  Studies of the effects of medical interventions increasingly take place in
distributed research settings using data from multiple clinical data sources
including electronic health records and administrative claims. In such
settings, privacy concerns typically prohibit sharing of individual patient
data, and instead, analyses can only utilize summary statistics from the
individual databases. In the specific but very common context of the Cox
proportional hazards model, we show that standard meta analysis methods then
lead to substantial bias when outcome counts are small. This bias derives
primarily from the normal approximations that the methods utilize. Here we
propose and evaluate methods that eschew normal approximations in favor of
three more flexible approximations: a skew-normal, a one-dimensional grid, and
a custom parametric function that mimics the behavior of the Cox likelihood
function. In extensive simulation studies we demonstrate how these
approximations impact bias in the context of both fixed-effects and (Bayesian)
random-effects models. We then apply these approaches to three real-world
studies of the comparative safety of antidepressants, each using data from four
observational healthcare databases.
