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On the Multivariate Asymptotic Distribution of Sequential Chi-Square Statistics

Published online by Cambridge University Press:  01 January 2025

James H. Steiger ,
Alexander Shapiro  and
Michael W. Browne
James H. Steiger*
Affiliation:
University of British Columbia
Alexander Shapiro
Affiliation:
University of South Africa
Michael W. Browne
Affiliation:
University of South Africa
*
Please send requests for reprints to James H. Steiger, Department of Psychology, 2136 West Mall, University of British Columbia, Vancouver, B.C. CANADA V6T IW5.
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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-square difference tests are asymptotically independent. Some Monte Carlo evidence on the applicability of the theory is provided.

Keywords

Asymptotic distribution theorysequential Chi-square tests
Type
Original Paper
Information
Psychometrika , Volume 50 , Issue 3 , September 1985 , pp. 253 - 263
Copyright
Copyright © 1985 The Psychometric Society

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Footnotes

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.

References

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Petrov, B. N., Csaki, F. (Eds.), Second International Symposium on Information Theory, Budapest: Akademiai Kaido.Google Scholar
Browne, M. W. (1968). A comparison of factor analytic techniques. Psychometrika, 33, 267–334.CrossRefGoogle ScholarPubMed
Browne, M. W. (1968). Gauss-Seidel computing procedures for a family of factor analytic solutions, Princeton, N.J.: Educational Testing Service.CrossRefGoogle 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 Hawkins, D. M. (Eds.), Topics in applied multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Cliff, N. (1983). Some cautions concerning the application of causal modeling methods. Multivariate Behavioral Research, 18, 115–126.CrossRefGoogle ScholarPubMed
Cudeck, R., Browne, M. W. (1983). Cross-validation of covariance structures. Multivariate Behavioral Research, 18, 147–167.CrossRefGoogle ScholarPubMed
Fishman, G. S. (1976). Sampling from the gamma distribution on a computer. Communications of the Association of Computing Machinery, 19, 407–409.CrossRefGoogle Scholar
Guttman, L. (1954). A new approach to factor analysis: The radex. In Lazarsfeld, P. F. (Eds.), 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Klingenberg, W. (1978). A course in differential geometry, New York: Springer-Verlag.CrossRefGoogle Scholar
Schwartz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.Google Scholar
Shapiro, A. (1983). 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.CrossRefGoogle 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.Google Scholar
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.CrossRefGoogle Scholar
Tucker, L. R., Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38, 1–10.CrossRefGoogle Scholar