On the relationship between item response theory and factor analysis of discretized variables. (English) Zbl 0628.62104

Equivalence of marginal likelihood of the two-parameter normal ogive model in item response theory (IRT) and factor analysis of dichotomized variables (FA) was formally proved. The basic result on the dichotomous variables was extended to multicategory cases, both ordered and unordered categorical data. Pair comparison data arising from multiple-judgment sampling were discussed as a special case of the unordered categorical data. A taxonomy of data for the IRT and FA models was also attempted.


62P15 Applications of statistics to psychology
62H25 Factor analysis and principal components; correspondence analysis
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[1] Akaike, H. (1980). Likelihood and the Bayes procedure. In J. M. Bernardo et al. (Eds.),Bayesian statistics. Valencia University Press. · Zbl 0471.62033
[2] Andersen, E. B. (1980).Discrete statistical models with social science applications. Amsterdam: North-Holland. · Zbl 0423.62001
[3] Basu, D. (1977). On elimination of nuisance parameters.Journal of the American Statistical Association, 22, 229–243. · Zbl 0395.62003
[4] Bartholomew, D. J. (1983). Latent variable models for ordered categorical data.Journal of Econometrics.22, 229–243.
[5] Bartholomew, D. J. (1985, July). A unified view of factor analysis, latent structure analysis and scaling. Invited paper given at the 4-th European Meeting of the Psychometric Society and Classification Societies, Cambridge.
[6] Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories.Psychometrika, 37, 29–51. · Zbl 0233.62016
[7] Bock, R. D. (1975).Multivariate statistical methods in behavioral research. New York: McGraw Hill. · Zbl 0398.62086
[8] Bock, R. D. (1984, June).Full information item factor analysis. Paper presented at the Joint Meeting of the Classification Society and the Psychometric Society, Sant Barbara.
[9] Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm.Psychometrika, 46, 443–459.
[10] Bock, R. D., & Jones, L. V. (1968).The measurement and prediction of judgment and choice. San Francisco: Holden Day.
[11] Bock, R. D., & Lieberman, M. (1970). Fitting a response model forn dichotomously scored items.Psychometrika, 35, 283–319. · Zbl 0202.19101
[12] Carroll, J. D., & Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via anN-way generalization of ”Eckart-Young” decomposition.Psychometrika, 35, 283–319. · Zbl 0202.19101
[13] Christoffersson, A. (1975). Factor analysis of dichotomized variables.Psychometrika, 40, 5–32. · Zbl 0322.62063
[14] Coombs, C. H. (1964).A theory of data. New York: Wiley.
[15] Cox, D. R. (1966). Some procedure connected with the logistic qualitative response curve. In F. N. David (Ed.),Research papers in statistics: Festschrift for J. Neyman (pp. 55–71). New York: Wiley. · Zbl 0158.17808
[16] de Leeuw, J. (1983). Models and methods for the analysis of correlation coefficients.Journal of Econometrics, 22, 113–137.
[17] De Soete, G., & Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model.Psychometrika, 48, 553–566. · Zbl 0524.62112
[18] De Soete, G., Carroll, J. D., & DeSarbo, W. S. (in press). The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data.Journal of Mathematical Psychology. · Zbl 0622.62108
[19] Gifi, A. (1981).Non-linear multivariate analysis. Unpublished manuscript, University of Leiden, Department of Data Theory.
[20] Greenacre, M. (1984).Theory and application of correspondence analysis. London: Academic Press. · Zbl 0555.62005
[21] Heiser, W. J. (1981).Unfolding analysis of proximity data. Unpublished doctoral Dissertation, University of Leiden.
[22] Heiser, W., & de Leeuw, J. (1981). Multidimensional mapping of preference data.Mathematique et Sciences Humaines, 19, 39–96.
[23] Johnson, N. L., & Kotz, S. (1974).Distribution in Statistics: Multivariate distributions. New York: Wiley. · Zbl 0274.62015
[24] Jöreskog, K. G. (1970). A general method for analysis of covariance structures.Biometrika, 57, 239–251. · Zbl 0195.48801
[25] Lazarsfeld, P. F., & Henry, N. (1968).Latent structure analysis. New York: Houghton Mifflin. · Zbl 0182.52201
[26] Lord, F. M., & Novick, M. R. (1968).Statistical theories of mental test scores. Reading, MA: Addison Wesley. · Zbl 0186.53701
[27] Luce, R. D. (1959).Individual choice behavior: A theoretical analysis. New York: Wiley. · Zbl 0093.31708
[28] Mislevy, R. J. (in press). Recent developments in factor analysis of categorical variables.Journal of Educational Statistics.
[29] Muthén, B. (1978). Contributions to factor analysis of dichotomous variables.Psychometrika, 43, 551–560. · Zbl 0394.62042
[30] Muthén, B. (1979). A structural probit model with latent variables.Journal of the American Statistical Association, 24, 807–811. · Zbl 0448.62052
[31] Muthén, B. (1983). Latent variable structural equation modeling with categorical data.Journal of Econometrics, 22, 43–65.
[32] Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators.Psychometrika, 49, 115–132.
[33] Muthén, B., & Christoffersson, A. (1981). Simultaneous factor analysis of dichotomous variables in several groups.Psychometrika, 46, 407–419. · Zbl 0482.62050
[34] Ramsay, J. O. (1982). Some statistical approaches to multidimensional scaling.Journal of the Royal Statistical Society, Series A, 145, 285–312. · Zbl 0487.62003
[35] Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores.Psychometrika Monograph No. 17, 34(4, Pt. 2).
[36] Schönemann, P. H., & Wang, M. M. (1972). An individual difference model for the multidimensional analysis of preference data.Psychometrika, 37, 275–309.
[37] Sjoberg, L. (1967). Successive categories scaling of paired comparisons.Psychometrika, 32, 297–308.
[38] Takane, Y. (1980a). Analysis of categorizing behavior by a quantification method.Behaviormetrika, 8, 75–86.
[39] Takane, Y. (1980b). Maximum likelihood estimation in the generalized case of Thurstone’s model of comparative judgment.Japanese Psychological Rsearch, 22, 188–196.
[40] Takane, Y. (1983a, July). 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.
[41] Takane, Y. (1983b, June). An item response model for multidimensional analysis of multiple-choice data. Handbook for the talk given at the Annual Meeting of the Psychometric Society, Los Angeles.
[42] Takane, Y. (1985, June). Probabilistic multidimensional pair comparison models that take into account systematic individual differences. Transcript of the talk given at the 50-th Anniversary Meeting of the Psychometric Society, Nashville, TN.
[43] Takane, Y., & Carroll, J. D. (1981). Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities.Psychometrika, 46, 389–405.
[44] Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model.Psychometrika, 47, 175–186. · Zbl 0494.62095
[45] Thurstone, L. L. (1927). A law of comparative judgments.Psychological Review, 34, 273–286.
[46] Thurstone, L. L. (1945). The prediction of choice.Psychometrika, 10, 237–253. · Zbl 0060.31213
[47] Weinberg, S. L. Carroll, J. D., & Cohen, H. S. (1984). Confidence regions for INDSCAL using the jackknife and bootstrap techniques.Psychomertrika, 49, 475–491.
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