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A Beta Unfolding Model for Continuous Bounded Responses

Published online by Cambridge University Press:  01 January 2025

Yvonnick Noel
Yvonnick Noel*
Affiliation:
University of Brittany, Rennes 2
*
Requests for reprints should be sent to Yvonnick Noel, University of Brittany Rennes 2, Department of Psychology, Place du Recteur Henri Le Moal, CS 24307, 35043 Rennes Cedex, France. E-mail: yvonnick.noel@uhb.fr
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Abstract

An unfolding model for continuous bounded responses is proposed, derived both from a hypothetical interpolation response mechanism and from the hypothesis of two opposite sources of item refusal being collapsed. These two sources of refusal are made explicit in a three-component Dirichlet response model and then collapsed to obtain a (two-component) beta response model. The two natural parameters of the beta are interpreted as acceptance and refusal parameters and expressed as functions of person-item distances on a latent continuum. The potentially bimodal shape of the beta is exploited to model chaotic response choices among ambivalent subjects.

Keywords

item response theoryunfolding analysisbeta distributioncontinuous bounded responsesbimodality

Information

Type
Original Paper
Information
Psychometrika , Volume 79 , Issue 4 , October 2014 , pp. 647 - 674
Copyright
Copyright © 2013 The Psychometric Society

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Footnotes

An R script for estimating the BUM parameters is available upon request from the author.

