Covariate-adjusted statistical dependence representation through partial copulas: bounds and new insights

TL;DR

This paper explores partial copulas as a nonlinear extension of partial correlation, providing bounds and insights into covariate-adjusted dependence, with implications for causal inference.

Contribution

It demonstrates how partial copulas can represent covariate-adjusted dependence and establishes their properties and potential applications in causal inference.

Findings

Partial copulas act as nonlinear analogues of partial correlation.

Dependence properties of conditional copulas constrain partial copula forms.

Simulation shows partial copulas effectively describe covariate-adjusted dependence.

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.