Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Biometrical Letters

The Journal of Polish Biometric Society

2 Issues per year

Open Access
See all formats and pricing
More options …

Analysis of multivariate repeated measures data using a MANOVA model and principal components

Mirosław Krzysko / Tadeusz Smiałowski / Waldemar Wołynski
  • Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznan, Poland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2014-12-20 | DOI: https://doi.org/10.2478/bile-2014-0008


In this paper we consider a set of T repeated measurements on p characteristics on each of n individuals. The n individuals themselves may be divided and randomly assigned to K groups. These data are analyzed using a mixed effect MANOVA model, assuming that the data on an individual have a covariance matrix which is a Kronecker product of two positive definite matrices. Results are illustrated on a data set obtained from experiments with varieties of winter rye.

Keywords: multivariate repeated measures data (doubly multivariate data); Kronecker product covariance structure; maximum likelihood estimates; mixed MANOVA model; principal component analysis


  • Arnold S. F. (1979): Linear models with exchangeably distributed errors. Journal of American Statistical Association 74: 194-199.CrossrefGoogle Scholar

  • Deregowski K., Krzysko M. (2009): Principal component analysis in the case of multivariate repeated measures data. Biometrical Letters 46: 163-172.Google Scholar

  • Florek K., Łukaszewicz J., Perkal J., Steinhaus H. and Zubrzycki S. (1951): Sur la liaison et la division des points d’un ensemble fini. Colloquium Mathematicum 2: 282-285.Google Scholar

  • Geisser S., Greenhouse S. (1958): An extension of Box’s results on the use of the F distribution in multivariate analysis. Annals of Mathematical Statistics 29: 885-891.CrossrefGoogle Scholar

  • Giri N. C. (1996): Multivariate Statistical Analysis. Marcel Dekker, Inc., New York.Google Scholar

  • Khatri C. G. (1962): Conditions for Wishartness and independence of second degree polynomials in normal vector. Annals of Mathematical Statistics 33: 1002-1007.CrossrefGoogle Scholar

  • Kruskal J. B. (1956): On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc. 7: 48âAS50.Google Scholar

  • Lancaster P., Tismenetsky M. (1985): The Theory of Matrices, Second Edition: With Applications. Academic Press, Orlando.Google Scholar

  • Mathew T. (1989): MANOVA in the multivariate components of variance model. Journal of Multivariate Analysis 29: 30-38.CrossrefGoogle Scholar

  • Naik D. N., Rao S. (2001): Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix. J. Appl. Statist. 28: 91-105.CrossrefGoogle Scholar

  • Ortega J. M. (1987): Matrix Theory: A Second Course. Plenum Press, New York.Google Scholar

  • Reinsel G. (1982): Multivariate repeated measurements or growth curve models with multivariate random-effects covariance structure. Journal of American Statistical Association 77: 190-195.CrossrefGoogle Scholar

  • Roy A., Khattree R. (2005a): Discrimination and classification with repeated measures data under different covariance structures. Communications in Statistics - Simulation and Computation 34: 167-178.CrossrefGoogle Scholar

  • Roy A., Khattree R. (2005b): On discrimination and classification with multivariate repeated measures data. Journal of Statistical Planning and Inference 134: 462-485.Web of ScienceGoogle Scholar

  • Roy A., Khattree R. (2008): Classification rules for repeated measures data from biomedical research. In: Khattree R., Naik D. N. (eds) Computational methods in biomedical research. Chapman and Hall/CRC: 323-370.Google Scholar

  • Srivastava M. S., von Rosen T., von Rosen D. (2008): Models with a Kronecker product covariance structure: estimation and testing. Math. Methods Stat. 17(4): 357-370.CrossrefGoogle Scholar

  • Ukalski K., Smiałowski T. (2011): Multivariate analysis of data from preliminary trials with winter rye. Biuletyn Instytutu Hodowli i Aklimatyzacji Roslin 260/261: 251-262. Google Scholar

About the article

Published Online: 2014-12-20

Published in Print: 2014-12-01

Citation Information: Biometrical Letters, Volume 51, Issue 2, Pages 103–114, ISSN (Online) 1896-3811, DOI: https://doi.org/10.2478/bile-2014-0008.

Export Citation

© by Mirosław Krzysko. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in