PARELLA model] (HB-88-937-SW). Heymans Bulletins Psychologische Instituten R. U. Groningen.Google Scholar
Laird, N. (1978). Nonparametric maximum likelihood estimation of a mixing distribution. Journal of the American Statistical Association, 73, 805–811.CrossRefGoogle Scholar
Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B, 44, 226–226.CrossRefGoogle Scholar
Mislevy, R. J., & Bock, R. D. (1982). Biweight estimates of latent ability. Journal of Educational and Psychological Measurement, 42, 725–737.CrossRefGoogle Scholar
Mokken, R. J. (1970). A theory and procedure of scale analysis, 'S-Gravenhage: Mouton.Google Scholar
Molenaar, I. W. (1983). Some improved diagnostics for failure of the Rasch Model. Psychometrika, 48, 49–57.CrossRefGoogle Scholar
Molenaar, I. W., & Hoijtink, H. (1990). The many null distributions of person fit indices. Psychometrika, 55, 75–105.CrossRefGoogle Scholar
Ramsay, J. O. (1975). Solving implicit equations in psychometric data analysis. Psychometrika, 40, 337–360.CrossRefGoogle Scholar
Rapoport, R. B., Abramowitz, A. I., & McGlennon, J. (1986). The life of the parties, Lexington: The University Press of Kentucky.Google Scholar
Sanathana, L., & Blumenthal, S. (1978). The logistic model and estimation of latent structure. Journal of the American Statistical Association, 73, 794–799.CrossRefGoogle Scholar
Stroud, A. H., & Sechrest, D. (1966). Gaussian quadrature formulas, Englewood Cliffs, NJ: Prentice Hall.Google Scholar
Takane, Y. (1983). Choice model analysis of the pick any/n type of binary data. Handout for the talk given at the European Psychometric and Classification Meeting, Jouy-en-Josas, France.Google Scholar
Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model. Psychometrika, 47, 175–186.CrossRefGoogle Scholar
van den Wollenberg, A. L. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123–140.CrossRefGoogle Scholar
van Schuur, W. H. (1984). Structure in political beliefs, Amsterdam: CT Press.Google Scholar
Wainer, H., & Wright, B. D. (1980). Robust estimation of ability in the Rasch model. Psychometrika, 45, 373–391.CrossRefGoogle Scholar
Wright, B. D., & Douglas, G. A. (1977). Best procedures for sample free item analysis. Applied Psychological Measurement, 1, 281–294.CrossRefGoogle Scholar