Abstract
A latent variable model for observed variables representing frequencies is proposed. The data type for the model is a subjects by variables two-way frequency table. The model has two groups of latent variables. The first group of latent variables represents the characteristics of subjects and corresponds to common factors in factor analysis. On the other hand, each of latent variables in the second group is related to one of the manifest variables and corresponds to a specific factor in factor analysis. The manifest variables in the model, when given the values of common latent variables, follow the negative binomial distributions. The latent variables in the first and second groups are integrated out of the model. The parameters in the model are estimated by the marginal maximum likelihood method, using a kind of the EM algorithm. The communality, specificity, and reliability for an observed variable are defined.
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References
Archer, C.O. & Jennrich, R.I. (1973). Standard errors for rotated factor loadings. Psyckometrika, 38, 581–592.
Bishop, Y.M.M., Fienberg, S.E. & Holland, P.W. (1975). Discrete multivariate analysis: Theory and practice. Cambridge, Massachusetts: MIT Press.
Bock, R.D. & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: An application of an EM algorithm. Psyckometrika, 46, 443–459.
Clogg, C.C. (1986). Discussion of paper by L.A. Goodman (1986) Some useful extension of the usual correspondence analysis approach and the usual log-linear models approach in the analysis of contingency tables. International Statistical Review, 54, 243–309.
Engel, J. (1984). Models for response data showing extra-Poisson variation. Statistica Neerlandica, 38, 159–167.
Gardner, W., Mulvey, E.P. & Shaw, E.G. (1995). Regression analysis of counts and rates: Poisson, overdispersed Poisson, and negative binomial models. Psychological Bulletin, 118, 392–404.
Goodman, L.A. (1985). The analysis of cross-classified data having ordered and/or unordered categories: Association models, correlation models, and asymmetry models for contingency tables with or without missing entries. Annals of Statistics, 13, 10–69.
Goodman, L.A. (1986). Some useful extension of the usual correspondence analysis approach and the usual log-linear models approach in the analysis of contingency tables (with discussion). International Statistical Review, 54, 243–309.
Goodman, L.A. (1991). Measures, models, and graphical displays in the analysis of cross-classified data (with discussion). Journal of the American Statistical Association, 86, 1085–1138.
Harman, H. (1976). Modern factor analysis (3rd ed.). Chicago: University of Chicago Press.
Jansen, M.G.H. & Van Duijn, M.A.J. (1992). Extensions of Rasch’s multiplicative Poisson model. Psyckometrika, 57, 405–414.
Jöreskog, K.G. (1978). Structural analysis of covariance and correlation matrices. Psyckometrika, 43, 443–447.
Jöreskog, K.G. & Sörbom, D. (1976). Statistical models and methods for test-retest situations. In D.N.M. De Gruijter and L.J.Th. van der Kamp (Eds.), Advances in psychological and educational measurement (pp. 135-157). Wiley: New York.
Lawless, J.F. (1987). Negative binomial and mixed Poisson regression. The Canadian Journal of Statistics, 15, 209–225.
Lord, F.M. & Novick, M.R. (1968). Statistical theories of mental test scores. Reading, Mass.: Addison-Wesley.
Louis, T.A. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, B, 44, 226–233.
Meredith, W. (1971). Poisson distributions of error in mental test theory. British Journal of Mathematical and Statistical Psychology, 24, 49–82.
Mislevy, R.J. (1984). Estimating latent distributions. Psyckometrika, 49, 359–381.
Mislevy, R.J. & Sheehan, K.M. (1989). Information matrices in latent-variable models. Journal of Educational Statistics, 14, 335–359.
Moustaki, I. (1996). A latent trait and a latent class model for mixed observed variables. British Journal of Mathematical and Statistical Psychology, 49, 313–334.
Ogasawara, H. (1996a). A proposal of Poisson factor analysis. Abstracts of the 26th International Congress of Psychology (p. 421).
Ogasawara, H. (1996b). Rasch’s multiplicative Poisson model with covariates. Psyckometrika, 61, 73–92.
Ogasawara, H. (1998a). A factor analysis model for a mixture of various types of variables. Bekaviormetrika, 25, 1–12.
Ogasawara, H. (1998b). A log-bilinear model with latent variables. Bekaviormetrika, 25, 95–110.
Rao, C.R. (1973). Linear statistical inference and its applications (2nd ed.). New-York: Wiley.
Rasch, G. (1980). Probabilistic models for some intelligence and attainment tests. Chicago: The University of Chicago Press, (original work published in 1960).
Sammel, M.D., Louise, M.R. & Legier, J.M. (1997). Latent variable models for mixed discrete and continuous outcomes. Journal of Royal Statistical Society, B, 59, 667–678.
Silvey, S.D. (1975). Statistical inference. London: Chapman & Hall.
Tsutakawa, R.K. (1988). Mixed model for analyzing geographic variability in mortality rates. Journal of the American Statistical Association, 83, 37–42.
Van Duijn, M.AJ. (1993). Mixed models for repeated count data. Leiden: DSWO Press.
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Ogasawara, H. Negative Binomial Factor Analysis. Behaviormetrika 26, 235–250 (1999). https://doi.org/10.2333/bhmk.26.235
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DOI: https://doi.org/10.2333/bhmk.26.235