Nonparametric testing of conditional independence by means of the partial copula

TL;DR

This paper introduces a nonparametric method for testing conditional independence using the partial copula transform, enabling the application of various independence tests with asymptotic validity and demonstrating good small sample performance.

Contribution

It proposes a novel partial copula transform-based approach for conditional independence testing that is simple, general, and asymptotically valid for a wide range of test statistics.

Findings

Method works well with large samples, asymptotically ignoring estimation effects.

Simulation shows good performance in small samples.

Applicable with many independence tests like Kendall's tau and Hoeffding's statistic.

Abstract

We propose a new method to test conditional independence of two real random variables $Y$ and $Z$ conditionally on an arbitrary third random variable $X$. %with $F_{.|.}$ representing conditional distribution functions, The partial copula is introduced, defined as the joint distribution of $U=F_{Y|X}(Y|X)$ and $V=F_{Z|X}(Z|X)$. We call this transformation of $(Y,Z)$ into $(U,V)$ the partial copula transform. It is easy to show that if $Y$ and $Z$ are continuous for any given value of $X$, then $Y\ind Z|X$ implies $U\ind V$. Conditional independence can then be tested by (i) applying the partial copula transform to the data points and (ii) applying a test of ordinary independence to the transformed data. In practice, $F_{Y|X}$ and $F_{Z|X}$ will need to be estimated, which can be done by, e.g., standard kernel methods. We show that under easily satisfied conditions, and for a very large class of test statistics for independence which includes the covariance, Kendall's tau, and Hoeffding's test statistic, the effect of this estimation vanishes asymptotically. Thus, for large samples, the estimation can be ignored and we have a simple method which can be used to apply a wide range of tests of independence, including ones with consistency for arbitrary alternatives, to test for conditional independence. A simulation study indicates good small sample performance. Advantages of the partial copula approach compared to competitors seem to be simplicity and generality.