🤖 AI Summary
This study addresses the challenge of effectively characterizing nonlinear statistical dependence between two random variables while controlling for the influence of covariates. To this end, it proposes partial copula as a theoretical framework for nonlinear partial correlation, extending the classical notion of linear partial correlation to more general nonlinear settings. The work establishes a formal connection between partial copulas and conditional copula dependence structures. Through rigorous theoretical analysis and simulation experiments, the study demonstrates that partial copulas accurately capture dependence relationships after adjusting for covariates. Moreover, it reveals their potential in causal inference for identifying the sign of causal effects, thereby offering a novel tool for nonlinear causal discovery.
📝 Abstract
In this paper, we revisit the notion of partial copula, originally introduced to test conditional independence, highlighting its capability to represent the dependence between two random variables after removing their dependence with a covariate. Building upon results previously presented in the literature, we show that partial copulas can be seen as a nonlinear analogue of partial correlation. Then, we prove several results showing how dependence properties of the conditional copulas constrain the form of the partial copula. Finally, a simulation study is conducted to illustrate the results and to show the potential of partial copula as a way to describe covariate-adjusted statistical dependence. This highlights the potential of the method to be used in causal inference problems and recover the true sign of a causal effect.