References

Aitchison, J., & Shen, S. (1980). Logistic-normal distributions: some properties and uses. Biometrika, 2, 261–272.CrossRefGoogle Scholar
Andrich, D. (1989). A probabilistic IRT model for unfolding preference data. Applied Psychological Measurement, 13, 193–216.CrossRefGoogle Scholar
Andrich, D. (1995). Hyperbolic cosine latent trait models for unfolding direct responses and pairwise preferences. Applied Psychological Measurement, 19, 269–290.CrossRefGoogle Scholar
Andrich, D., & Luo, G. (1993). A hyperbolic cosine latent trait model for unfolding dichotomous single-stimulus responses. Applied Psychological Measurement, 17, 253–276.CrossRefGoogle Scholar
Andrich, D., & Styles, I.M. (1998). The structural relationship between attitude and behaviour statements from the unfolding perspective. Psychological Methods, 3, 454–469.CrossRefGoogle Scholar
Byrd, R.H., Lu, P., Nocedal, J., & Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing, 16, 1190–1208.CrossRefGoogle Scholar
Coombs, C. (1950). Psychological scaling without a unit of measurement. Psychological Review, 57, 145–158.CrossRefGoogle ScholarPubMed
Davison, M., Robbins, S., & Swanson, D. (1978). Stage structure in objective moral judgments. Developmental Psychology, 14, 137–146.CrossRefGoogle Scholar
DeMars, C.E., & Erwin, T.D. (2003). Revising the scale of intellectual development: application of an unfolding model. Journal of College Student Development, 44, 168–184.CrossRefGoogle Scholar
Ferrari, S.L.P., & Cribari-Neto, F. (2004). Beta regression for modeling rates and proportions. Journal of Applied Statistics, 10, 1–18.Google Scholar
Greenacre, M., & Browne, M. (1986). An efficient alternating least-squares algorithm to perform multidimensional unfolding. Psychometrika, 51, 241–250.CrossRefGoogle Scholar
Gupta, R.D., & Kundu, D.K. (1999). Generalized exponential distributions. Australian & New Zealand Journal of Statistics, 41, 173–188.CrossRefGoogle Scholar
Herrnstein, R. (1961). Relative and absolute strength of response as a function of frequency of reinforcement. Journal of the Experimental Analysis of Behavior, 4, 267–272.CrossRefGoogle ScholarPubMed
Hoijtink, H. (1993). PARELLA: a parametric approach to parallelogram analysis. Quantitative Methods, 43, 93–107.Google Scholar
Kotz, S., Johnson, N. (1982). Beta distributions. In Kotz, S., & Johnson, N. (Eds.), Encyclopedia of statistical sciences (pp. 228–229). New York: Wiley.Google Scholar
Luce, R.D. (1959). Individual choice behavior. New York: Wiley.Google Scholar
McKelvie, S. (1978). Graphic rating scales: how many categories?. British Journal of Psychology, 69, 185–202.CrossRefGoogle Scholar
Mislevy, R. (1984). Estimating latent distribution. Psychometrika, 49, 359–381.CrossRefGoogle Scholar
Noel, Y. (1999). An ordination approach to the quantification of smokers’ readiness to change. Swiss Journal of Psychology, 58, 123–133.Google Scholar
Noel, Y. (1999). Recovering latent unimodal patterns of change by unfolding analysis: application to smoking cessation. Psychological Methods, 4, 173–191.CrossRefGoogle Scholar
Noel, Y. (2009, August). When extreme responses are substantial: a generalized beta response model of behavior change. Paper presented at the 42nd Annual Meeting of the Society for Mathematical Psychology, Amsterdam, The Netherlands.Google Scholar
Noel, Y., & Dauvier, B. (2007). A beta item response model for continuous bounded responses. Applied Psychological Measurement, 31, 47–73.CrossRefGoogle Scholar
R Core Team (2013). R: a language and environment for statistical computing [Computer software manual]. Vienna, Austria. Available from http://www.R-project.org/ (ISBN 3-900051-07-0).Google Scholar
Ramsay, J. (1980). The joint analysis of direct ratings, pairwise preferences, and dissimilarities. Psychometrika, 45, 149–165.CrossRefGoogle Scholar
Roberts, J.S., Donoghue, J.R., & Laughlin, J.E. (1998). The generalized graded unfolding model: a general parametric item response model for unfolding graded responses. Princeton: Educational Testing Service. (RR-98-32).Google Scholar
Roberts, J.S., Donoghue, J.R., & Laughlin, J.E. (2000). A general item response theory model for unfolding unidimensional polytomous responses. Applied Psychological Measurement, 24, 3–32.CrossRefGoogle Scholar
Roberts, J.S., & Laughlin, J.E. (1996). A unidimensional item response model for unfolding responses from a graded disagree-agree response scale. Applied Psychological Measurement, 20, 231–255.CrossRefGoogle Scholar
Roberts, J.S., Lin, Y., & Laughlin, J.E. (2001). Computerized adaptive testing with the generalized graded unfolding model. Applied Psychological Measurement, 25, 177–196.CrossRefGoogle Scholar
Roberts, J.S., Rost, J., & Macready, G.B., (2000b). An unfolding mixture model to explore the latitude of acceptance concept in attitude measurement. Invited paper presented at the Fifth International Conference on Social Science Methodology, Cologne, Germany.Google Scholar
Samejima, F. (1973). Homogeneous case of the continuous response model. Psychometrika, 38, 203–219.CrossRefGoogle Scholar
Samejima, F. (1974). Normal ogive model on the continuous response level in the multidimensional latent space. Psychometrika, 39, 111–121.CrossRefGoogle Scholar
van Schuur, W.H. (1993). Nonparametric unidimensional unfolding for multicategory data. Political Analysis, 4, 41–74.CrossRefGoogle Scholar
Smithson, M., & Verkuilen, J. (2005). Beta regression: practical issues in estimation (supplementary material). Retrieved from http://supp.apa.org/psycarticles/supplemental/met_11_1_54/met_smithson1631_supplementalmaterial.doc.Google Scholar
Smithson, M., & Verkuilen, J. (2006). A better Lemon-Squeezer? Maximum likelihood regression with beta-distributed dependent variables. Psychological Methods, 11, 54–71.CrossRefGoogle ScholarPubMed
Tamhane, A.C., Ankenman, B., & Yang, Y. (2002). The beta distribution as a latent response model for ordinal data (I): estimation of location and dispersion parameters. Journal of Statistical Computation and Simulation, 72, 473–494.CrossRefGoogle Scholar
Thurstone, L. (1927). A law of comparative judgement. Psychological Review, 34, 278–286.CrossRefGoogle Scholar
van der Maas, H., & Molenaar, P. (1992). Stagewise cognitive development: an application of catastrophe theory. Psychological Review, 99, 395–417.CrossRefGoogle ScholarPubMed
Verhelst, H., & Verstralen, H. (1993). A stochastic unfolding model derived from the partial credit model. Quantitative Methods, 43, 73–92.Google Scholar
Verkuilen, J., & Smithson, M. (2012). Mixed and mixture regression models for continuous bounded responses using the beta distribution. Journal of Educational and Behavioral Statistics, 37, 82–113.CrossRefGoogle Scholar
Volet, S.E., & Chalmers, D. (1992). Investigation of qualitative differences in university students’ learning goals based on an unfolding model of stage development. British Journal of Educational Psychology, 62, 17–34.CrossRefGoogle Scholar
Wang, T., & Zeng, L. (1998). Item parameter estimation for a continuous response model using an EM algorithm. Applied Psychological Measurement, 22, 333–344.CrossRefGoogle Scholar
Woodruff, D., & Hanson, B., (1996). Estimation of item response models using the EM algorithm for finite mixtures. (ACT research report series No. 96-6). Iowa City, IA: ACT, Inc.Google Scholar
Wright, B., & Masters, G. (1982). Rating scale analysis. Chicago: MESA Press.Google Scholar