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Comparisons of Power for Some Exact Multinomial Significance Tests

Published online by Cambridge University Press:  01 January 2025

Joan M. Gurian ,
Jerome Cornfield  and
James E. Mosimann
Joan M. Gurian
Affiliation:
National Heart Institute
Jerome Cornfield
Affiliation:
National Heart Institute
James E. Mosimann
Affiliation:
National Institute of Neurological Diseases and Blindness National Institutes of Health
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Abstract

Two small-sample tests (proposed by Tate and Clelland and by Chapanis respectively) of hypotheses about the parameters of the multinomial distribution, where

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(x|p) = n!\prod\limits_{i = 1}^k {\frac{{p_i^{x_i } }}{{x_i !}}} $$\end{document}
are described. A proof is given that a test of the form
\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R = \frac{{\smallint f(x|p)g(p)dp}}{{f(x|p_0 )}}$$\end{document}
, where g(p) ≥ 0 and p0 is the vector of p's specified by the hypothesis, is admissible. It is shown that the Tate and Clelland test is obtained by setting g(p) equal to a constant for all p and is therefore admissible. The Chapanis test is shown to have power less than or equal to the power of the Tate and Clelland test for (k = n = 3) and (k = 3, n = 4).

Type
Original Paper
Information
Psychometrika , Volume 29 , Issue 4 , December 1964 , pp. 409 - 419
Copyright
Copyright © 1964 Psychometric Society

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References

Chapanis, A. An exact multinomial one-sample test of significance. Psychol. Bull., 1962, 59, 306–310.CrossRefGoogle Scholar
Gridgeman, N. T. Exact multinomial significance tests: Note on a paper by Alphonse Chapanis. Psychol. Bull., 1964, 61, 239–240.CrossRefGoogle ScholarPubMed
Mosimann, J. E. On the compound multinomial distribution, the multivariateβ-distribution, and correlations among proportions. Biometrika, 1962, 49, 65–82.Google Scholar
Neyman, J. and Pearson, E. S. On the problem of the most efficient tests of statistical hypotheses. Phil. Trans. roy. Soc., London, Series A, 231, 289–337.Google Scholar
Tate, N. W. and Clelland, R. C. Nonparametric and shortcut statistics, Danville, Ill.: Interstate Printers, 1957.Google Scholar
Wald, A. Contributions to the theory of statistical estimation and testing hypotheses. Ann. math. Statist., 1939, 10, 299–326.CrossRefGoogle Scholar
Whittaker, E. T. and Watson, G. M. A course of modern analysis (4th ed.), Cambridge, Eng.: Univ. Press, 1940.Google Scholar
Wilks, S. S. Mathematical statistics, New York: Wiley, 1962.Google Scholar