Hostname: page-component-5f745c7db-nzk4m Total loading time: 0 Render date: 2025-01-06T07:50:04.034Z Has data issue: true hasContentIssue false
  • English
  • Français

The Mixed Model for Multivariate Repeated Measures: Validity Conditions and an Approximate Test

Published online by Cambridge University Press:  01 January 2025

Robert J. Boik
Robert J. Boik*
Affiliation:
Department of Mathematical Sciences, Montana State University
*
Requests for reprints should be sent to Robert J. Boik, Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717.
Get access Rights & Permissions [Opens in a new window]

Abstract

Repeated measures on multivariate responses can be analyzed according to either of two models: a doubly multivariate model (DMM) or a multivariate mixed model (MMM). This paper reviews both models and gives three new results concerning the MMM. The first result is, primarily, of theoretical interest; the second and third have implications for practice. First, it is shown that, given multivariate normality, a condition called multivariate sphericity of the covariance matrix is both necessary and sufficient for the validity of the MMM analysis. To test for departure from multivariate sphericity, the likelihood ratio test can be employed. The second result is an approximation to the null distribution of the likelihood ratio test statistic, useful for moderate sample sizes. Third, for situations satisfying multivariate normality, but not multivariate sphericity, a multivariate ε correction factor is derived. The ε correction factor generalizes Box's ε and can be used to construct an adjusted MMM test.

Keywords

likelihood ratio testmatrix quadratic formmultivariate sphericityWishart distribution
Type
Original Paper
Information
Psychometrika , Volume 53 , Issue 4 , December 1988 , pp. 469 - 486
Copyright
Copyright © 1988 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.)

Footnotes

I am grateful to an anonymous referee for carefully attending to the mathematical details of this paper.

