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The percentage bend correlation coefficient

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Abstract

A well-known result is that the usual correlation coefficient,ρ, is highly nonrobust: very slight changes in only one of the marginal distributions can alterρ by a substantial amount. There are a variety of methods for correcting this problem. This paper identifies one particular method which is useful in psychometrics and provides a simple test for independence. It is not recommended that the new test replace the usual test ofH 0:ρ = 0, but the new test has important advantages over the usual test in terms of both Type I errors and power.

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References

  • Ammann, L. P. (1993). Robust singular value decompositions: A new approach to projection pursuit.Journal of the American Statistical Association, 88, 505–514.

    Google Scholar 

  • Bickel, P. J., & Lehmann, E. L. (1976). Descriptive statistics for nonparametric models III. Dispersion.Annals of Statistics, 4, 1139–1158.

    Google Scholar 

  • Blair, R. C., & Lawson, S. B. (1982). Another look at the robustness of the product-moment correlation coefficient to population nonnormality.Florida Journal of Educational Research, 24, 11–15.

    Google Scholar 

  • Davies, P. L. (1987). Asymptotic behavior ofS-estimates of multivariate location parameters and dispersion matrices.Annals of Statistics, 76, 1269–1292.

    Google Scholar 

  • Devlin, S. J., Gnanadesikan, R., & Kettenring, J. R. (1981). Robust estimation of dispersion matrices and principal components.Journal of the American Statistical Association, 76, 354–362.

    Google Scholar 

  • Duncan, G. T., & Layard, M. W. (1973). A monte-carlo study of asymptotically robust tests for correlation.Biometrika, 60, 551–558.

    Google Scholar 

  • Edgell, S. E., & Noon, S. M. (1984). Effect of violation of normality on thet test of the correlation coefficient.Psychological Bulletin, 95, 576–583.

    Google Scholar 

  • Freedman, D. A., & Diaconis, P. (1982). On inconsistentM-estimators.Annals of Statistics, 10, 454–461.

    Google Scholar 

  • Gleason, J. R. (1993). Understanding elongation: The scale contaminated normal family.Journal of the American Statistical Association, 88, 327–337.

    Google Scholar 

  • Golberg, K. M., & Iglewicz, B. (1992). Bivariate extensions of the boxplot.Technometrics, 34, 307–320.

    Google Scholar 

  • Hall, P., Martin, M. A., & Schucany, W. R. (1989). Better nonparametric bootstrap confidence intervals for the correlation coefficient.Journal of Statistical Computation and Simulation, 33, 161–172.

    Google Scholar 

  • Hampel, F. R., Ronchetti, E. M. Rousseeuw, P. J., & Stahel, W. A. (1986).Robust Statistics. New York: Wiley.

    Google Scholar 

  • Hoaglin, D. C. (1985). Summarizing shape numerically: Theg-and-h distributions. In D. Hoaglin, F. Mosteller & J. Tukey (Eds.),Exploring data tables, trends and shapes. New York: Wiley.

    Google Scholar 

  • Hoaglin, D. C., Mosteller, F., & Tukey, J. W. (1983).Understanding robust and exploratory data analysis. New York: Wilkey.

    Google Scholar 

  • Hogg, R. V. (1974). Adaptive robust procedures: A partial review and some suggestions for future applications and theory (with discussions).Journal of the American Statistical Association, 69, 909–927.

    Google Scholar 

  • Huber, P. J. (1981).Robust statistics. New York: Wiley.

    Google Scholar 

  • Iglewicz, B. (1983). Robust scale estimators and confidence intervals for location. In D. Hoaglin, F. Mosteller & J. Tukey (Eds.),Understanding robust and exploratory data analysis (pp. 404–431). New York: Wiley.

    Google Scholar 

  • Kafadar, K. (1982). Using biweightM-estimates in the two-sample problem Part I: Symmetric populations.Communications in Statistics—Theory and Methods, 11, 1883–1901.

    Google Scholar 

  • Kowalski, C. J. (1972). On the effects of nonnormality on the distribution of the sample product-moment correlation coefficient.Applied Statistics, 21, 1–12.

