Caveats on Using Firth's Penalization in the Model‐Based Regression Standardization for Rare Diseases

ABSTRACT

Model‐based regression standardization, also known as the parametric g‐formula, is widely used to estimate marginal effect measures. However, in rare disease settings, the small number of observed events relative to the number of covariates can lead to (quasi‐)complete separation, resulting in non‐convergent estimates in the regression models. Firth's penalized likelihood estimates are a common solution to this issue, ensuring finite parameter estimates even for separated data. While effective for estimating regression coefficients, Firth's method introduces bias into model‐based regression standardization because of its tendency to shrink the predicted probabilities to 0.5, leading to discrepancies between the predicted and observed event rates. We examined the implications of applying Firth's method to model‐based regression standardization, illustrating its potential bias through an empirical study on surgical site infections (SSI) in orthopedic surgeries. We also proposed two ad hoc corrections (i.e., Firth's logistic regression with intercept correction and added covariate) to mitigate this bias and evaluated these methods via simulation studies, comparing them with propensity score‐based approaches. Finally, we applied the proposed method to assess the association between SSI, a rare disease, and smoking status in a clinical database of orthopedic surgeries.