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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.
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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

Information

Type
Original Paper
Information
Psychometrika , Volume 53 , Issue 4 , December 1988, pp. 469 - 486
Copyright
Copyright © 1988 The Psychometric Society

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Footnotes

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

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