7. Tests on Covariance Matrices

  1. Alvin C. Rencher

Published Online: 27 MAR 2003

DOI: 10.1002/0471271357.ch7

Methods of Multivariate Analysis, Second Edition

Methods of Multivariate Analysis, Second Edition

How to Cite

Rencher, A. C. (2002) Tests on Covariance Matrices, in Methods of Multivariate Analysis, Second Edition, John Wiley & Sons, Inc., New York, NY, USA. doi: 10.1002/0471271357.ch7

Author Information

  1. Brigham Young University, USA

Publication History

  1. Published Online: 27 MAR 2003
  2. Published Print: 22 FEB 2002

ISBN Information

Print ISBN: 9780471418894

Online ISBN: 9780471271352



  • sphericity;
  • chi-square approximation;
  • uniformity;
  • compound symmetry;
  • intraclass correlation;
  • Box's M-test;
  • F approximation;
  • Wilks' test statistic;
  • canonical correlations


In this chapter, we consider the pattern of the variances and covariances. Tests on covariance matrices may be carried out to check assumptions pertaining to other tests and procedures. The tests, most of which are based on the likelihood ratio, can be summarized in three basic types: (1) tests on the structure of the covariance matrix or correlation matrix, (2) tests comparing two or more covariance matrices, and (3) testing the hypothesis that two or more of the random variables are independent. The first of these, tests on the structure, include an arbitrary pattern, independent variables with a common variance, and a common variance and a common covariance.

Most techniques in this chapter are illustrated with examples using real data. The problems at the end of the chapter call for derivations of some of the techniques and provide further illustrations with real data.