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On the multivariate asymptotic distribution of sequential Chi-square statistics

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Abstract

The multivariate asymptotic distribution of sequential Chi-square test statistics is investigated. It is shown that: (a) when sequential Chi-square statistics are calculated for nested models on the same data, the statistics have an asymptotic intercorrelation which may be expressed in closed form, and which is, in many cases, quite high; and (b) sequential Chi-squaredifference tests are asymptotically independent. Some Monte Carlo evidence on the applicability of the theory is provided.

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References

  • Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B. N. Petrov & F. Csaki (Eds.),Second International Symposium on Information Theory. Budapest: Akademiai Kaido.

    Google Scholar 

  • Browne, M. W. (1968a). A comparison of factor analytic techniques.Psychometrika, 33, 267–334.

    Google Scholar 

  • Browne, M. W. (1968b). Gauss-Seidel computing procedures for a family of factor analytic solutions (Research Bulletin RB-68-61). Princeton, N.J.: Educational Testing Service.

    Google Scholar 

  • Browne, M. W. (1974). Generalized least squares in the analysis of covariance structures.South African Statistical Journal, 8, 1–24.

    Google Scholar 

  • Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.),Topics in applied multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press.

    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 

  • Cliff, N. (1983). Some cautions concerning the application of causal modeling methods.Multivariate Behavioral Research, 18, 115–126.

    Google Scholar 

  • Cudeck, R., & Browne, M. W. (1983). Cross-validation of covariance structures.Multivariate Behavioral Research, 18, 147–167.

    Google Scholar 

  • Fishman, G. S. (1976). Sampling from the gamma distribution on a computer.Communications of the Association of Computing Machinery, 19, 407–409.

    Google Scholar 

  • Guttman, L. (1954). A new approach to factor analysis: The radex. In P. F. Lazarsfeld (Ed.),Mathematical thinking in the social sciences (pp. 216–348). Glencoe, IL: Free Press.

    Google Scholar 

  • Hogg, R. V. (1961). On the resolution of statistical hypotheses.Journal of the American Statistical Association, 56, 978–989.

    Google Scholar 

  • Johnson, N. L., & Kotz, S. (1970).Continuous univariate distributions—2. Boston: Houghton-Mifflin.

    Google Scholar 

  • Kendall, M. G., & Stuart, A. (1979).The advanced theory of statistics: Vol. 2 (4th ed.) London: Griffin.

    Google Scholar 

  • Kennedy, W. J., & Gentle, J. E. (1980).Statistical computing. New York: Marcel Dekker.

    Google Scholar 

  • Kinderman, A. J., & Ramage, J. G. (1976). Computer generation of normal random variables.Journal of the American Statistical Association, 71, 893–896.

    Google Scholar 

  • Klingenberg, W. (1978).A course in differential geometry. New York: Springer-Verlag.

    Google Scholar 

  • Schwartz, G. (1978). Estimating the dimension of a model.Annals of Statistics, 6, 461–464.

    Google Scholar 

  • Shapiro, A. (1983a). Asymptotic distribution theory in the analysis of covariance structures (a unified approach).South African Statistical Journal, 17, 33–81.

    Google Scholar 

  • Shapiro, A. (1984). A note on the consistency of estimators in the analysis of moment structures.British Journal of Mathematical and Statistical Psychology, 37, 84–88.

    Google Scholar 

  • Shapiro, A. (1985). Asymptotic equivalence of minimum discrepancy function estimators to G.L.S. estimators.South African Statistical Journal, 19, 73–81.

    Google Scholar 

  • Steiger, J. H., & Lind, J. C. (1980). Statistically-based tests for the number of common factors. Paper presented at the annual meeting of the Psychometric Society, Iowa City, IA.

  • Stroud, T. W. (1972). Fixed alternatives and Wald's formulation of the noncentral asymptotic behavior of the likelihood ratio statistic.Annals of Mathematical Statistics, 43, 447–454.

    Google Scholar 

  • Tucker, L. R., & Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis.Psychometrika, 38, 1–10.

    Google Scholar 

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This research was carried out while the first author was Visiting Professor in the Department of Statistics in the University of South Africa, and was supported in part by a research grant (NSERC #67-4640) from the National Sciences and Engineering Council of Canada to the first author. The support of both of these organizations is acknowledged with gratitude.

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Steiger, J.H., Shapiro, A. & Browne, M.W. On the multivariate asymptotic distribution of sequential Chi-square statistics. Psychometrika 50, 253–263 (1985). https://doi.org/10.1007/BF02294104

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  • DOI: https://doi.org/10.1007/BF02294104

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