Abstract
A general procedure is provided for comparing correlation coefficients between optimal linear composites. The procedure allows computationally efficient significance tests on independent or dependent multiple correlations, partial correlations, and canonical correlations, with or without the assumption of multivariate normality. Evidence from some Monte Carlo studies on the effectiveness of the methods is also provided.
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Steiger, J. H. (1980).K-sample pattern hypotheses on correlation matrices by the method of generalized least squares. University of British Columbia, Institute of Applied Mathematics and Statistics Research Bulletin 80-2.
Steiger, J. H. (June 2, 1982). A robust large-sample procedure for comparing dependent correlations. Paper presented at the annual Spring Meeting of the Psychometric Society.
References
Browne, M. W. (1977). The analysis of patterned correlation matrices by generalized least squares.British Journal of Mathematical and Statistical Psychology, 30, 113–124.
Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.),Topics in applied multivariate analysis. Cambridge: Cambridge University Press.
Devlin, S. J., Gnanadesikan, R., & Kettenring, J. R. (1976). Some multivariate applications of elliptical distributions. In S. Ideka (Ed.),Essays in probability and statistics. Tokyo: Shinko Tsusho.
Duncan, G. T., & Layard, M. W. J. (1973). A Monte Carlo study of asymptotically robust tests for correlation coefficients.Biometrika, 60, 551–558.
Hsu, P. L. (1949). The limiting distribution of functions of sample means and application to testing hypotheses.Proceedings of the First Berkely Symposium on Mathematical Statistics and Probability, 359–402.
Isserlis, L. (1916). On certain probable errors and correlation coefficients of multiple frequency distributions with skew regression.Biometrika, 11, 185–190.
Knuth, D. E. (1969).The art of computer programming. Vol. 2. Reading, Mass.: Addison-Wesley.
Jöreskog, K. G. (1967). Some contributions to maximum likelihood factor analysis.Psychometrika, 32, 443–482.
Lord, F. M. (1975). Automated hypothesis tests and standard errors for nonstandard problems.The American Statistician, 29, 56–59.
Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications.Biometrika, 57, 519–530.
Mardia, K. V. (1974). Applications of some measures of multivariate skewness to testing normality and to robustness studies.Sankhya, Series B,36, 115–128.
Mardia, K. V., & Zemroch, P. J. (1975). Algorithm AS84: Measures of multivariate skewness and kurtosis.Applied Statistics, 24, 262–264.
Muirhead, R. J., & Waternaux, C. M. (1980). Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal distributions.Biometrika, 67, 31–43.
Muirhead, R. J. (1982).Aspects of multivariate statistical theory. New York: Wiley.
Olkin, I., & Siotani, M. (1976). Asymptotic distribution of functions of a correlation matrix. Technical Report No. 6, Laboratory for Quantitative Research in Education. Stanford University, 1964. Reprinted in S. Ideka (Ed.)Essays in probability and statistics, Tokyo, Shinko Tsusho.
Pearson, K., & Filon, L. N. G. (1898). Mathematical contributions to the theory of evolution: IV. On the probable error of frequency constants and on the influence of random selection of variation and correlation.Philosophical Transactions of the Royal Society of London, Series, A, 191, 229–311.
Rao, C. R. (1973).Linear statistical inference and its applications (2nd Ed.). New York: Wiley.
Ross, G. J. S. (1970). The efficient use of function minimisation in nonlinear maximum likelihood estimation.Applied Statistics, 19, 205–221.
Schuenemeyer, J. H., & Bargmann, R. E. (1978). Maximum eccentricity as a union-intersection test statistic in multivariate analysis.Journal of Multivariate Analysis, 8, 268–273.
Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance structures (a unified approach).South African Statistical Journal, 17, 33–81.
Steiger, J. H. (1979). MULTICORR: A computer program for fast, accurate, small-sample testing of correlational pattern hypotheses.Educational and Psychological Measurement, 39, 677–680.
Steiger, J. H. (1980a). Tests for comparing elements of a correlation matrix.Psychological Bulletin, 87, 245–251.
Steiger, J. H. (1980b). Testing pattern hypotheses on correlation matrices: Alternative statistics and some empirical results.Multivariate Behavioral Research, 15, 335–352.
Steiger, J. H., & Hakstian, A. R. (1982). The asymptotic distribution of elements of a correlation matrix: Theory and application.British Journal of Mathematical and Statistical Psychology, 35, 208–215.
Steiger, J. H., & Hakstian, A. R. (1983). A historical note on the asymptotic distribution of correlations.British Journal of Mathematical and Statistical Psychology, 36, 157.
Venables, W. (1976). Some implications of the union-intersection principle for tests of sphericity.Journal of Multivariate Analysis, 6, 175–190.
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This research was supported in part by an operating grant (#67-4640) to the first author from the National Sciences and Engineering Research Council of Canada. The authors would also like to acknowledge the helpful comments and encouragement of Alexander Shapiro, Stanley Nash, and Ingram Olkin.
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Steiger, J.H., Browne, M.W. The comparison of interdependent correlations between optimal linear composites. Psychometrika 49, 11–24 (1984). https://doi.org/10.1007/BF02294202
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DOI: https://doi.org/10.1007/BF02294202