Limited information goodness-of-fit testing in multidimensional contingency tables. (English) Zbl 1306.62477

Summary: We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. 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 asymptotically normal consistent estimator. For a widely used item response model, when \(r\) is small and multidimensional tables are sparse, the proposed statistics have accurate empirical type I errors, unlike Pearson’s \(X^2\). For this model in nonsparse situations, the proposed statistics are also more powerful than \(X^2\). In addition, the proposed statistics are asymptotically chi-square when applied to subtables, and can be used for a piecewise goodness-of-fit assessment to determine the source of misfit in poorly fitting models.


62P15 Applications of statistics to psychology
62H15 Hypothesis testing in multivariate analysis
62H17 Contingency tables


Full Text: DOI


[1] Agresti, A. (2002). Categorical data analysis (2nd edn.). New York: Wiley. · Zbl 1018.62002
[2] Bartholomew, D.J., & Knott, M. (1999). Latent variable models and factor analysis (2nd edn.). London: Arnold. · Zbl 1066.62528
[3] Bartholomew, D.J., & Leung, S.O. (2002). A goodness-of-fit test for sparse 2p contingency tables. British Journal of Mathematical and Statistical Psychology, 55, 1–5.
[4] Bartholomew, D.J., & Tzamourani, P. (1999). The goodness of fit of latent trait models in attitude measurement. Sociolological Methods and Research, 27, 525–46.
[5] Bentler, P.M. (1995). EQS. Encino, CA: Multivariate Software.
[6] Bishop, Y.M.M., Fienberg, S.E., & Holland, P.W. (1975). Discrete multivariate analysis. Cambridge, MA: MIT Press. · Zbl 0332.62039
[7] Bock, R.D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37, 29–1. · Zbl 0233.62016
[8] Bock, R.D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–59.
[9] 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, 173–94.
[10] Christoffersson, A. (1975). Factor analysis of dichotomized variables. Psychometrika, 40, 5–2. · Zbl 0322.62063
[11] Cochran, W.G. (1952). The X2 test of goodness of fit. Annals of Mathematical Statistics, 23, 315–45. · Zbl 0047.13105
[12] Collins, L.M., Fidler, P.L., Wugalter, S.E., & Long, J. (1993). Goodness-of-fit testing for latent class models. Multivariate Behavioral Research, 28, 375–89. · Zbl 02305847
[13] Diener, E., Emmons, R.A., Larsen, R.J., & Griffin, S. (1985). The Satisfaction with Life Scale. Journal of Personality Assessment, 49, 71–5.
[14] Drasgow, F., Levine, M.V., Tsien, S., Williams, B., & Mead, A. (1995). Fitting polytomous item response theory models to multiple-choice tests. Applied Psychological Measurement, 19, 143–65.
[15] D’Zurilla, T.J., Nezu, A.M., & Maydeu-Olivares, A. (2002). Manual of the social problem-solving inventory-Revised. North Tonawanda, NY: Multi-Health Systems.
[16] Fraser, C., & McDonald, R.P. (1988). NOHARM: Least squares item factor analysis. Multivariate Behavioral Research, 23, 267–69.
[17] Glas, C.A.W. (1988). The derivation of some tests for the Rasch model from the multinomial distribution. Psychometrika, 53, 525–46. · Zbl 0718.62267
[18] Glas, C.A.W. (1999). Modification indices for the 2-PL and the nominal response model. Psychometrika, 64, 273–94. · Zbl 1291.62207
[19] Glas, C.A.W., & Verhelst, N.D. (1989). Extensions of the partial credit model. Psychometrika, 54, 635–59. · Zbl 0732.62104
[20] Godambe, V.P. (Ed.) (1991). Estimating functions. Oxford: Oxford University Press. · Zbl 0745.00006
[21] Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman & Hall. · Zbl 0990.62517
[22] Jöreskog, K.G. (1994). On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika, 59, 381–89. · Zbl 0830.62059
[23] Jöreskog, K.G., & Moustaki, I. (2001). Factor analysis of ordinal variables: A comparison of three approaches. Multivariate Behavioral Research, 36, 347–87.
