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.
Bickel, P. J., & Lehmann, E. L. (1976). Descriptive statistics for nonparametric models III. Dispersion.Annals of Statistics, 4, 1139–1158.
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.
Davies, P. L. (1987). Asymptotic behavior ofS-estimates of multivariate location parameters and dispersion matrices.Annals of Statistics, 76, 1269–1292.
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.
Duncan, G. T., & Layard, M. W. (1973). A monte-carlo study of asymptotically robust tests for correlation.Biometrika, 60, 551–558.
Edgell, S. E., & Noon, S. M. (1984). Effect of violation of normality on thet test of the correlation coefficient.Psychological Bulletin, 95, 576–583.
Freedman, D. A., & Diaconis, P. (1982). On inconsistentM-estimators.Annals of Statistics, 10, 454–461.
Gleason, J. R. (1993). Understanding elongation: The scale contaminated normal family.Journal of the American Statistical Association, 88, 327–337.
Golberg, K. M., & Iglewicz, B. (1992). Bivariate extensions of the boxplot.Technometrics, 34, 307–320.
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.
Hampel, F. R., Ronchetti, E. M. Rousseeuw, P. J., & Stahel, W. A. (1986).Robust Statistics. New York: Wiley.
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.
Hoaglin, D. C., Mosteller, F., & Tukey, J. W. (1983).Understanding robust and exploratory data analysis. New York: Wilkey.
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.
Huber, P. J. (1981).Robust statistics. New York: Wiley.
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.
Kafadar, K. (1982). Using biweightM-estimates in the two-sample problem Part I: Symmetric populations.Communications in Statistics—Theory and Methods, 11, 1883–1901.
Kowalski, C. J. (1972). On the effects of nonnormality on the distribution of the sample product-moment correlation coefficient.Applied Statistics, 21, 1–12.
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.
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.
Lupuhaä, H. P. (1989). On the relation betweenS-estimators andM-estimators of multivariate location and covariance.Annals of Statistics, 17, 1662–1683.
Marrona, R. (1976). RobustM-estimators of multivariate location and scatter.Annals of Statistics, 4, 51–67.
Marrona, R., & Morgenthaler, S. (1986). Robust regression through robust covariances.Communications in Statistics—Theory and Methods, 15, 1347–1365.
Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures.Psychological Bulletin, 105, 156–166.
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.
Mosteller, F., & Tukey, J. W. (1977).Data analysis and regression. Reading, MA: Addison-Wesley.
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.
Rasmussen, J. L. (1989). Computer-intensive correlational analysis: Bootstrap and approximate randomization techniques.British Journal of Mathematical and Statistical Psychology, 42, 103–111.
Rousseeuw, P. J., & Leroy, A. M. (1987).Robust regression & outlier detection. New York: Wiley.
Serfling, R. J. (1980).Approximation theorems of mathematical statistics. New York: Wiley.
Shoemaker, L. H., & Hettmansperger, T. P. (1982). Robust estimates and tests for the one- and two-sample scale models.Biometrika, 69, 47–54.
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.
Staudte, R. G., & Sheather, S. J. (1990).Robust estimation and testing. New York: Wiley.
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.
Wainer, H., & Thissen, D. (1976). Three steps toward robust regression.Psychometrika, 41, 9–34.
Wilcox, R. R. (1990). Comparing the means of two independent groups.Biometrical Journal, 32, 771–780.
Wilcox, R. R. (1991). Bootstrap inferences about the correlation and variances of paired data.British Journal of Mathematical and Statistical Psychology, 44, 379–382.
Wilcox, R. R. (1992a). Comparing one-stepM-estimators of location corresponding to two independent groups.Psychometrika, 57, 141–154.
Wilcox, R. R. (1992b). Robust generalizations of classical test reliability and Cronbach's alpha.British Journal of Mathematical and Statistical Psychology, 45, 239–254.
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