Fig. 1

A directed acyclic graph (DAG) showing the causal assumption of the observational data and confounding caused by alternative pathways through the unobserved (U) confounders and through hospital (H). H: hospital. Z: treatment preference as instrument: proportion of treated patients within each hospital. T: treatment. C: patient characteristics. PS: propensity score. Y: outcome. U: unobserved confounders