The Annals of Applied Statistics

An empirical study of the maximal and total information coefficients and leading measures of dependence

Abstract

In exploratory data analysis, we are often interested in identifying promising pairwise associations for further analysis while filtering out weaker ones. This can be accomplished by computing a measure of dependence on all variable pairs and examining the highest-scoring pairs, provided the measure of dependence used assigns similar scores to equally noisy relationships of different types. This property, called equitability and previously formalized, can be used to assess measures of dependence along with the power of their corresponding independence tests and their runtime.

Here we present an empirical evaluation of the equitability, power against independence, and runtime of several leading measures of dependence. These include the two recently introduced and simultaneously computable statistics ${\mbox{MIC}_{e}}$, whose goal is equitability, and ${\mbox{TIC}_{e}}$, whose goal is power against independence.

Regarding equitability, our analysis finds that ${\mbox{MIC}_{e}}$ is the most equitable method on functional relationships in most of the settings we considered. Regarding power against independence, we find that ${\mbox{TIC}_{e}}$ and Heller and Gorfine’s ${S^{\mathrm{DDP}}}$ share state-of-the-art performance, with several other methods achieving excellent power as well. Our analyses also show evidence for a trade-off between power against independence and equitability consistent with recent theoretical work. Our results suggest that a fast and useful strategy for achieving a combination of power against independence and equitability is to filter relationships by ${\mbox{TIC}_{e}}$ and then to rank the remaining ones using ${\mbox{MIC}_{e}}$. We confirm our findings on a set of data collected by the World Health Organization.

Article information

Source
Ann. Appl. Stat. Volume 12, Number 1 (2018), 123-155.

Dates
Revised: August 2017
First available in Project Euclid: 9 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1520564467

Digital Object Identifier
doi:10.1214/17-AOAS1093

Citation

Reshef, David N.; Reshef, Yakir A.; Sabeti, Pardis C.; Mitzenmacher, Michael. An empirical study of the maximal and total information coefficients and leading measures of dependence. Ann. Appl. Stat. 12 (2018), no. 1, 123--155. doi:10.1214/17-AOAS1093. https://projecteuclid.org/euclid.aoas/1520564467

