Annals of Statistics

Partial distance correlation with methods for dissimilarities

Gábor J. Székely and Maria L. Rizzo

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Abstract

Distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension. Properties, extensions and applications of distance correlation have been discussed in the recent literature, but the problem of defining the partial distance correlation has remained an open question of considerable interest. The problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space. For the definition of partial distance correlation, we introduce a new Hilbert space where the squared distance covariance is the inner product. We define the partial distance correlation statistics with the help of this Hilbert space, and develop and implement a test for zero partial distance correlation. Our intermediate results provide an unbiased estimator of squared distance covariance, and a neat solution to the problem of distance correlation for dissimilarities rather than distances.

Article information

Source
Ann. Statist., Volume 42, Number 6 (2014), 2382-2412.

Dates
First available in Project Euclid: 20 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1413810731

Digital Object Identifier
doi:10.1214/14-AOS1255

Mathematical Reviews number (MathSciNet)
MR3269983

Zentralblatt MATH identifier
1309.62105

Subjects
Primary: 62Hxx: Multivariate analysis [See also 60Exx] 62H20: Measures of association (correlation, canonical correlation, etc.) 62H15: Hypothesis testing
Secondary: 62Gxx: Nonparametric inference

Keywords
Independence multivariate partial distance correlation dissimilarity energy statistics

Citation

Székely, Gábor J.; Rizzo, Maria L. Partial distance correlation with methods for dissimilarities. Ann. Statist. 42 (2014), no. 6, 2382--2412. doi:10.1214/14-AOS1255. https://projecteuclid.org/euclid.aos/1413810731


