Electronic Journal of Statistics

Asymptotic independence of correlation coefficients with application to testing hypothesis of independence

Zhengjun Zhang, Yongcheng Qi, and Xiwen Ma

Full-text: Open access

Abstract

This paper first proves that the sample based Pearson’s product-moment correlation coefficient and the quotient correlation coefficient are asymptotically independent, which is a very important property as it shows that these two correlation coefficients measure completely different dependencies between two random variables, and they can be very useful if they are simultaneously applied to data analysis. Motivated from this fact, the paper introduces a new way of combining these two sample based correlation coefficients into maximal strength measures of variable association. Second, the paper introduces a new marginal distribution transformation method which is based on a rank-preserving scale regeneration procedure, and is distribution free. In testing hypothesis of independence between two continuous random variables, the limiting distributions of the combined measures are shown to follow a max-linear of two independent χ2 random variables. The new measures as test statistics are compared with several existing tests. Theoretical results and simulation examples show that the new tests are clearly superior. In real data analysis, the paper proposes to incorporate nonlinear data transformation into the rank-preserving scale regeneration procedure, and a conditional expectation test procedure whose test statistic is shown to have a non-standard limit distribution. Data analysis results suggest that this new testing procedure can detect inherent dependencies in the data and could lead to a more meaningful decision making.

Article information

Source
Electron. J. Statist., Volume 5 (2011), 342-372.

Dates
First available in Project Euclid: 10 May 2011

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1305034906

Digital Object Identifier
doi:10.1214/11-EJS610

Mathematical Reviews number (MathSciNet)
MR2802047

Zentralblatt MATH identifier
1274.62401

Subjects
Primary: 62H20: Measures of association (correlation, canonical correlation, etc.) 62G10: Hypothesis testing
Secondary: 62G30: Order statistics; empirical distribution functions

Keywords
Dependence measures rank-preserving scale regeneration distribution free test Fisher’s Z-transformation tests gamma tests χ2 tests order statistics conditional expectation

Citation

Zhang, Zhengjun; Qi, Yongcheng; Ma, Xiwen. Asymptotic independence of correlation coefficients with application to testing hypothesis of independence. Electron. J. Statist. 5 (2011), 342--372. doi:10.1214/11-EJS610. https://projecteuclid.org/euclid.ejs/1305034906


Export citation

References

  • [1] Baek, E. G., and Brock, W. A. (1992). A nonparametric test for independence of a multivariate time series., Statistica Sinica, 2, 137–156.
  • [2] Balakrishnan, N. and Cohen, A. C. (1991). Order Statistics & Inference: Estimation Methods. Academic, Press.
  • [3] Blum, J.R., Kiefer, J. and Rosenblatt, M. (1961). Distribution free tests of independence based on the sample distribution function., Ann. Math. Statist. 32, 485–498.
  • [4] Centers for Disease Control and Prevention (CDC). (2007). Behavior Risk Factor Surveillance Survey (BRFSS); State Cigarette Taxes:, http://www.tobaccofreekids.org/research/factsheets/pdf/0099.pdf
  • [5] Christensen R. and Bedrick, E. J. (1997). Testing the independence assumption in linear models., Journal of the American Statistical Association, 92, 1006–1016.
  • [6] Cromwell, J. B., Labys, W. C., and Terraza, M. (1994)., Univariate tests for time series models. Sage Publications.
  • [7] Cromwell, J. B., Hannan, M. J., Labys, W. C., and Terraza, M. (1999)., Multivariate tests for time series models. Sage Publications.
  • [8] Deheuvels, P. (1983). Point processes and multivariate extreme values., Jour. of Multi. Anal. 13, pp. 257–272.
  • [9] Diaconis, P. and Efron, B. (1985). Testing for independence in a two-way table., Annals of Statistics, 13, 845–874.
  • [10] Doksum, K. and Samarov, A. (1995). Nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression., The Annals of Statistics, 23, 1443–1473
  • [11] Drouet-Mari, D. and Kotz, S. (2001)., Correaltion and dependence. Imperial College Press.
  • [12] Fisher, R. A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population., Biometrika, 10, 507–521.
  • [13] Fisher, R. A. (1921). On the “Probable Error” of a Coefficient of Correlation Deduced from a Small Sample., Metron, 1, 3–32.
  • [14] Freeman, J. and Modarres, R. (2005). Efficiency of test for independence after Box-Cox transformation., Journal of Multivariate Analysis, 95, 107–118.
  • [15] Heffernan, J. E., Tawn, J. A., and Zhang, Z. (2007). Asymptotically (in)dependent multivariate maxima of moving maxima processes., Extremes, 10, 57–82.
  • [16] Hoeffding, W. (1948). A nonparametric test of independence., Ann. Math. Statist. 19 (1948), 546–557.
  • [17] Hong, Y. (1996). Testing for independence between two covariance stationary time series., Biometrika. 83, 615–625.
  • [18] Hotelling, H. (1953). New light on the correlation coefficient and its transform., Journal of the Royal Statistical Society, Series B 15, 193–232.
  • [19] Ikeda, S. and Matsunawa, T. (1970). On asymptotic independence of order statistics., Ann. Inst. Statist. Math., 22, 435–449.
  • [20] Joe, H. (1997)., Multivariate models and dependence concepts, Chapman & Hall.
  • [21] Kanji, G. K. (1999). 100 Statistical Tests. Sage, Publications.
  • [22] Kendall, M. (1938). A new measure of rank correlation, Biometrika, 30, 81–89.
  • [23] Smith, R. L. and Weissman, I. (1996). Characterization and estimation of the multivariate extremal index. Unpublished manuscript, University of North Carolina., http://www.stat.unc.edu/postscript/rs/extremal.pdf
  • [24] Spearman, C. (1904). The proof and measurement of association between two things., Amer. J. Psychol., 15, 72–101.
  • [25] Special Report: (2008). Higher cigarette taxes: reduce smoking, save lives, save money., http://www.tobaccofreekids.org/reports/prices/
  • [26] Székely, G. J., Rizzo, M. L., and Bakirov, N. K. (2007). Measuring and testing dependence by correlation of distances., Annal. Statist. 35, 2769–2794.
  • [27] Thode, H. C. Jr. (2002)., Testing for Normality. Marcel Dekker.
  • [28] Wang, Y. (2010). Quotient correlation of dependent random variables. Manuscript. Department of Statistics, University of, Michigan.
  • [29] Wilcox, R. R. (1995). Testing the hypothesis of independence between two sets of variates., Multivariate Behavioral Research. 30, No. 2, Pages 213–225.
  • [30] Zhang, Z. (2008). Quotient correlation: A sample based alternative to Pearson’s correlation., Annals of Statistics. 36(2), 1007–1030.
  • [31] Zhang, Z. (2009). On approximating max-stable processes and constructing extremal copula functions., Statistical Inference for Stochastic Processes, 12, 89–114.
  • [32] Zhang, Z. and Smith, R.L. (2004). The behavior of multivariate maxima of moving maxima processes., Journal of Applied Probability, 41, 1113–1123.