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.

翻译：本文重新审视了最初为检验条件独立性而引入的偏Copula概念，强调其表征两个随机变量在剔除与某个协变量的依赖关系后剩余依赖性的能力。基于文献中已有的结果，我们证明偏Copula可视为偏相关的非线性类比。随后，我们通过若干定理论证了条件Copula的依赖性质如何约束偏Copula的形态。最后，通过模拟研究展示相关结果，并说明偏Copula作为描述协变量调整后统计依赖性工具的潜力。这凸显了该方法在因果推断问题中的应用前景，以及其在识别因果效应真实方向方面的潜力。