Conditional Distance Correlation

Abstract

Statistical inference on conditional dependence is essential in many fields including genetic association studies and graphical models. The classic measures focus on linear conditional correlations and are incapable of characterizing nonlinear conditional relationship including nonmonotonic relationship. To overcome this limitation, we introduce a nonparametric measure of conditional dependence for multivariate random variables with arbitrary dimensions. Our measure possesses the necessary and intuitive properties as a correlation index. Briefly, it is zero almost surely if and only if two multivariate random variables are conditionally independent given a third random variable. More importantly, the sample version of this measure can be expressed elegantly as the root of a V or U-process with random kernels and has desirable theoretical properties. Based on the sample version, we propose a test for conditional independence, which is proven to be more powerful than some recently developed tests through our numerical simulations. The advantage of our test is even greater when the relationship between the multivariate random variables given the third random variable cannot be expressed in a linear or monotonic function of one random variable versus the other. We also show that the sample measure is consistent and weakly convergent, and the test statistic is asymptotically normal. By applying our test in a real data analysis, we are able to identify two conditionally associated gene expressions, which otherwise cannot be revealed. Thus, our measure of conditional dependence is not only an ideal concept, but also has important practical utility. Supplementary materials for this article are available online.

• Conditional independence test
• Local bootstrap
• U(V) process with random kernel

Additional information

Notes on contributors

Xueqin Wang

Xueqin Wang is Professor, Department of Statistical Science, School of Mathematics and Computational Science, Southern China Research Center of Statistical Science, Sun Yat-Sen University, Guangzhou 510275, China; Zhongshan School of Medicine, Sun Yat-Sen University, Guangzhou 510080, China; and Xinhua College, Sun Yat-Sen University, Guangzhou 510520, China (E-mail: [email protected]). Wenliang Pan (E-mail: [email protected]) is a Ph.D. student and Yuan Tian (E-mail: [email protected]) is a Master student, Department of Statistical Science, School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China. Wenhao Hu is a doctoral student, Department of Statistics, North Carolina State University, Raleigh, NC 27695 (E-mail: [email protected]). Heping Zhang is Susan Dwight Bliss Professor, Department of Biostatistics, Yale University School of Public Health, New Haven, CT 06520-8034 (E-mail: [email protected]). Wang’s research is partially supported by National Science Foundation of China (NSFC) for Excellent Young Scholar (11322108), NCET(12-0559), NSFC(11001280), and RFDP(20110171110037). Zhang’s research is partially supported by U.S. National Institute on Drug Abuse (R01 DA016750), a 1000-plan scholarship from Chinese Government, and International Collaborative Research Fund from NSFC(11328103).

Wenliang Pan

Xueqin Wang is Professor, Department of Statistical Science, School of Mathematics and Computational Science, Southern China Research Center of Statistical Science, Sun Yat-Sen University, Guangzhou 510275, China; Zhongshan School of Medicine, Sun Yat-Sen University, Guangzhou 510080, China; and Xinhua College, Sun Yat-Sen University, Guangzhou 510520, China (E-mail: [email protected]). Wenliang Pan (E-mail: [email protected]) is a Ph.D. student and Yuan Tian (E-mail: [email protected]) is a Master student, Department of Statistical Science, School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China. Wenhao Hu is a doctoral student, Department of Statistics, North Carolina State University, Raleigh, NC 27695 (E-mail: [email protected]). Heping Zhang is Susan Dwight Bliss Professor, Department of Biostatistics, Yale University School of Public Health, New Haven, CT 06520-8034 (E-mail: [email protected]). Wang’s research is partially supported by National Science Foundation of China (NSFC) for Excellent Young Scholar (11322108), NCET(12-0559), NSFC(11001280), and RFDP(20110171110037). Zhang’s research is partially supported by U.S. National Institute on Drug Abuse (R01 DA016750), a 1000-plan scholarship from Chinese Government, and International Collaborative Research Fund from NSFC(11328103).

