Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-08T12:22:06.722Z Has data issue: false hasContentIssue false

Tests of Sphericity of Normal Distributions and the Analysis of Repeated Measures Designs

Published online by Cambridge University Press:  01 January 2025

A. P. Grieve*
Affiliation:
CIBA-GEIGY AG
*
Requests for reprints should be sent to: A. P. Grieve, Mathematical Applications, CIBA-GEIGY AG, CH-4002 Basel, Switzerland.

Abstract

The locally best invariant test statistic for testing sphericity of normal distributions is shown to be a simple function of the Box/Geisser-Greenhouse degrees of freedom correction factor in a repeated measures design. Because of this relationship it provides a more intuitively appealing test of the necessary and sufficient conditions for valid F-tests in repeated measures analysis of variance than the likelihood ratio test. The properties of the two tests are compared and tables of the critical values of the Box/Geisser-Greenhouse correction factor are given.

Type
Original Paper
Copyright
Copyright © 1984 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. (1965). Handbook of Mathematical Functions, Washington: National Bureau of Standards.Google Scholar
Boik, R. J. (1981). A priori tests in repeated measures designs: effects of non-sphericity. Psychometrika, 46, 241255.CrossRefGoogle Scholar
Box, G. E. P. (1954). Some theorems on quadratic forms applied in the study of analysis of variance problems, II. Effect of inequality of variance and correlation between errors in the two-way classification. Annals of Mathematical Statistics, 25, 484498.CrossRefGoogle Scholar
Carter, E. M. & Srivastava, M. S. (1977). Monotonicity of the power functions of the modified likelihood ratio tests for homogeneity of variances and sphericity test. Journal of Multivariate Analysis, 7, 229233.CrossRefGoogle Scholar
Dixon, W. J. (1981). BMDP Statistical Software 1981, Los Angeles: University of California Press.Google Scholar
Durbin, J. & Watson, G. S. (1951). Testing for serial correlation in least squares regression II. Biometrika, 38, 159178.CrossRefGoogle ScholarPubMed
Durbin, J. & Watson, G. S. (1971). Testing for serial correlation in least squares regression III. Biometrika, 58, 119.Google Scholar
Geisser, S. & Greenhouse, S. W. (1958). An extension of Box's results on the use of theF-distribution in multivariate analysis. Annals of Mathematical Statistics, 29, 885891.CrossRefGoogle Scholar
Huynh, H. (1978). Some approximate tests for repeated measures designs. Psychometrika, 43, 161175.CrossRefGoogle Scholar
Huynh, H. & Feldt, L. S. (1970). Conditions under which mean square ratios in repeated measurements designs have exactF-distributions. Journal of the American Statistical Association, 65, 15821589.CrossRefGoogle Scholar
Huynh, H. & Feldt, L. S. (1976). Estimation of the Box correction for degrees of freedom from sample data in the randomized block and split-plot designs. Journal of Educational Statistics, 1, 6982.CrossRefGoogle Scholar
John, S. (1971). Some optimal multivariate tests. Biometrika, 58, 123127.Google Scholar
John, S. (1972). The distribution of a statistic used for testing sphericity of normal distributions. Biometrika, 59, 169173.CrossRefGoogle Scholar
John, S. (1976). Fitting sampling distribution agreeing in support and moments and tables of critical values of sphericity criterion. Journal of Multivariate Analysis, 6, 601607.CrossRefGoogle Scholar
McHugh, R. B., Sivanich, G., & Geisser, S. (1961). On the evaluation of changes as measured by psychometric test profiles. Psychological Reports, 7, 335344.CrossRefGoogle Scholar
Mauchly, J. W. (1940). Significance test for sphericity of a normaln-variate distribution. Annals of Mathematical Statistics, 11, 204209.CrossRefGoogle Scholar
Nagao, H. (1973). On some test criteria for covariance matrix. The Annals of Statistics, 1, 700709.CrossRefGoogle Scholar
Nagarsenker, B. N. (1976). Exact non-null distribution of the likelihood ratio criteria for covariance matrix. The Canadian Journal of Statistics, 4, 237254.CrossRefGoogle Scholar
Newmann, T. G. & Odell, P. L. (1971). The Generation of Random Variates, London: Charles Griffin and Co. Ltd..Google Scholar
Rogan, J. C., Keselman, H. J., & Mendoza, J. L. (1979). Analysis of repeated measurements. British Journal of Mathematical and Statistical Psychology, 32, 269286.CrossRefGoogle Scholar
Sugiura, N. (1972). Locally best invariant test for sphericity and the limiting distributions. Annals of Mathematical Statistics, 43, 13121316.CrossRefGoogle Scholar
Wallenstein, S., & Fleiss, J. L. (1979). Repeated measurements analysis of variance when the correlations have a certain pattern. Psychometrika, 44, 229233.CrossRefGoogle Scholar