    Google Scholar 

  • Lax, D. A. (1985). Robust estimators of scale: Finite-sample performance in long-tailed symmetric distributions.Journal of the American Statistical Association, 80, 736–741.

    Google Scholar 

  • Li, G., & Chen, Z. (1985). Projection-pursuit approach to robust dispersion and principal components: Primary theory and Monte Carlo.Journal of the American Statistical Association, 80, 759–766.

    Google Scholar 

  • Lupuhaä, H. P. (1989). On the relation betweenS-estimators andM-estimators of multivariate location and covariance.Annals of Statistics, 17, 1662–1683.

    Google Scholar 

  • Marrona, R. (1976). RobustM-estimators of multivariate location and scatter.Annals of Statistics, 4, 51–67.

    Google Scholar 

  • Marrona, R., & Morgenthaler, S. (1986). Robust regression through robust covariances.Communications in Statistics—Theory and Methods, 15, 1347–1365.

    Google Scholar 

  • Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures.Psychological Bulletin, 105, 156–166.

    Google Scholar 

  • Mickey, M. R., Dunn, O. J., & Clark, V. (1967). Note on the use of stepwise regression in detecting outliers.Computational Biomedical Research, 1, 105–111.

    Google Scholar 

  • Mosteller, F., & Tukey, J. W. (1977).Data analysis and regression. Reading, MA: Addison-Wesley.

    Google Scholar 

  • Ramberg, J. S., Tadikamalla, P. R., Dudewicz, E. J., & Mykytka, E. F. (1978). A probability distribution and its uses in fitting data.Technometrics, 21, 201–214.

    Google Scholar 

  • Rasmussen, J. L. (1989). Computer-intensive correlational analysis: Bootstrap and approximate randomization techniques.British Journal of Mathematical and Statistical Psychology, 42, 103–111.

    Google Scholar 

  • Rousseeuw, P. J., & Leroy, A. M. (1987).Robust regression & outlier detection. New York: Wiley.

    Google Scholar 

  • Serfling, R. J. (1980).Approximation theorems of mathematical statistics. New York: Wiley.

    Google Scholar 

  • Shoemaker, L. H., & Hettmansperger, T. P. (1982). Robust estimates and tests for the one- and two-sample scale models.Biometrika, 69, 47–54.

    Google Scholar 

  • Srivastava, M. S., & Awan, H. M. (1984). On the robustness of the correlation coefficient in sampling from a mixture of two bivariate normals.Communications in Statistics—Theory and Methods, 13, 371–382.

    Google Scholar 

  • Staudte, R. G., & Sheather, S. J. (1990).Robust estimation and testing. New York: Wiley.

    Google Scholar 

  • Tukey, J. W. (1960). A survey of sampling from contaminated normal distributions. In I. Olkin, et al. (Eds.),Contributions to probability and statistics. Stanford, CA: Stanford University Press.

    Google Scholar 

  • Wainer, H., & Thissen, D. (1976). Three steps toward robust regression.Psychometrika, 41, 9–34.

    Google Scholar 

  • Wilcox, R. R. (1990). Comparing the means of two independent groups.Biometrical Journal, 32, 771–780.

    Google Scholar 

  • Wilcox, R. R. (1991). Bootstrap inferences about the correlation and variances of paired data.British Journal of Mathematical and Statistical Psychology, 44, 379–382.

    Google Scholar 

  • Wilcox, R. R. (1992a). Comparing one-stepM-estimators of location corresponding to two independent groups.Psychometrika, 57, 141–154.

    Google Scholar 

  • Wilcox, R. R. (1992b). Robust generalizations of classical test reliability and Cronbach's alpha.British Journal of Mathematical and Statistical Psychology, 45, 239–254.

    Google Scholar 

  • Wilcox, R. R. (in press). Some results on a Winsorized correlation coefficient.British Journal of Mathematical and Statistical Psychology.

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Wilcox, R.R. The percentage bend correlation coefficient. Psychometrika 59, 601–616 (1994). https://doi.org/10.1007/BF02294395

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  • DOI: https://doi.org/10.1007/BF02294395

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