Fig. 1: Illustration of difference-in-differences (DiD) analysis.

a The rationale for DiD without the negative control outcome (NCO) calibration. It disentangles the racial/ethnic differences related to COVID-19 infection in PASC symptoms and conditions from the pre-infection observed racial/ethnic differences. b The directed acyclic graph (DAG) for the DiD without NCO calibration. Each node in the DAG represents each variable and the arrow symbol shows the causal effect. The left panel illustrates the parallel trends assumption for DiD. The right panel demonstrates how DiD without NCO calibration effectively blocks pathways from unmeasured confounders to PASC symptoms and conditions, provided the parallel trends assumption holds. c The DAG for the DiD with NCO calibration. The left panel illustrates a scenario where the parallel trends assumption is violated. The middle panel demonstrates that DiD without NCO calibration fails to block the pathway from unmeasured confounders to PASC symptoms and conditions. The right panel highlights that the DiD method with NCO calibration successfully eliminates the bias from unmeasured confounding.