A modified two-factor multivariate analysis of variance: Asymptotics and small sample approximations

Solomon W. Harrar, Arne C. Bathke

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


In this paper, we present results for testing main, simple and interaction effects in heteroscedastic two factor MANOVA models. In particular, we suggest modifications to the MANOVA sum of squares and cross product matrices to account for heteroscedasticity. Based on these modified matrices, we define some multivariate test statistics and derive their asymptotic distributions under non-normality for the null as well as non-null cases. Derivation of these results relies on the perturbation method and limit theorems for independently distributed random matrices. Based on the asymptotic distributions, we devise small sample approximations for the quantiles of the null distributions. The numerical accuracy of the large sample as well as small sample approximations are favorable. A real data set from a Smoking Cessation Trial is analyzed to illustrate the application of the methods.

Original languageEnglish
Pages (from-to)135-165
Number of pages31
JournalAnnals of the Institute of Statistical Mathematics
Issue number1
StatePublished - Feb 2012

Bibliographical note

Funding Information:
Acknowledgments The research of Solomon W. Harrar was supported by the 2007/2008 Small Grant Program of the University of Montana. Part of the manuscript was prepared during a research visit of the second author at the University of Montana, and the authors would like to express their gratitude to both departments for facilitating this visit. The authors are grateful to Dr. Kari J. Harris, principal investigator of the Greek Health Project, for making available the data analyzed in Sect. 4. The authors are also thankful to the editor, the associate editor and the two anonymous referees for their thoughtful suggestions and comments which have led to significant improvements over the original version of the manuscript.

Copyright 2019 Elsevier B.V., All rights reserved.


  • Heteroscedasticity
  • Local alternatives
  • Multivariate tests
  • Non-normality
  • Perturbation method

ASJC Scopus subject areas

  • Statistics and Probability


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