Abstract
Brien et al. (1984, Biometrika, 71, 545–554; 1988, Biometrika, 75, 469–476) have proposed, illustrated and discussed advantages of using Fisher's z-transforms for analyzing correlation structures of multinormal data. Chen and Mudholkar (1988, Austral. J. Statist., 31, 105–110) have studied the sum of squared z-transforms of sample correlations as a test statistic for complete independence. In this paper Brown's (1987, Ann. Probab., 15, 416–422) graph-theoretic characterization of the dependence structure of sample correlations is used to evaluate moments of the test statistic. These moments are then used to approximate its null distribution accurately over a broad range of parameters, including the case where the population dimension exceeds the sample size.
Similar content being viewed by others
References
Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis, Wiley, New York.
Arnold, S. F. (1973). Application of the theory of products of problems to certain patterned covariance matrices, Ann. Statist., 1, 682–699.
Brien, C. J., Venables, W. N., James, A. T. and Mayo, O. (1984). An analysis of correlation matrices: Equal correlations, Biometrika, 71, 545–554.
Brien, C. J., James, A. T. and Venables, W. N. (1988). An analysis of correlation matrices: Variables cross-classified by two factors, Biometrika, 75, 469–476.
Brown, T. C. (1987). Independent subsets of correlation and other matrices, Ann. Probab., 15, 416–422.
Chen, S. and Mudholkar, G. S. (1989). A remark on testing significance of an observed correlation matrix, Austral. J. Statist., 31, 105–110.
Gayen, A. K. (1951). The frequency distribution of the product-moment correlation coefficient in random samples of any size drawn from non-normal universes, Biometrika, 38, 219–247.
Mudholkar, G. S. (1983). Fisher's z-transformation, Encyclopedia of Statistical Sciences, (eds. S., Kotz, N. L., Johnson and C. B., Read), Vol. 3, 130–135, Wiley, New York.
Nagao, H. (1973). On some test criteria for covariance matrix, Ann. Statist., 1, 700–709.
Pearson, E. S. (1959) Note on an approximation to the distribution of non-central χ2, Biometrika, 46, 364.
Author information
Authors and Affiliations
About this article
Cite this article
Chen, S., Mudholkar, G.S. Null distribution of the sum of squared z-transforms in testing complete independence. Ann Inst Stat Math 42, 149–155 (1990). https://doi.org/10.1007/BF00050785
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00050785