Partial Distance Correlation

Abstract

Partial distance correlation measures association between two random vectors with respect to a third random vector, analogous to, but more general than (linear) partial correlation. Distance correlation characterizes independence of random vectors in arbitrary dimension. Motivation for the definition is discussed. We introduce a Hilbert space of U-centered distance matrices in which squared distance covariance is the inner product. Simple computation of the sample partial distance correlation and definitions of the population coefficients are presented. Power of the test for zero partial distance correlation is compared with power of the partial correlation test and the partial Mantel test.