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

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.

Figures (24)

• Figure 1: Scenario 11c: Gaussian copula with $\theta(z) = 1 - 2z$. Left: conditional scatter for $Z < 0.5$ (positive dependence). Center: conditional scatter for $Z > 0.5$ (negative dependence). Right: partial copula averaging over all $Z$. Color gradient represents $Z$; $\rho$ denotes Spearman's correlation.
• Figure 2: Scenario 1 (see Table 1). Each row corresponds to a copula family and parameter triplet $(\theta_{XZ}, \theta_{YZ}, \theta_{XY|Z})$. Each panel displays $\rho$ (Spearman) and $\tau$ (Kendall). Color gradient represents $Z$.
• Figure 3: Scenario 2 (see Table 1). Each row corresponds to a copula family and parameter triplet $(\theta_{XZ}, \theta_{YZ}, \theta_{XY|Z})$, where ind denotes independence.
• Figure 4: Scenario 3 (see Table 1). Each row corresponds to a copula family and parameter triplet $(\theta_{XZ}, \theta_{YZ}, \theta_{XY|Z})$, where ind denotes independence. (R) indicates 90-degree rotation.
• Figure 5: Scenario 4 (see Table 1). Each row corresponds to a copula family and parameter triplet $(\theta_{XZ}, \theta_{YZ}, \theta_{XY|Z})$, where ind denotes independence.

Theorems & Definitions (50)

• Definition 1
• Proposition 1: Probability Integral Transform
• Proposition 2
• Definition 2
• Remark 1
• Proposition 3
• proof
• Remark 2
• Corollary 1
• proof