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A Priori Tests in Repeated Measures Designs: Effects of Nonsphericity

Published online by Cambridge University Press:  01 January 2025

Robert J. Boik
Robert J. Boik*
Affiliation:
Temple University
*
Reprints are available from Robert J, Boik, Department of Statistics, Temple University, Philadelphia, PA 19122.
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Abstract

The validity conditions for univariate repeated measures designs are described. Attention is focused on the sphericity requirement. For a v degree of freedom family of comparisons among the repeated measures, sphericity exists when all contrasts contained in the v dimensional space have equal variances. Under nonsphericity, upper and lower bounds on test size and power of a priori, repeated measures, F tests are derived. The effects of nonsphericity are illustrated by means of a set of charts. The charts reveal that small departures from sphericity (.97 ≤ ε < 1.00) can seriously affect test size and power. It is recommended that separate rather than pooled error term procedures be routinely used to test a priori hypotheses.

Keywords

repeated measures designssphericity criterionmultiple contrasts
Type
Original Paper
Information
Psychometrika , Volume 46 , Issue 3 , September 1981 , pp. 241 - 255
Copyright
Copyright © 1981 The Psychometric Society

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Footnotes

Appreciation is extended to Milton Parnes for his insightful assistance.

References

Boik, R. J. Interactions in the analysis of variance: A procedure for interpretation and a Monte Carlo comparison of univariate and multivariate methods for repeated measures designs (Doctoral dissertation, Baylor University, 1975). Dissertation Abstracts International, 1975, 36, 2908–2908.Google Scholar
Boik, R. J. Interactions, partial interactions, and interaction contrasts in the analysis of variance. Psychological Bulletin, 1979, 86, 1084–1089.CrossRefGoogle Scholar
Boik, R. J., & Kirk, R. E. A general method for partitioning sums of squares of treatments and interactions in the analysis of variance. Educational and Psychological Measurement, 1977, 37, 1–9.CrossRefGoogle Scholar
Box, G. E. P. Some theorems on quadratic forms in the study of analysis of variance problems, I. Effect of inequality of variances and correlation between errors in the two-way classification. Annals of Mathematical Statistics, 1954, 25, 484–498.CrossRefGoogle Scholar
Bradu, D., & Gabriel, K. R. Simultaneous statistical inference on interactions in two way analysis of variance. Journal of the American Statistical Association, 1974, 69, 428–436.CrossRefGoogle Scholar
Cole, J. W., & Grizzle, J. E. Applications of multivariate analysis to repeated measurements experiments. Biometrics, 1966, 22, 810–828.CrossRefGoogle Scholar
Collier, R. O., Baker, F. B., Mandeville, G. K., & Hays, T. F. Estimates of test size for several procedures based on conventional variance ratios in the repeated measure design. Psychometrika, 1967, 32, 339–353.CrossRefGoogle Scholar
Danford, M. B., Hughes, H. M., & McNee, R. C. On the analysis of repeated-measurements experiments. Biometrics, 1960, 16, 547–565.CrossRefGoogle Scholar
Edgington, E. S. A new tabulation of statistical procedures used in APA journals. American Psychologist, 1974, 29, 25–26.CrossRefGoogle Scholar
Gabriel, K. R., Putter, J., & Wax, Y. Simultaneous confidence intervals for product-type interaction contrasts. Journal of the Royal Statistical Society, 1973, 35, 234–244.CrossRefGoogle Scholar
Geisser, S., & Greenhouse, S. W. An extension of Box's results on the use of the F distribution in multivariate analysis. Annals of Mathematical Statistics, 1958, 29, 885–891.CrossRefGoogle Scholar
Graybill, F. Variance heterogeneity in a randomized block design. Biometrics, 1954, 10, 516–520.CrossRefGoogle Scholar
Greenhouse, S. W., & Geisser, S. On methods in the analysis of profile data. Psychometrika, 1959, 24, 95–112.CrossRefGoogle Scholar
