5. Tests on One or Two Mean Vectors

  1. Alvin C. Rencher

Published Online: 27 MAR 2003

DOI: 10.1002/0471271357.ch5

Methods of Multivariate Analysis, Second Edition

Methods of Multivariate Analysis, Second Edition

How to Cite

Rencher, A. C. (2002) Tests on One or Two Mean Vectors, in Methods of Multivariate Analysis, Second Edition, John Wiley & Sons, Inc., New York, NY, USA. doi: 10.1002/0471271357.ch5

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



  • power of a test;
  • standardized distance;
  • null hypothesis;
  • alternative hypothesis;
  • degrees of freedom;
  • homogeneity of covariance matrices;
  • likelihood ratio test;
  • discriminant function;
  • experimentwise error rate;
  • matched pairs;
  • subvectors;
  • commensurate variables;
  • profile;
  • parallelism hypothesis;
  • levels hypothesis;
  • flatness hypothesis


Most tests in this chapter are based on Hotelling's T2 distribution, which is a multivariate extension of the univariate t. Univariate t-tests are reviewed to set the stage for the multivariate tests. A multivariate test has several advantages over testing each variable separately. For example, a multivariate test preserves the exact α level (Type I error rate), takes into account the correlations among the variables, and is often more powerful than the univariate tests. When the multivariate and univariate test results disagree, we should use the multivariate result.

A table of critical values of the Hotelling's T2 distribution is given in Appendix A. The table provides many useful insights into multivariate testing in general and the ways in which it differs fundamentally from univariate testing. Other properties of Hotelling's T2-test are discussed.

The two-sample Hotelling's T2-test compares two mean vectors for significant differences. Computation of T2 can be carried out using any one of four MANOVA test statistics (see Chapter 6) or R2 from multiple regression (see Chapter 10). Various tests on individual variables are discussed for use following rejection of the multivariate hypothesis by the Hotelling's T2-test.

The paired observation test is extended from the univariate to the multivariate case. The test for additional information examines a subset of variables to determine if they contribute a significant amount to T2. In profile analysis, we compare the means of the variables in a single sample or compare the profiles of the two mean vectors in two samples. Profile analysis is extended to repeated measures and growth curves in Chapter 6.

Examples using real data are provided for most techniques in this chapter, and the problems at the end of the chapter provide derivations of certain techniques in the chapter and additional illustrations using real data.