References

• Algeo, T. J. and Lyons, T. W. (2006). Mo–total organic carbon covariation in modern anoxic marine environments: Implications for analysis of paleoredox and paleohydrographic conditions. Paleoceanography 21 PA1016.
• Breiman, L. and Friedman, J. (1985). Estimating optimal transformations for multiple regression and correlation. J. Amer. Statist. Assoc. 80 580–598.
• Breiman, L., Friedman, J. H., Olshen, R. A. and Stone, C. J. (1984). Classification and Regression Trees. Wadsworth Advanced Books and Software, Belmont, CA.
• Caspi, A., Sugden, K., Moffitt, T. E., Taylor, A., Craig, I. W., Harrington, H., McClay, J., Mill, J., Martin, J., Braithwaite, A. and Poulton, R. (2003). Influence of life stress on depression: Moderation by a polymorphism in the 5-HTT gene. Science 301 386–389.
• Clayton, R. N. and Mayeda, T. K. (1996). Oxygen isotope studies of achondrites. Geochim. Cosmochim. Acta 60 1999–2017.
• Ding, A. A. and Li, Y. (2013). Copula correlation: An equitable dependence measure and extension of pearson’s correlation. Preprint. Available at arXiv:1312.7214.
• Emilsson, V., Thorleifsson, G., Zhang, B., Leonardson, A. S., Zink, F., Zhu, J., Carlson, S., Helgason, A., Bragi Walters, G., Gunnarsdottir, S. et al. (2008). Genetics of gene expression and its effect on disease. Nature 452 423–428.
• Gill, T. et al. (2002). Obesity in the pacific: Too big to ignore. World Health Organization Regional Office for the Western Pacific, Secretariat of the Pacific Community.
• Gorfine, M., Heller, R. and Heller, Y. (2012). Comment on “Detecting novel associations in large data sets.” Unpublished. Available at http://www.math.tau.ac.il/~ruheller/Papers/science6.pdf.
• Gretton, A., Bousquet, O., Smola, A. and Schölkopf, B. (2005). Measuring statistical dependence with Hilbert–Schmidt norms. In Algorithmic Learning Theory 63–77. Springer, Berlin.
• Gretton, A., Fukumizu, K., Teo, C. H., Le, S., Schölkopf, B. and Smola, A. J. (2008). A kernel statistical test of independence. In Advances in Neural Information Processing Systems 585–592.
• Heller, R., Heller, Y. and Gorfine, M. (2013). A consistent multivariate test of association based on ranks of distances. Biometrika 100 503–510.
• Heller, R., Heller, Y., Kaufman, S., Brill, B. and Gorfine, M. (2016). Consistent distribution-free $k$-sample and independence tests for univariate random variables. J. Mach. Learn. Res. 17 1–54.
• Hoeffding, W. (1948). A non-parametric test of independence. Ann. Math. Stat. 546–557.
• Huo, X. and Szekely, G. J. (2014). Fast computing for distance covariance. Preprint. Available at arXiv:1410.1503.
• Jaakkola, T. S. and Haussler, D. (1999). Probabilistic kernel regression models. In AISTATS.
• Jiang, B., Ye, C. and Liu, J. S. (2015). Nonparametric k-sample tests via dynamic slicing. J. Amer. Statist. Assoc. 110 642–653.
• Kinney, J. B. and Atwal, G. S. (2014). Equitability, mutual information, and the maximal information coefficient. Proc. Natl. Acad. Sci. USA 111 3354–3359.
• Kraskov, A., Stogbauer, H. and Grassberger, P. (2004). Estimating mutual information. Phys. Rev. E 69 066138.
• Linfoot, E. H. (1957). An informational measure of correlation. Inf. Control 1 85–89.
• Lopez-Paz, D., Hennig, P. and Schölkopf, B. (2013). The randomized dependence coefficient. In Advances in Neural Information Processing Systems 1–9.
• Moon, Y.-I., Rajagopalan, B. and Lall, U. (1995). Estimation of mutual information using kernel density estimators. Phys. Rev. E 52 2318–2321.
• Murrell, B., Murrell, D. and Murrell, H. (2014). R2-equitability is satisfiable. Proc. Natl. Acad. Sci. USA 111 E2160–E2160. Available at http://www.pnas.org/content/early/2014/04/29/1403623111.
• Paninski, L. (2003). Estimation of entropy and mutual information. Neural Comput. 15 1191–1253.
• Rényi, A. (1959). On measures of dependence. Acta Math. Hungar. 10 441–451.
• Reshef, D. N., Reshef, Y. A., Sabeti, P. C. and Mitzenmacher, M. (2018a). Appendix to “An empirical study of the maximal and total information coefficients and leading measures of dependence.” DOI:10.1214/17-AOAS1093SUPPA.
• Reshef, D. N., Reshef, Y. A., Sabeti, P. C. and Mitzenmacher, M. (2018b). Supplement to “An empirical study of the maximal and total information coefficients and leading measures of dependence.” DOI:10.1214/17-AOAS1093SUPPB.
• Reshef, D. N., Reshef, Y. A., Finucane, H. K., Grossman, S. R., McVean, G., Turnbaugh, P. J., Lander, E. S., Mitzenmacher, M. and Sabeti, P. C. (2011). Detecting novel associations in large data sets. Science 334 1518–1524.
• Reshef, D., Reshef, Y., Mitzenmacher, M. and Sabeti, P. (2013). Equitability analysis of the maximal information coefficient, with comparisons. Preprint. Available at arXiv:1301.6314.
• Reshef, D. N., Reshef, Y. A., Mitzenmacher, M. and Sabeti, P. C. (2014). Cleaning up the record on the maximal information coefficient and equitability. Proc. Natl. Acad. Sci. USA 111 E3362–E3363. Available at http://www.pnas.org/content/early/2014/08/07/1408920111.
• Reshef, Y. A., Reshef, D. N., Sabeti, P. C. and Mitzenmacher, M. (2015). Equitability, interval estimation, and statistical power. Available at arXiv:1505.02212.
• Reshef, Y. A., Reshef, D. N., Finucane, H. K., Sabeti, P. C. and Mitzenmacher, M. (2016). Measuring dependence powerfully and equitably. J. Mach. Learn. Res. 17 Paper No. 212, 63.
• 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.
• Simon, N. and Tibshirani, R. (2012). Comment on “Detecting novel associations in large data sets”. Unpublished. Available at http://statweb.stanford.edu/tibs/reshef/comment.pdf.
• Speed, T. (2011). A correlation for the 21st century. Science 334 1502–1503.
• Szekely, G. J. and Rizzo, M. L. (2009). Brownian distance covariance. Ann. Appl. Stat. 3 1236–1265.
• Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288.
• Wang, X., Jiang, B. and Liu, J. S. (2017). Generalized R-squared for detecting dependence. Biometrika 104 129–139.

Supplemental materials

• Appendix: Supplementary methods and figures. Details of the methods, parameter choices, and supplemental figures referenced in the main text.
• Empirical Supplement: Full results of all analyses. The full set of results for all analyses presented, as well as additional, complementary analyses.