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References

  • [1] Baba, K., Shibata, R. and Sibuya, M. (2004). Partial correlation and conditional correlation as measures of conditional independence. Aust. N.Z. J. Stat. 46 657–664.
    Mathematical Reviews (MathSciNet): MR2115961
  • [2] Cailliez, F. (1983). The analytical solution of the additive constant problem. Psychometrika 48 305–308.
    Mathematical Reviews (MathSciNet): MR721286
    Digital Object Identifier: doi:10.1007/BF02294026
  • [3] Cox, T. F. and Cox, M. A. A. (2001). Multidimensional Scaling, 2nd ed. Chapman & Hall, London.
    Mathematical Reviews (MathSciNet): MR1335449
  • [4] Dueck, J., Edelmann, D., Gneiting, T. and Richards, D. (2012). The affinely invariant distance correlation. Preprint. Available at arXiv:1210.2482.
    Mathematical Reviews (MathSciNet): MR3263106
    Digital Object Identifier: doi:10.3150/13-BEJ558
    Project Euclid: euclid.bj/1411134461
  • [5] Feuerverger, A. (1993). A consistent test for bivariate dependence. Int. Stat. Rev. 61 419–433.
  • [6] Goslee, S. C. and Urban, D. L. (2007). The ecodist package for dissimilarity-based analysis of ecological data. J. Stat. Softw. 22 1–19.
  • [7] Gower, J. C. (1966). Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53 325–338.
    Mathematical Reviews (MathSciNet): MR214224
    Digital Object Identifier: doi:10.1093/biomet/53.3-4.325
  • [8] Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2722294
  • [9] Huber, J. (1981). Partial and semipartial correlation—A vector approach. Two-Year Coll. Math. J. 12 151–153.
  • [10] Josse, J. and Holmes, S. (2013). Measures of dependence between random vectors and tests of independence. Literature review. Available at arXiv:1307.7383.
  • [11] Kim, S. (2012). ppcor: Partial and semi-partial (part) correlation. R package version 1.0. Available at http://CRAN.R-project.org/package=ppcor.
  • [12] Kong, J., Klein, B. E. K., Klein, R., Lee, K. and Wahba, G. (2012). Using distance correlation and SS-ANOVA to assess associations of familial relationships, lifestyle factors, diseases, and mortality. Proc. Natl. Acad. Sci. 109 20352–20357.
  • [13] Legendre, P. (2000). Comparison of permutation methods for the partial correlation and partial Mantel tests. J. Stat. Comput. Simul. 67 37–73.
    Mathematical Reviews (MathSciNet): MR1815171
    Digital Object Identifier: doi:10.1080/00949650008812035
  • [14] Legendre, P. and Legendre, L. (2012). Numerical Ecology, 3rd English ed. Elsevier, Amsterdam.
  • [15] Li, R., Zhong, W. and Zhu, L. (2012). Feature screening via distance correlation learning. J. Amer. Statist. Assoc. 107 1129–1139.
    Mathematical Reviews (MathSciNet): MR3010900
    Digital Object Identifier: doi:10.1080/01621459.2012.695654
  • [16] Lyons, R. (2013). Distance covariance in metric spaces. Ann. Probab. 41 3284–3305.
    Mathematical Reviews (MathSciNet): MR3127883
    Digital Object Identifier: doi:10.1214/12-AOP803
    Project Euclid: euclid.aop/1378991840
  • [17] Mantel, N. (1967). The detection of disease clustering and a generalized regression approach. Cancer Res. 27 209–220.
  • [18] Mardia, K. V. (1978). Some properties of classical multi-dimensional scaling. Comm. Statist. Theory Methods 7 1233–1241.
    Mathematical Reviews (MathSciNet): MR514645
    Digital Object Identifier: doi:10.1080/03610927808827707
  • [19] Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
    Mathematical Reviews (MathSciNet): MR560319
  • [20] Oksanen, J., Guillaume Blanchet, F., Kindt, R., Legendre, P., Minchin, P. R., O’Hara, R. B., Simpson, G. L., Solymos, P., Stevens, M. H. H. and Wagner, H. (2013). vegan: Community ecology package. R package version 2.0-7. Available at http://CRAN.R-project.org/package=vegan.
  • [21] Piepho, H. P. (2005). Permutation tests for the correlation among genetic distances and measures of heterosis. Theor. Appl. Genet. 111 95–99.
  • [22] R Core Team (2013). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Available at http://www.R-project.org/.
  • [23] Reif, J. C., Melchinger, A. E., Xia, X. C., Warburton, M. L., Hoisington, D. A., Vasal, S. K., Srinivasan, G., Bohn, M. and Frisch, M. (2003). Genetic distance based on simple sequence repeats and heterosis in tropical maize populations. Crop Sci. 43 1275–1282.
  • [24] Rizzo, M. L. and Székely, G. J. (2013). pdcor: Partial distance correlation. R package version 1.0.0.
  • [25] Rizzo, M. L. and Székely, G. J. (2014). energy: E-statistics (energy statistics). R package version 1.6.1. Available at http://CRAN.R-project.org/package=energy.
  • [26] Schoenberg, I. J. (1935). Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert.” Ann. of Math. (2) 36 724–732.
    Mathematical Reviews (MathSciNet): MR1503248
    Digital Object Identifier: doi:10.2307/1968654
  • [27] Sejdinovic, D., Sriperumbudur, B., Gretton, A. and Fukumizu, K. (2013). Equivalence of distance-based and RKHS-based statistics in hypothesis testing. Ann. Statist. 41 2263–2291.
    Mathematical Reviews (MathSciNet): MR3127866
    Digital Object Identifier: doi:10.1214/13-AOS1140
    Project Euclid: euclid.aos/1383661264
  • [28] Smouse, P. E., Long, J. C. and Sokal, R. R. (1986). Multiple regression and correlation extensions of the Mantel test of matrix correspondence. Syst. Zool. 35 7–632.
  • [29] Stamey, T. A., Kabalin, J. N., McNeal, J. E., Johnstone, I. M., Freiha, F., Redwine, E. A. and Yang, N. (1989). Prostate specific antigen in the diagnosis and treatment of adenocarcinoma of the prostate. II. Radical prostatectomy treated patients. J. Urol. 141 1076–1083.
  • [30] Székely, G. J. and Rizzo, M. L. (2009). Brownian distance covariance. Ann. Appl. Stat. 3 1236–1265.
    Mathematical Reviews (MathSciNet): MR2752127
    Digital Object Identifier: doi:10.1214/09-AOAS312
    Project Euclid: euclid.aoas/1267453933
  • [31] Székely, G. J. and Rizzo, M. L. (2012). On the uniqueness of distance covariance. Statist. Probab. Lett. 82 2278–2282.
    Mathematical Reviews (MathSciNet): MR2979766
  • [32] Székely, G. J. and Rizzo, M. L. (2013). The distance correlation $t$-test of independence in high dimension. J. Multivariate Anal. 117 193–213.
    Mathematical Reviews (MathSciNet): MR3053543
    Digital Object Identifier: doi:10.1016/j.jmva.2013.02.012
  • [33] Székely, G. J. and Rizzo, M. L. (2013). Energy statistics: A class of statistics based on distances. J. Statist. Plann. Inference 143 1249–1272.
    Mathematical Reviews (MathSciNet): MR3055745
    Digital Object Identifier: doi:10.1016/j.jspi.2013.03.018
  • [34] Székely, G. J., Rizzo, M. L. and Bakirov, N. K. (2007). Measuring and testing dependence by correlation of distances. Ann. Statist. 35 2769–2794.
    Mathematical Reviews (MathSciNet): MR2382665
    Digital Object Identifier: doi:10.1214/009053607000000505
    Project Euclid: euclid.aos/1201012979
  • [35] Torgerson, W. S. (1958). Theory and Methods of Scaling. Wiley, New York.
  • [36] Wermuth, N. and Cox, D. R. (2013). Concepts and a case study for a flexible class of graphical Markov models. In Robustness and Complex Data Structures 331–350. Springer, Heidelberg.
    Mathematical Reviews (MathSciNet): MR3135888
    Digital Object Identifier: doi:10.1007/978-3-642-35494-6_20
  • [37] Young, G. and Householder, A. S. (1938). Discussion of a set of points in terms of their mutual distances. Psychometrika 3 19–22.

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