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Limited Information Goodness-of-fit Testing in Multidimensional Contingency Tables

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  • Albert Maydeu-Olivares

    ()

  • Harry Joe

Abstract

We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables of arbitrary dimensions. These statistics are quadratic forms in marginal residuals up to order r. They are asymptotically chi-square under the null hypothesis when parameters are estimated using any consistent and asymptotically normal estimator. We show that when r is small (r = 2) the proposed statistics have more accurate empirical Type I errors and are more powerful than Pearson´s X2 for a widely used item response model. Also, we show that the proposed statistics are asymptotically chi-squared under the null hypothesis when applied to subtables.

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Bibliographic Info

Article provided by Springer in its journal Psychometrika.

Volume (Year): 71 (2006)
Issue (Month): 4 (December)
Pages: 713-732

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Handle: RePEc:spr:psycho:v:71:y:2006:i:4:p:713-732

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Related research

Keywords: multivariate discrete data; categorical data analysis; multivariate multinomial distribution; composite likelihood; item response theory; Lisrel;

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References

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  1. Cees Glas, 1999. "Modification indices for the 2-PL and the nominal response model," Psychometrika, Springer, vol. 64(3), pages 273-294, September.
  2. Maydeu-Olivares, Albert & Joe, Harry, 2005. "Limited- and Full-Information Estimation and Goodness-of-Fit Testing in 2n Contingency Tables: A Unified Framework," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1009-1020, September.
  3. Teugels, Jozef L, 1990. "Some representations of the multivariate Bernoulli and binomial distributions," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 256-268, February.
  4. R. Bock & Murray Aitkin, 1981. "Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm," Psychometrika, Springer, vol. 46(4), pages 443-459, December.
  5. R. Darrell Bock, 1972. "Estimating item parameters and latent ability when responses are scored in two or more nominal categories," Psychometrika, Springer, vol. 37(1), pages 29-51, March.
  6. Bengt Muthén, 1978. "Contributions to factor analysis of dichotomous variables," Psychometrika, Springer, vol. 43(4), pages 551-560, December.
  7. C. Glas & N. Verhelst, 1989. "Extensions of the partial credit model," Psychometrika, Springer, vol. 54(4), pages 635-659, September.
  8. Cees Glas, 1988. "The derivation of some tests for the rasch model from the multinomial distribution," Psychometrika, Springer, vol. 53(4), pages 525-546, December.
  9. Albert Maydeu-Olivares, 2001. "Limited information estimation and testing of Thurstonian models for paired comparison data under multiple judgment sampling," Psychometrika, Springer, vol. 66(2), pages 209-227, June.
  10. Alberto Maydeu, 2003. "Limited Information Estimation and Testing of Discretized Multivariate Normal Structural Models," Working Papers Economia wp03-03, Instituto de Empresa, Area of Economic Environment.
  11. David Thissen & Lynne Steinberg, 1986. "A taxonomy of item response models," Psychometrika, Springer, vol. 51(4), pages 567-577, December.
  12. Geoff Masters, 1982. "A rasch model for partial credit scoring," Psychometrika, Springer, vol. 47(2), pages 149-174, June.
  13. Anders Christoffersson, 1975. "Factor analysis of dichotomized variables," Psychometrika, Springer, vol. 40(1), pages 5-32, March.
  14. Karl Jöreskog, 1994. "On the estimation of polychoric correlations and their asymptotic covariance matrix," Psychometrika, Springer, vol. 59(3), pages 381-389, September.
  15. Bengt Muthén, 1984. "A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators," Psychometrika, Springer, vol. 49(1), pages 115-132, March.
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Cited by:
  1. Harry Joe & Alberto Maydeu-Olivares, 2010. "A General Family of Limited Information Goodness-of-Fit Statistics for Multinomial Data," Psychometrika, Springer, vol. 75(3), pages 393-419, September.
  2. Javier Revuelta, 2010. "Estimating Difficulty from Polytomous Categorical Data," Psychometrika, Springer, vol. 75(2), pages 331-350, June.
  3. Alberto Maydeu-Olivares & Rosa Montaño, 2013. "How Should We Assess the Fit of Rasch-Type Models? Approximating the Power of Goodness-of-Fit Statistics in Categorical Data Analysis," Psychometrika, Springer, vol. 78(1), pages 116-133, January.
  4. Jochen Ranger & Jorg-Tobias Kuhn, 2012. "A flexible latent trait model for response times in tests," Psychometrika, Springer, vol. 77(1), pages 31-47, January.
  5. Kim, Sung-Ho & Choi, Hyemi & Lee, Sangjin, 2009. "Estimate-based goodness-of-fit test for large sparse multinomial distributions," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1122-1131, February.
  6. G. Iliopoulos & M. Kateri & I. Ntzoufras, 2009. "Bayesian Model Comparison for the Order Restricted RC Association Model," Psychometrika, Springer, vol. 74(4), pages 561-587, December.
  7. Joe, Harry & Lee, Youngjo, 2009. "On weighting of bivariate margins in pairwise likelihood," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 670-685, April.
  8. Vassilis Vasdekis & Silvia Cagnone & Irini Moustaki, 2012. "A Composite Likelihood Inference in Latent Variable Models for Ordinal Longitudinal Responses," Psychometrika, Springer, vol. 77(3), pages 425-441, July.
  9. Li Cai, 2010. "A Two-Tier Full-Information Item Factor Analysis Model with Applications," Psychometrika, Springer, vol. 75(4), pages 581-612, December.
  10. Silvia cagnone & Stefania Mignani, 2007. "Assessing the goodness of fit of a latent variable model for ordinal data," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 337-361.

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