Propensity score calibration (PSC) can be used to adjust for unmeasured confounders using a cross- sectional validation study that lacks information on the disease outcome (Y), under a strong surrogacy assumption. Using directed acyclic graphs and path analysis, the authors developed a formula to predict the presence and magnitude of the bias of PSC in the simplest setting of a binary exposure (T) and 1 confounder (X) that are observed in the main study and 1 confounder (C) that is observed in the validation study only. PSC bias is predicted on the basis of parameters that can be estimated from the data and a single unidentifiable parameter, the relative risk (RR) associated with C (RRCY). The authors simulated 1,000 cohort studies each with a Poisson- distributed outcome Y, varying parameter values over a wide range. When using the true parameter for RRCY, the formula predicts PSC bias almost perfectly in this simple setting (correlation with observed bias over 24 scenarios assessed: r = 0.998). The authors conclude that the bias from PSC observed in certain scenarios can be estimated from the imbalance in C between treated and untreated persons, after adjustment for X, in the validation study and assuming a range of plausible values for the unidentifiable RRCY.
