Abstract
As a multivariate model of the number of events, Rasch's multiplicative Poisson model is extended such that the parameters for individuals in the prior gamma distribution have continuous covariates. The parameters for individuals are integrated out and the hyperparameters in the prior distribution are estimated by a numerical method separately from difficulty parameters that are treated as fixed parameters or random variables. In addition, a method is presented for estimating parameters in Rasch's model with missing values.
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References
Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B. N. Petrov & F. Csaki (Eds.),Second International Symposium on Information Theory (pp. 267–281). Budapest: Akademiai Kiado.
Böckenholt, U. (1993a). Estimating latent distributions in recurrent choice data.Psychometrika, 58, 489–509.
Böckenholt, U. (1993b). A latent-class regression approach for the analysis of recurrent choice data.British Journal of Mathematical and Statistical Psychology, 46, 95–118.
Consul, P. C., & Jain, G. C. (1973). A generalization of the Poisson distribution.Technometrics, 15, 791–799.
Engel, J. (1984). Models for response data showing extra-Poisson variation.Statistica Neerlandica, 38, 159–167.
Gulliksen, H. (1950). The reliability of speeded tests.Psychometrika, 15, 259–269.
Jansen, M. G. H. (1986). A Bayesian version of Rasch's multiplicative Poisson model for the number of errors on an achievement test.Journal of Educational Statistics, 11, 147–160.
Jansen, M. G. H., & Snijders, T. A. B. (1991). Comparison of Bayesian estimation procedures for two-way contingency tables without interaction.Statistica Neerlandica, 45, 51–65.
Jansen, M. G. H., & Van Duijn, M. A. J. (1992). Extensions of Rasch's multiplicative Poisson model.Psychometrika, 57, 405–414.
Jorgenson, D. W. (1961). Multiple regression analysis of a Poisson process.American Statistical Association Journal, 56, 235–245.
Lawless, J. F. (1987). Negative bionomial and mixed Poisson regression.The Canadian Journal of Statistics, 15(3), 209–225.
Little, R. J. A., & Rubin, D. B. (1987).Statistical analysis with missing data. New York: Wiley.
Lord, F. M., & Novick, M. R. (1968).Statistical theories of mental test scores. Reading, MA: Addison-Wesley.
Meredith, W. (1971). Poisson distributions of error in mental test theory.British Journal of Mathematical and Statistical Psychology, 24, 49–82.
Nelson, J. F. (1984). The Dirichlet-Gamma-Poisson model of repeated events.Sociological Methods and Research, 12, 347–373.
Nelson, J. F. (1985). Multivariate gamma-Poisson models.Journal of the American Statistical Association, 80, 828–834.
Ogasawara, H. (1989). Covariance structure analysis of continuously changing populations.Behaviormetrika, 25, 15–33.
Ogasawara, H. (1992). Models of the number of errors using structured parameters in a generalized Poisson distribution and the Polya-Eggenberger distribution.Japanese Journal of Behaviormetrics, 19(2), 1–13. (In Japanese)
Owen, R. J. (1969).A Bayesian analysis of Rasch's multiplicative Poisson model (ETS Research Bulletin 69-64). Princeton, NJ: Educational Testing Service.
Rasch, G. (1973). Two applications of the multiplicative Poisson models in road accidents statistics.Bulletin of the International Statistical Institute, Proceedings of the 39th session, Vienna, 31–43.
Rasch, G. (1980).Probabilistic models for some intelligence and attainment tests. Chicago: The University of Chicago Press. (Original work published in 1960)
Van Duijn, M. A. J. (1993).Mixed models for repeated count data. Leiden: DSWQ Press.
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The author is now affiliated with the Otara University of Commerce.
The author is grateful to Yoshio Takane, Haruo Yanai, Eiji Muraki, the editor and referees for their careful readings and helpful suggestions on earlier versions of this paper. Part of this work was presented at the third European Congress of Psychology at Tampere, Finland in 1993.
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Ogasawara, H. Rasch's multiplicative poisson model with covariates. Psychometrika 61, 73–92 (1996). https://doi.org/10.1007/BF02296959
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DOI: https://doi.org/10.1007/BF02296959