Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-11T14:50:11.757Z Has data issue: false hasContentIssue false
  • English
  • Français

A General Family of Limited Information Goodness-of-Fit Statistics for Multinomial Data

Published online by Cambridge University Press:  01 January 2025

Harry Joe  and
Alberto Maydeu-Olivares
Harry Joe
Affiliation:
Department of Statistics, University of British Columbia
Alberto Maydeu-Olivares*
Affiliation:
Faculty of Psychology, University of Barcelona
*
Requests for reprints should be sent to Alberto Maydeu-Olivares, Faculty of Psychology, University of Barcelona, P. Valle de Hebrón, 171, 08035 Barcelona, Spain. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Maydeu-Olivares and Joe (J. Am. Stat. Assoc. 100:1009–1020, 2005; Psychometrika 71:713–732, 2006) introduced classes of chi-square tests for (sparse) multidimensional multinomial data based on low-order marginal proportions. Our extension provides general conditions under which quadratic forms in linear functions of cell residuals are asymptotically chi-square. The new statistics need not be based on margins, and can be used for one-dimensional multinomials. We also provide theory that explains why limited information statistics have good power, regardless of sparseness. We show how quadratic-form statistics can be constructed that are more powerful than X2 and yet, have approximate chi-square null distribution in finite samples with large models. Examples with models for truncated count data and binary item response data are used to illustrate the theory.

Keywords

categorical data analysiscell-focusingdiscrete dataitem response theoryoverdispersionoverlapping cellsPoisson modelsquadratic form statisticsRasch modelsscore testsparse contingency tableszero-inflation
Type
Original Paper
Information
Psychometrika , Volume 75 , Issue 3 , September 2010 , pp. 393 - 419
Copyright
Copyright © 2010 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

This research has been by grant SEJ2006-08204 from the Spanish Ministry of Education, and an NSERC Canada Discovery Grant. We are grateful to the referees and associate editor for comments leading to improvements. Also, we thank Virginia Yue Chen for some early numerical investigations.

References

Bishop, Y.M.M., Fienberg, S.E., Holland, P.W. (1975). Discrete multivariate analysis, Cambridge: MIT Press.Google Scholar
Bock, R.D., Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197.Google Scholar
Böckenholt, U. (1999). An INAR(1) negative multinomial regression model for longitudinal count data. Psychometrika, 64, 53–67.Google Scholar
Browne, M.W. (1984). Asymptotically distribution free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.Google Scholar
Cai, L., Maydeu-Olivares, A., Coffman, D.L., Thissen, D. (2006). Limited information goodness of fit testing of item response theory models for sparse 2p tables. British Journal of Mathematical and Statistical Psychology, 59, 173–194.Google Scholar
Cochran, W.G. (1952). The X 2 test of goodness of fit. Annals of Mathematical Statistics, 23, 315–345.Google Scholar
Consul, P.C. (1989). Generalized Poisson distribution: properties and applications, New York: Dekker.Google Scholar
Cressie, N., Holland, P.W. (1983). Characterizing the manifest probabilities of latent trait models. Psychometrika, 48, 129–141.Google Scholar
Eubank, R. (1997). Testing goodness of fit with multinomial data. Journal of the American Statistical Association, 92, 1084–1093.Google Scholar
Glas, C.A.W. (1988). The derivation of some tests for the Rasch model from the multinomial distribution. Psychometrika, 53, 525–546.Google Scholar
Glas, C.A.W., Verhelst, N.D. (1989). Extensions of the partial credit model. Psychometrika, 54, 635–659.Google Scholar
Glas, C.A.W., Verhelst, N.D. (1995). Testing the Rasch model. In Fischer, G.H., Molenaar, I.W. (Eds.), Rasch models. Their foundations, recent developments and applications (pp. 69–96). New York: Springer.Google Scholar
Hall, P. (1985). Tailor-made tests of goodness of fit. Journal of the Royal Statistical Society B, 47, 125–131.Google Scholar
Joe, H., Zhu, R. (2005). Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution. Biometrical Journal, 47, 219–229.Google Scholar
Kendall, M., Stuart, A. (1979). The advanced theory of statistics, London: Griffin.Google Scholar
Khatri, C.G. (1966). A note on a MANOVA model applied to problems in growth curve. Annals of the Institute of Statistical Mathematics, 18, 75–86.Google Scholar
Lee, A.H., Wang, K., Yau, K.K.W. (2001). Analysis of zero-inflated Poisson data incorporating extent of exposure. Biometrical Journal, 43, 963–975.Google Scholar
Mavridis, D., Moustaki, I., Knott, M. (2007). Goodness-of-fit measures for latent variable models for binary data. In Lee, S.-Y. (Eds.), Handbook of latent variable and related models (pp. 135–161). Amsterdam: Elsevier.Google Scholar
Maydeu-Olivares, A. (2001). Multidimensional item response theory modeling of binary data: Large sample properties of NOHARM estimates. Journal of Educational and Behavioral Statistics, 26, 49–69.Google Scholar
Maydeu-Olivares, A. (2006). Limited information estimation and testing of discretized multivariate normal structural models. Psychometrika, 71, 57–77.Google Scholar
Maydeu-Olivares, A., Joe, H. (2005). Limited and full information estimation and goodness-of-fit testing in 2n contingency tables: A unified framework. Journal of the American Statistical Association, 100, 1009–1020.Google Scholar
Maydeu-Olivares, A., Joe, H. (2006). Limited information goodness-of-fit testing in multidimensional contingency tables. Psychometrika, 71, 713–732.Google Scholar
Maydeu-Olivares, A., Joe, H. (2008). An overview of limited information goodness-of-fit testing in multidimensional contingency tables. In Shigemasu, K., Okada, A., Imaizumi, T., Hoshino, T. (Eds.), New trends in psychometrics (pp. 253–262). Tokyo: Universal Academy Press.Google Scholar
Rao, C.R. (1973). Linear statistical inference and its applications, New York: Wiley.Google Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests, Copenhagen: Danmarks Pedagogiske Institut.Google Scholar
Rayner, J.C.W., Best, D.J. (1989). Smooth tests of goodness of fit, New York: Oxford University Press.Google Scholar
Rayner, J.C.W., Best, D.J. (1990). Smooth tests of goodness of fit: An overview. International Statistical Review, 58, 9–17.Google Scholar
Reiser, M. (1996). Analysis of residuals for the multinomial item response model. Psychometrika, 61, 509–528.Google Scholar
Reiser, M. (2008). Goodness-of-fit testing using components based on marginal frequencies of multinomial data. British Journal of Mathematical and Statistical Psychology, 61, 331–360.Google Scholar
Reiser, M., Lin, Y. (1999). A goodness of fit test for the latent class model when expected frequencies are small. In Sobel, M., Becker, M. (Eds.), Sociological methodology 1999 (pp. 81–111). Boston: Blackwell.Google Scholar
Tjur, T. (1982). A connection between Rasch’s item analysis model and a multiplicative Poisson model. Scandinavian Journal of Statistics, 9, 23–30.Google Scholar
Van Duijn, M.A.J., Jansen, M.G.H. (1995). Repeated count data: some extensions of the rasch Poisson counts model. Journal of Educational and Behavioral Statistics, 20, 241–258.Google Scholar
White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–26.Google Scholar
Yuan, K.-H., Bentler, P.M. (1997). Mean and covariance structure analysis: Theoretical and practical improvements. Journal of the American Statistical Association, 92, 767–774.Google Scholar