Comparing logistic and log-binomial models for causal mediation analyses of binary mediators and rare binary outcomes: evidence to support cross-checking of mediation results in practice

Background

In the binary outcome framework to causal mediation, closed-form expressions introduced by Valeri and VanderWeele for the natural direct and indirect effect odds ratios (ORs) are established from a logistic outcome model by invoking several approximations that hold under the rare-disease assumption. Such ORs are expected to be close to corresponding effects on the risk ratio (RR) scale based on a log-binomial outcome model, however new insight indicates that this is not always verified. The objective was to report on mediation results from these two models when the incidence of the outcome was <10%.

Methods

Standard (approximate) ORs and RRs were estimated using data on a cohort of asthmatic pregnant women from Québec (Canada) and their babies. Prematurity and low birthweight were the mediator and outcome variables, respectively, and two binary exposure variables were considered: treatment to inhaled corticosteroids and placental abruption. Exact closed-form effects expressed on the OR scale were also derived and estimated using a SAS code we provide. A study based on two simulation scenarios was subsequently devised to supplement on the substantive findings.

Results

Many approximate ORs and RRs estimated from our cohort analyses did not closely agree. Approximate ORs were systematically observed farther from RRs in comparison with exact ORs, possibly leading to different conclusions regarding the null hypothesis. Exact OR estimates were very close to RR estimates for exposure to inhaled corticosteroids, but less so for placental abruption. The approximate OR estimator was found to exhibit important bias and undercoverage in the simulation scenario which featured a strong mediator-outcome relationship.

Conclusions

Logistic and log-binomial outcome models can yield dissimilar binary-binary mediation effects even if the outcome incidence is small marginally. Large discrepancies between approximate ORs and RRs may indicate invalid inference for these ORs. Exact OR estimates can be obtained for validation or to replace RRs if the log-binomial model exhibits convergence problems.