References

Anderson, T. W. (1984). An introduction to multivariate statistical analysis 2nd ed.,, New York: John Wiley.Google Scholar
Barcikowski, R. S., & Robey, R. R. (1984). Decisions in single group repeated measures analysis: Statistical tests and three computer packages. The American Statistician, 38, 148–150.CrossRefGoogle Scholar
Bock, R. D. (1975). Multivariate statistical methods in behavioral research, New York: McGraw-Hill.Google Scholar
Boik, R. J. (1981). A priori test in repeated measures designs: Effects of nonsphericity. Psychometrika, 46, 241–255.CrossRefGoogle Scholar
Box, G. E. P. (1949). A general distribution theory for a class of likelihood criteria. Biometrika, 36, 317–346.CrossRefGoogle ScholarPubMed
Box, G. E. P. (1954). Some theorems on quadratic forms applied in the study of analysis of variance problems, II. Effects of inequality of variance and correlation between errors in the two-way classification. Annals of Mathematical Statistics, 25, 484–498.CrossRefGoogle Scholar
Collier, R. O., Baker, F. B., Mandeville, G. K., & Hayes, T. F. (1967). Estimates of test size for several test procedures based on conventional variance ratios in the repeated measures design. Psychometrika, 32, 339–353.CrossRefGoogle ScholarPubMed
Davidson, M. L. (1972). Univariate versus multivariate tests in repeated measures experiments. Psychological Bulletin, 77, 446–452.CrossRefGoogle Scholar
Driscoll, M. G., & Gundberg, W. R. (1986). A history of the development of Craig's theorem. The American Statistician, 40, 65–70.CrossRefGoogle Scholar
Greenhouse, S. W., & Geisser, S. (1959). On methods in the analysis of profile data. Psychometrika, 24, 95–112.CrossRefGoogle Scholar
Hand, D. J., Taylor, C. C. (1987). Multivariate analysis of variance and repeated measures, New York: Chapman and Hall.CrossRefGoogle Scholar
Henderson, H. V., & Searle, S. R. (1979). Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics. The Canadian Journal of Statistics, 7, 65–81.CrossRefGoogle Scholar
Henderson, H. V., & Searle, S. R. (1981). The vec-permutation matrix, the vec operator and Kronecker products: A review. Linear and Multilinear Algebra, 9, 271–288.CrossRefGoogle Scholar
Huynh, H. (1978). Some approximate tests for repeated measures designs. Psychometrika, 43, 161–175.CrossRefGoogle Scholar
Huynh, H., & Feldt, L. S. (1970). Conditions under which mean square ratios in repeated measures designs have exactF-distributions. Journal of the American Statistical Association, 65, 1582–1589.CrossRefGoogle Scholar
Huynh, H., & Feldt, L. S. (1976). Estimation of the Box correction for degrees of freedom from sample data in randomized block and split-plot designs. Journal of Educational Statistics, 1, 69–82.CrossRefGoogle Scholar
Huynh, H., & Feldt, L. S. (1980). Performance of traditionalF tests in repeated measures designs under covariance heterogeneity. Communications in Statistics—Theory and Methods, 9, 61–74.CrossRefGoogle Scholar
Keselman, H. J., Rogan, J. C., Mendoza, J. L., & Breen, L. J. (1980). Testing the validity conditions of repeated measuresF tests. Psychological Bulletin, 87, 479–480.CrossRefGoogle Scholar
Khatri, C. G. (1959). On conditions for the forms of typeXAX' to be distributed independently or to obey Wishart distribution. Calcutta Statistical Association Bulletin, 8, 162–168.CrossRefGoogle Scholar
Laha, R. G. (1956). On the stochastic independence of two second degree polynomial statistics in normally distributed variates. Annals of Mathematical Statistics, 27, 790–796.CrossRefGoogle Scholar
MacRae, E. C. (1974). Matrix derivatives with an application to an adaptive linear decision problem. The Annals of Statistics, 2, 337–346.CrossRefGoogle Scholar
Magnus, J. R., & Neudecker, H. (1979). The commutation matrix: Some properties and applications. The Annals of Statistics, 7, 381–394.CrossRefGoogle Scholar
Mauchly, J. W. (1940). Significance test for sphericity of a normal-n-variate distribution. Annals of Mathematical Statistics, 11, 204–209.CrossRefGoogle Scholar
Mendoza, J. L. (1980). A significance test for multisample sphericity. Psychometrika, 45, 495–498.CrossRefGoogle Scholar
Mendoza, J. L., Toothaker, L. E., & Crain, B. R. (1976). Necessary and sufficient conditions forF ratios in theL × J × K factorial design with two repeated factors. Journal of the American Statistical Association, 71, 992–993.Google Scholar
Mendoza, J. L., Toothaker, L. E., & Nicewander, W. A. (1974). A Monte Carlo comparison of the univariate and multivariate methods for the groups by trials repeated measures design. Multivariate Behavioral Research, 9, 165–178.CrossRefGoogle Scholar
Mitzel, H. C., & Games, P. A. (1981). Circularity and multiple comparisons in repeated measures designs. British Journal of Mathematical and Statistical Psychology, 34, 253–259.CrossRefGoogle Scholar
Muirhead, R. J. (1982). Aspects of multivariate statistical theory, New York: John Wiley.CrossRefGoogle Scholar
Muller, K. E., & Barton, C. N. (1985). Approximate power for repeated measures ANOVA lacking sphericity. American Statistical Association 1985 Proceedings of the Statistical Computing Section, (pp. 413–418).Google Scholar
O'Brien, R. G., & Kaiser, M. K. (1985). MANOVA method for analyzing repeated measures designs: An extensive primer. Psychological Bulletin, 97, 316–333.CrossRefGoogle ScholarPubMed
Rao, C. R. (1951). An asymptotic expansion of the distribution of Wilks' criterion. Bulletin of the International Statistics Institute, 33, 177–180.Google Scholar
Reinsel, G. (1982). Multivariate repeated-measurement or growth curve models with multivariate random-effects covariance structure. Journal of the American Statistical Association, 77, 190–195.CrossRefGoogle Scholar
Rouanet, H., & Lépine, D. (1970). Comparison between treatments in a repeated-measurement design: ANOVA and multivariate methods. British Journal of Mathematical and Statistical Psychology, 23, 147–163.CrossRefGoogle Scholar
Roy, S. N. (1953). On a heuristic method of test construction and its use in multivariate analysis. Annals of Mathematical Statistics, 24, 220–238.CrossRefGoogle Scholar
Scheffé, H. (1956). A “mixed model” for the analysis of variance. Annals of Mathematical Statistics, 27, 23–36.CrossRefGoogle Scholar
SPSS (1983). SPSSX: User's guide, New York: McGraw Hill.Google Scholar
Srivastava, M. S., & Khatri, C. G. (1979). An introduction to multivariate statistics, New York: Elsevier North Holland.Google Scholar
Timm, N. H. (1980). Multivariate analysis of variance of repeated measures. In Krishnaiah, P. R. (Eds.), Handbook of statistics, Vol. 1 (pp. 41–87). New York: North Holland.Google Scholar
Thomas, D. R. (1983). Univariate repeated measures techniques applied to multivariate data. Psychometrika, 48, 451–464.CrossRefGoogle Scholar
Wang, C. M. (1983). On the analysis of multivariate repeated measures designs. Communications in Statistics—Theory and Methods, 12, 1647–1659.CrossRefGoogle Scholar
Wilks, S. S. (1932). Certain generalisations in the analysis of variance. Biometrika, 24, 471–494.CrossRefGoogle Scholar