Analysis of Within- and Across-Subject Correlations

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


In some fields of applications, response variables are measured on k(k > 1) independent samples for each experimental subject. For this type of situation, within-subject and across-subject correlation matrices are defined and methods of analysis are discussed.

The maximum likelihood estimators for the two different correlation matrices are obtained, and the exact test for within-subject correlation and two approximate tests for across-subject correlation are proposed.

Simulation studies for bivariate distributions suggest that the estimators are satisfactory although the across-subject correlation coefficients are somewhat under estimated. The studies also showed that the two approximate tests are adequate in terms of the size and power. Other properties of the estimators and the tests are discussed.

Key words and phrases

Multivariate Model Within-Subject Correlation Matrix Across-Subject Correlation Matrix Estimation Testing Computer Simulation 


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  1. T. W. Anderson (1984). An Introduction to Multivariate Statistical Analysis. Wiley, New York.zbMATHGoogle Scholar
  2. M. S. Bartlett (1950). ‘Fitting a Straight Line When Both Variables are Subject to Error’ Biometrics, 5, 207–212.MathSciNetCrossRefGoogle Scholar
  3. J. Berkson (1950). ‘Are there two regressions?’ J. Am. Statist. Assoc., 45, 164–180.zbMATHCrossRefGoogle Scholar
  4. R. D. Bock and A. C. Petersen (1975). ‘A Multivariate Correction for Attenuation’ Biometrika, 62, 673–678.MathSciNetzbMATHCrossRefGoogle Scholar
  5. W. G. Cochran (1968). ‘Errors of Measurement in Statistics’ Technometrics, 10, 637–666.zbMATHCrossRefGoogle Scholar
  6. J. Durbin (1954). ‘Errors in Variables’ Rev. Inter. Statist. Inst. 22, 23–32 (1954).MathSciNetCrossRefGoogle Scholar
  7. R. C. Geary, R.C. (1949). ‘Determination of Linear Relations Between Systematic Parts of Variables With Errors of Observation the Variance of Which are Unknown’ Econometrika, 17, 30–58.MathSciNetCrossRefGoogle Scholar
  8. F. E. Grubbs (1948). ‘On Estimating Precision of Measuring Instruments and Product Variability’ J. Am. Statist. Assoc., 43, 243–264.zbMATHCrossRefGoogle Scholar
  9. M. J. R. Healy (1958). ‘Variations Within Individuals in Human Biology’ Human Biology, 30, 210–218.Google Scholar
  10. L. Kish (1962). ‘Studies of Interviewer Variance for Attitudinal Variables’ J. Am. Statist. Assoc. 57, 92–115.MathSciNetCrossRefGoogle Scholar
  11. D. V. Lindley (1947). ‘Regression Lines and the Linear Functional Relationship’ J. Roy. Statist. Soc. Suppl. 9, 218–244.MathSciNetCrossRefGoogle Scholar
  12. A. Madansky (1959). ‘The Fitting of Straight Lines When Both Variables are Subject to Error’ J. Am. Statist. Assoc., 54, 173–205.MathSciNetzbMATHCrossRefGoogle Scholar
  13. J. R. Magnus and H. Neudecker (1979). ‘The Commutation matrix: Some properties and Applications’ Annals of Statistics, 7, 381–394.MathSciNetzbMATHCrossRefGoogle Scholar
  14. R. J. Muirhead (1982). Aspects of Statistical Multivariate Theory, page 157. New York: John Wiley and Sons.CrossRefGoogle Scholar
  15. I. Olkin and J. W. Pratt (1958). Unbiased Estimation of Certain Correlation Coefficients. Annals of Mathematical Statistics, 29, 201–211.MathSciNetCrossRefGoogle Scholar
  16. S. J. Press (1979). ‘Matrix Intraclass Covariance Matrices With Applications in Agriculture’ Technical Report No. 49, Department of Statistics, University of California, Riverside.Google Scholar
  17. C. R. Rao (1947). ‘Large Sample Tests of Statistical Hypotheses Concerning Several Parameters With Applications to Problems of Estimation’ Proc. Camb. Phil. Soc. 44, 50–57.Google Scholar
  18. O. Reiersal (1950). ‘Identiflability of a Linear Relation Between Variables Which are Subject to Error’ Econometrika, 18, 375–389.CrossRefGoogle Scholar
  19. J. W. Tukey (1951). ‘Components in Regression’ Biometrics, 7, 33–69.CrossRefGoogle Scholar
  20. A. Wald (1940). ‘The Fitting of Straight Lines if Both Variables are Subject to Error’ Annals Math. Statist., 11, 284–300.zbMATHCrossRefGoogle Scholar
  21. A. Wald (1943). ‘Test of Statistical Hypothesis Concerning Several Parameters When the Number of Observations is large. Trans. Am. Math. Soc., 54, 426–482.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1987

Authors and Affiliations

  • Sung C. Choi
    • 1
  • Vernon M. Chinchilli
    • 1
  1. 1.Department of Biostatistics Medical College of VirginiaVirginia Commonwealth UniversityRichmondUSA

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