Simulating correlated multivariate nonnormal distributions: Extending the fleishman power method

Psychometrika volume 64, pages25–35(1999)Cite this article

Abstract

A procedure for generating multivariate nonnormal distributions is proposed. Our procedure generates average values of intercorrelations much closer to population parameters than competing procedures for skewed and/or heavy tailed distributions and for small sample sizes. Also, it eliminates the necessity of conducting a factorization procedure on the population correlation matrix that underlies the random deviates, and it is simpler to code in a programming language (e.g., FORTRAN). Numerical examples demonstrating the procedures are given. Monte Carlo results indicate our procedure yields excellent agreement between population parameters and average values of intercorrelation, skew, and kurtosis.

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References

  1. Blair, R.C. (1987).RANGEN. Boca Raton, FL: IBM.

    Google Scholar 

  2. Blair, R.C., & Higgins, J.J. (1985). Comparison of the power of the paired samplest test to that of the Wilcoxon's signed-ranks test under various population shapes.Psychological Bulletin, 97, 119–127.

    Google Scholar 

  3. Burr, I.W. (1942). Cumulative frequency functions.Annals of Mathematical Statistics, 13, 215–232.

    Google Scholar 

  4. Fleishman, A. I. (1978). A method for simulating non-normal distributions.Psychometrika, 43, 521–532.

    Google Scholar 

  5. Harwell, M.R., & Serlin, R.C. (1988). An experimental study of a proposed test of non-parametric analysis of covariance.Psychological Bulletin, 104(2), 268–281.

    Google Scholar 

  6. Harwell, M.R., & Serlin, R.C. (1989). A nonparametric test statistic for the general linear model.Journal of Educational Statistics, 14(4), 351–371.

    Google Scholar 

  7. Iman, R.L., & Conover, W.J. (1979). The use of the rank transform in regression.Technometrics, 21(4), 499–509.

    Google Scholar 

  8. Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation.Biometrika, 36, 149–176.

    Google Scholar 

  9. Kaiser, H.F., & Dickman, K. (1962). Sample and population score matrices and sample correlation matrices from an arbitrary population correlation matrix.Psychometrika, 27, 179–182.

    Google Scholar 

  10. Knapp, T.R., & Swoyer, V.H. (1967). Some empirical results concerning the power of Bartlett's test of the significance of a correlation matrix.American Educational Research Journal, 4(1), 13–17.

    Google Scholar 

  11. Levy, K.J. (1980). A Monte Carlo study of analysis of covariance under violations of the assumptions of normality and equal regression slopes.Educational and Psychological Measurement, 40, 835–840.

    Google Scholar 

  12. Olejnik, S.F., & Algina, J. (1984). Parametric ANCOVA and the rank transform ANCOVA when the data are conditionally non-normal and heteroscedastic.Journal of Educational Statistics, 9(2), 129–150.

    Google Scholar 

  13. Olejnik, S.F., & Algina, J. (1987). An analysis of statistical power for parametric ANCOVA and rank transform ANCOVA.Communications in Statistics: Theory and Methods, 16(7), 1923–1949.

    Google Scholar 

  14. Ramberg, J.S., & Schmeiser, B.W. (1974). An approximate method for generating asymmetric random variables.Communications of the ACM, 17, 78–82.

    Google Scholar 

  15. Sawilowsky, S. S., Kelley, D. L., Blair, R. C., & Markman, B. S. (1994). Meta-analysis and the Solomon four-group design.Journal of Experimental Education, 62(4), 361–376.

    Google Scholar 

  16. Schmeiser, B. W., & Deutch, S. J. (1977). A versatile four parameter family of probability distributions suitable for simulation.AIIE Transactions, 9, 176–182.

    Google Scholar 

  17. Seamen, S., Algina, J., & Olejnik, S. F. (1985). Type I error probabilities and power of rank and parametric ANCOVA procedures.Journal of Educational Statistics, 10(4), 345–367.

    Google Scholar 

  18. Tadikamalla, P. R. (1980). On simulating nonnormal distributions.Psychometrika, 45, 273–279.

    Google Scholar 

  19. Tadikamalla, P. R., & Johnson, N. L. (1979). Systems of frequency curves generated by transformations of logistic variables (Mimeo Series No. 1126). Chapel Hill, NC: University of North Carolina at Chapel Hill, Department of Statistics.

    Google Scholar 

  20. Tukey, J. W. (1960). The practical relationship between the common transformations of percentages of counts and of amounts (Technical Report 36). Princeton, NJ: Princeton University, Statistical Techniques Research Group.

    Google Scholar 

  21. Vale, D. C., & Maurelli, V. A. (1983). Simulating multivariate nonnormal distributions.Psychometrika, 48, 465–471.

    Google Scholar 

  22. Visual Numerics. (1994). IMSL Math/Library: FORTRAN subroutines for mathematical applications, Volume II. Houston, TX: Author.

    Google Scholar 

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  1. #351 EDUC, College of Education, Wayne State University, 48202, Detroit, MI

    Todd C. Headrick & Shlomo S. Sawilowsky

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  1. Todd C. Headrick
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  2. Shlomo S. Sawilowsky
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Correspondence to Shlomo S. Sawilowsky.

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Headrick, T.C., Sawilowsky, S.S. Simulating correlated multivariate nonnormal distributions: Extending the fleishman power method. Psychometrika 64, 25–35 (1999). https://doi.org/10.1007/BF02294317

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Key words

  • simulations
  • pseudo-random numbers
  • correlated data
  • nonnormality