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Blus Residuals in Multivariate Linear Models

  • Vernon M. Chinchilli
Part of the Theory and Decision Library book series (TDLB, volume 8)

Abstract

The examination of residuals is always an important aspect of fitting data to statistical models in terms of identifying influential observations and detecting violations of assumptions. The latter use is difficult to perform for the ordinary residuals in the univariate linear model because these residuals are not independent. This led researchers to consider alternative sets of residuals, such as the best linear, unbiased, scalar-type variance (BLUS) residuals. In this article the definition of BLUS residuals is extended to multivariate models. The extension is relatively straightforward for the multivariate analysis of variance (MANOVA) model, but not for the generalized multivariate analysis of variance (GMANOVA) model and the mixed MANOVA-GMANOVA model. For each of the GMANOVA and mixed MANOVA-GMANOVA models two sets of BLUS residuals arise naturally, namely, a “between” set and a “within” set.

Key words and phrases

MANOVA GMANOVA residual analysis 

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References

  1. Anscombe, F.J. and Tukey, J.W. (1963). ‘The Examination and Evaluation of Residuals,’ Technometrics 5, 29–37.MathSciNetCrossRefGoogle Scholar
  2. Brown, M.B. and Forsythe, A.B. (1974). ‘Robust Tests for the Equality of Variance,’ Journal of the American Statistical Association 69, 364–367.CrossRefzbMATHGoogle Scholar
  3. Chinchilli, V.M. and Elswick, R.K. (1985). ‘A Mixture of the MANOVA and GMANOVA Models,’ Communications in Statistics — Theory & Methods 14, 3075–3089.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Danford, M.B., Hughes, H.M., and McNee, R.C. (1960). ‘On the Analysis of Repeated-Measurements Experiments,’ Biometrics 16, 547–565.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Das Gupta, S. (1977). ‘Two Problems in Multivariate Analysis: BLUS Residuals and Testability of Linear Hypothesis,’ Annals of the Institute of Statistical Mathematics, 29A, 35–41.CrossRefGoogle Scholar
  6. Freund, R.J., Vail, R.W., and Clunies-Ross, C.W. (1961). ‘Residual Analysis,’ Journal of the American Statistical Association 56, 98–104.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Grizzle, J.E. and Allen, D.M. (1969). ‘Analysis of Growth and Dose Response Curves,’ Biometrics 25, 357–381.CrossRefGoogle Scholar
  8. Grossman, S.I. and Styan, G.P.H. (1972). ‘Uncorrected Regression Residuals and Singular Values,’ Journal of the American Statistical Association 67, 672–673.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Khatri, C.G. (1966). ‘A Note on a MANOVA Model applied to problems in growth curves,’ Annals of the Institute of Statistical Mathematics 18, 75–86.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Mardia, K.V. (1980). ‘Tests of Univariate and Multivariate Normality,’ in Handbook of Statistics, Volume I (P.R. Krishnaiah, editor). North-Holland: New York, 279–320.Google Scholar
  11. Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. Wiley: New York.CrossRefzbMATHGoogle Scholar
  12. Potthoff, R.F. and Roy, S.N. (1964). ‘A Generalized Multivariate Analysis of Variance Model Useful Especially for Growth Curve Problems,’ Biometrika 51, 313–326.MathSciNetzbMATHGoogle Scholar
  13. Rao, C.R. (1965). ‘Theory of Least Squares When the Parameters Are Stochastic and Its Application to the Analysis of Growth Curves,’ Biometrika, 52, 447–458.MathSciNetzbMATHGoogle Scholar
  14. Rao, C.R. (1967). ‘Least Squares Theory Using an Estimated Dispersion Matrix and Its Application to Measurement of Signals,’ in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume I (L.M. LeCam and J. Neyman, editors). University of California Press: Berkeley, California, 355–372.Google Scholar
  15. Theil, H. (1965). ‘The Analysis of Disturbances in Regression Analysis,’ Journal of the American Statistical Association 60, 1067–1079.MathSciNetCrossRefGoogle Scholar
  16. Theil, H. (1968). ‘A Simplification of the BLUS Procedure for analyzing regression disturbances,’ Journal of the American Statistical Association 63, 242–251.MathSciNetCrossRefGoogle Scholar
  17. Zyskind, G. (1963). ‘A Note on Residual Analysis,’ Journal of the American Statistical Association 58, 1125–1132.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1987

Authors and Affiliations

  • Vernon M. Chinchilli
    • 1
  1. 1.Department of Biostatistics, Medical College of VirginiaVirginia Commonwealth UniversityRichmondUSA

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