[24] Jöreskog, K.G., & Sörbom, D. (2001). LISREL 8. Chicago: Scientific Software.
[25] Koehler, K., & Larntz, K. (1980). An empirical investigation of goodness-of-fit statistics for sparse multinomials. Journal of the American Statistical Association, 75, 336–44. · Zbl 0442.62025
[26] Kramp, U. (2006). Effects of the number of response options on personality rating scales. Unpublished doctoral dissertation. University of Barcelona.
[27] Lee, S.Y., Poon, W.Y., & Bentler, P.M. (1995). A two-stage estimation of structural equation models with continuous and polytomous variables. British Journal of Mathematical and Statistical Psychology, 48, 339–58. · Zbl 0858.62100
[28] Lord, F.M., & Novick, M.R. (1968). Statistical theories of mental test scores. Reading, MA: Addison-Wesley. · Zbl 0186.53701
[29] Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–74. · Zbl 0493.62094
[30] Maydeu-Olivares, A. (2001a). Limited information estimation and testing of Thurstonian models for paired comparison data under multiple judgment sampling. Psychometrika, 66, 209–28. · Zbl 1293.62249
[31] Maydeu-Olivares, A. (2001b). Multidimensional item response theory modeling of binary data: Large sample properties of NOHARM estimates. Journal of Educational and Behavioral Statistics, 26, 49–9.
[32] Maydeu-Olivares, A. (2005). Further empirical results on parametric vs.non-parametric IRT modeling of Likert-type personality data. Multivariate Behavioral Research, 40, 275–93.
[33] Maydeu-Olivares, A. (2006). Limited information estimation and testing of discretized multivariate normal structural models. Psychometrika, 71, 57–7. · Zbl 1306.62476
[34] 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, 1009–020. · Zbl 1117.62398
[35] Muthén, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43, 551–60. · Zbl 0394.62042
[36] Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115–32.
[37] Muthén, B. (1993). Goodness of fit with categorical and other nonnormal variables. In K.A. Bollen, & J.S. Long (Eds.), Testing structural equation models (pp. 205–34). Newbury Park, CA: Sage.
[38] Muthén, L., & Muthén, B. (2001). MPLUS. Los Angeles: Muthén & Muthén.
[39] Rao, C.R. (1973). Linear statistical inference and its applications. New York: Wiley. · Zbl 0256.62002
[40] Reiser, M. (1996). Analysis of residuals for the multinomial item response model. Psychometrika, 61, 509–28. · Zbl 0863.62086
[41] Reiser, M., & Lin, Y. (1999). A goodness-of-fit test for the latent class model when expected frequencies are small. In M. Sobel, & M. Becker (Eds.), Sociological methodology 1999 (pp. 81–11). Boston: Blackwell.
[42] Reiser, M., & VandenBerg, M. (1994). Validity of the chi-square test in dichotomous variable factor analysis when expected frequencies are small. British Journal of Mathematical and Statistical Psychology, 47, 85–07.
[43] Samejima, F. (1969). Calibration of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement, No. 17.
[44] Schott, J.R. (1997). Matrix analysis for statistics. New York: Wiley. · Zbl 0872.15002
[45] Teugels, J.L. (1990). Some representations of the multivariate Bernoulli and binomial distributions. Journal of Multivariate Analysis, 32, 256–68. · Zbl 0697.62042
[46] Thissen, D., & Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51, 567–77. · Zbl 0646.62098
[47] Tollenaar, N., & Mooijaart, A. (2003). Type I errors and power of the parametric bootstrap goodness-of-fit test: Full and limited information. British Journal of Mathematical and Statistical Psychology, 56, 271–88.
[48] van der Linden, W.J., & Hambleton, R.K. (Eds.) (1997). Handbook of modern item response theory. New York: Springer-Verlag. · Zbl 0872.62099
[49] Zhao, Y., & Joe, H. (2005). Composite likelihood estimation in multivariate data analysis. Canadian Journal of Statistics, 33, 335–56. · Zbl 1077.62045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.