Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2025-01-04T03:30:18.417Z Has data issue: false hasContentIssue false
  • English
  • Français

Rasch's Multiplicative Poisson Model with Covariates

Published online by Cambridge University Press:  01 January 2025

Haruhiko Ogasawara
Haruhiko Ogasawara*
Affiliation:
Railway Technical Research Institute
*
Request for reprints should be sent to Haruhiko Ogasawara, Department of Information and Management Science, Otaru University of Commerce, 3-5-21, Midori, Otaru 047 JAPAN.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

negative binomial distributionPoisson regressionRasch's multiplicative Poisson modelmissing valuesDirichlet distribution
Type
Original Paper
Information
Psychometrika , Volume 61 , Issue 1 , March 1996 , pp. 73 - 92
Copyright
© 1996 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.)

Footnotes

1

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.

References

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Petrov, B. N., Csaki, F. (Eds.), Second International Symposium on Information Theory (pp. 267–281). Budapest: Akademiai Kiado.Google Scholar
Böckenholt, U. (1993). Estimating latent distributions in recurrent choice data. Psychometrika, 58, 489–509.CrossRefGoogle Scholar
Böckenholt, U. (1993). A latent-class regression approach for the analysis of recurrent choice data. British Journal of Mathematical and Statistical Psychology, 46, 95–118.CrossRefGoogle Scholar
Consul, P. C., Jain, G. C. (1973). A generalization of the Poisson distribution. Technometrics, 15, 791–799.CrossRefGoogle Scholar
Engel, J. (1984). Models for response data showing extra-Poisson variation. Statistica Neerlandica, 38, 159–167.CrossRefGoogle Scholar
Gulliksen, H. (1950). The reliability of speeded tests. Psychometrika, 15, 259–269.CrossRefGoogle ScholarPubMed
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.Google Scholar
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.CrossRefGoogle Scholar
Jansen, M. G. H., Van Duijn, M. A. J. (1992). Extensions of Rasch's multiplicative Poisson model. Psychometrika, 57, 405–414.CrossRefGoogle Scholar
Jorgenson, D. W. (1961). Multiple regression analysis of a Poisson process. American Statistical Association Journal, 56, 235–245.CrossRefGoogle Scholar
Lawless, J. F. (1987). Negative bionomial and mixed Poisson regression. The Canadian Journal of Statistics, 15(3), 209–225.CrossRefGoogle Scholar
Little, R. J. A., Rubin, D. B. (1987). Statistical analysis with missing data, New York: Wiley.Google Scholar
Lord, F. M., Novick, M. R. (1968). Statistical theories of mental test scores, Reading, MA: Addison-Wesley.Google Scholar
Meredith, W. (1971). Poisson distributions of error in mental test theory. British Journal of Mathematical and Statistical Psychology, 24, 49–82.CrossRefGoogle Scholar
Nelson, J. F. (1984). The Dirichlet-Gamma-Poisson model of repeated events. Sociological Methods and Research, 12, 347–373.CrossRefGoogle Scholar
Nelson, J. F. (1985). Multivariate gamma-Poisson models. Journal of the American Statistical Association, 80, 828–834.CrossRefGoogle Scholar
Ogasawara, H. (1989). Covariance structure analysis of continuously changing populations. Behaviormetrika, 25, 15–33.CrossRefGoogle Scholar
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)Google Scholar
Owen, R. J. (1969). A Bayesian analysis of Rasch's multiplicative Poisson model, Princeton, NJ: Educational Testing Service.Google Scholar
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.Google Scholar
Rasch, G. (1980). Probabilistic models for some intelligence and attainment tests, Chicago: The University of Chicago Press.Google Scholar
Van Duijn, M. A. J. (1993). Mixed models for repeated count data, Leiden: DSWQ Press.Google Scholar