Wenhao Hu

Xueqin Wang is Professor, Department of Statistical Science, School of Mathematics and Computational Science, Southern China Research Center of Statistical Science, Sun Yat-Sen University, Guangzhou 510275, China; Zhongshan School of Medicine, Sun Yat-Sen University, Guangzhou 510080, China; and Xinhua College, Sun Yat-Sen University, Guangzhou 510520, China (E-mail: [email protected]). Wenliang Pan (E-mail: [email protected]) is a Ph.D. student and Yuan Tian (E-mail: [email protected]) is a Master student, Department of Statistical Science, School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China. Wenhao Hu is a doctoral student, Department of Statistics, North Carolina State University, Raleigh, NC 27695 (E-mail: [email protected]). Heping Zhang is Susan Dwight Bliss Professor, Department of Biostatistics, Yale University School of Public Health, New Haven, CT 06520-8034 (E-mail: [email protected]). Wang’s research is partially supported by National Science Foundation of China (NSFC) for Excellent Young Scholar (11322108), NCET(12-0559), NSFC(11001280), and RFDP(20110171110037). Zhang’s research is partially supported by U.S. National Institute on Drug Abuse (R01 DA016750), a 1000-plan scholarship from Chinese Government, and International Collaborative Research Fund from NSFC(11328103).

Yuan Tian

Xueqin Wang is Professor, Department of Statistical Science, School of Mathematics and Computational Science, Southern China Research Center of Statistical Science, Sun Yat-Sen University, Guangzhou 510275, China; Zhongshan School of Medicine, Sun Yat-Sen University, Guangzhou 510080, China; and Xinhua College, Sun Yat-Sen University, Guangzhou 510520, China (E-mail: [email protected]). Wenliang Pan (E-mail: [email protected]) is a Ph.D. student and Yuan Tian (E-mail: [email protected]) is a Master student, Department of Statistical Science, School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China. Wenhao Hu is a doctoral student, Department of Statistics, North Carolina State University, Raleigh, NC 27695 (E-mail: [email protected]). Heping Zhang is Susan Dwight Bliss Professor, Department of Biostatistics, Yale University School of Public Health, New Haven, CT 06520-8034 (E-mail: [email protected]). Wang’s research is partially supported by National Science Foundation of China (NSFC) for Excellent Young Scholar (11322108), NCET(12-0559), NSFC(11001280), and RFDP(20110171110037). Zhang’s research is partially supported by U.S. National Institute on Drug Abuse (R01 DA016750), a 1000-plan scholarship from Chinese Government, and International Collaborative Research Fund from NSFC(11328103).

Heping Zhang

Xueqin Wang is Professor, Department of Statistical Science, School of Mathematics and Computational Science, Southern China Research Center of Statistical Science, Sun Yat-Sen University, Guangzhou 510275, China; Zhongshan School of Medicine, Sun Yat-Sen University, Guangzhou 510080, China; and Xinhua College, Sun Yat-Sen University, Guangzhou 510520, China (E-mail: [email protected]). Wenliang Pan (E-mail: [email protected]) is a Ph.D. student and Yuan Tian (E-mail: [email protected]) is a Master student, Department of Statistical Science, School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China. Wenhao Hu is a doctoral student, Department of Statistics, North Carolina State University, Raleigh, NC 27695 (E-mail: [email protected]). Heping Zhang is Susan Dwight Bliss Professor, Department of Biostatistics, Yale University School of Public Health, New Haven, CT 06520-8034 (E-mail: [email protected]). Wang’s research is partially supported by National Science Foundation of China (NSFC) for Excellent Young Scholar (11322108), NCET(12-0559), NSFC(11001280), and RFDP(20110171110037). Zhang’s research is partially supported by U.S. National Institute on Drug Abuse (R01 DA016750), a 1000-plan scholarship from Chinese Government, and International Collaborative Research Fund from NSFC(11328103).