Hotelling, H. The generalization of “Students ratio. Annals of Mathematical Statistics, 1931, 2, 360–378.CrossRefGoogle Scholar
Huynh, H. Some approximate tests for repeated measures designs. Psychometrika, 1978, 43, 161–175.CrossRefGoogle Scholar
Huynh, H., & Feldt, L. S. Conditions under which mean square ratios in repeated measurements designs have exact F distributions. Journal of the American Statistical Association, 1970, 65, 1582–1589.CrossRefGoogle Scholar
Huynh, H., & Feldt, L. S. Estimation of the Box correction for degrees of freedom from sample data in the randomized block and split-plot designs. Journal of Educational Statistics, 1976, 1, 69–82.CrossRefGoogle Scholar
Huynh, H., & Mandeville, G. K. Validity conditions in repeated measures designs. Psychological Bulletin, 1979, 86, 964–973.CrossRefGoogle Scholar
International Mathematical and Statistical Libraries, Edition 6. International Mathematical and Statistical Libraries, Inc., Houston, 1977.Google Scholar
Johnson, D. E. Some new multiple comparison procedures for the two-way AOV model with interaction. Biometrics, 1976, 32, 929–934.CrossRefGoogle Scholar
Keselman, H. J., Rogan, J. C., Mendoza, J. L., & Breen, L. J. Testing the validity conditions of repeated measures F tests. Psychological Bulletin, 1980, 87, 479–481.CrossRefGoogle Scholar
Kirk, R. E. Experimental design: Procedures for the behavioral sciences, 1968, Belmont, California: Brooks/Cole.Google Scholar
Mauchley, J. W. Significance test for sphericity of a normal n-variate distribution. Annals of Mathematical Statistics, 1940, 11, 204–209.CrossRefGoogle Scholar
Maxwell, S. E. Pairwise multiple comparisons in repeated measures designs. Journal of Educational Statistics, 1980, 5, 269–287.CrossRefGoogle Scholar
Mendoza, J. L., Toothaker, L. E., & Crain, B. R. Necessary and sufficient conditions for F ratios in the L ×J ×K factorial design with two repeated factors. Journal of the American Statistical Association, 1976, 71, 992–993.Google Scholar
Mendoza, J. L., Toothaker, L. E., & Nicewander, W. A. A Monte Carlo comparison of the univariate and multivariate methods for the groups by trials repeated measures design. Multivariate Behavioral Research, 1974, 9, 165–177.CrossRefGoogle Scholar
Mudholkar, G. S., & Subbaiah, P. Unequal precision multiple comparisons for randomized block design under nonstandard conditions. Journal of the American Statistical Association, 1976, 71, 429–434.CrossRefGoogle Scholar
Ramsey, P. H. Exact Type I error rates for robustness of Student's t test with unequal variances. Journal of Educational Statistics, 1980, 5, 337–349.CrossRefGoogle Scholar
Rogan, J. C., Keselman, H. J., & Mendoza, J. L. Analysis of repeated measurements. British Journal of Mathematical and Statistical Psychology, 1979, 32, 269–286.CrossRefGoogle Scholar
Rouanet, H., & Lepine, D. Comparison between treatments in a repeated measures design: ANOVA and multivariate methods. The British Journal of Mathematical and Statistical Psychology, 1970, 23, 147–163.CrossRefGoogle Scholar
Roy, S. N. On a heuristic method of test construction and its use in multivariate analysis. Annals of Mathematical Statistics, 1953, 24, 220–238.CrossRefGoogle Scholar
Scheffé, H. A “mixed model” for the analysis of variance. Annals of Mathematical Statistics, 1956, 27, 23–26.CrossRefGoogle Scholar
Stoloff, P. H. Correcting for heterogeneity of covariance for repeated measures designs of the analysis of variance. Educational and Psychological Measurement, 1970, 30, 909–924.CrossRefGoogle Scholar
Tiku, M. L. Laguerre series forms of non-central x 2 and F distributions. Biometrika, 1965, 52, 415–427.Google Scholar
Wilks, S. S. Certain generalizations in the analysis of variance. Biometrika, 1932, 24, 471–494.CrossRefGoogle Scholar
Wilson, K. The sampling distributions of conventional, conservative and corrected F-ratios in repeated measurements designs with heterogeneity of covariance. Journal of Statistical Computation and Simulation, 1975, 3, 201–215.CrossRefGoogle Scholar
Winer, B. J. Statistical Principles in experimental design, 2nd Ed., New York: McGraw Hill, 1971.Google Scholar