A copula-based portrayal of the collider bias

This article proposes to use copulas to characterize the collider bias that concerns the non-substantive change in the causal dependence between variables before and after conditioning on their common effect (collider). This copula-based portrayal allows scholars (1) to capture the sophisticated (e.g., asymmetric or heavy-tail) causal dependence structure that is usually not evidenced by a summative causal effect estimate, such as the regression coefficient based on a well-matched sample; (2) to focus on the causal dependence structure that is insensitive to the influences from the marginal distributions; and (3) to directly and formally test the significance of change in the causal dependence structure using the Cramér–von Mises statistic. Both simulation and real data examples are presented, which suggest that copulas can be a handy tool for practical researchers to describe the collider bias.