Determine the outcome rarity required for unbiased Cox difference-method estimation

Determine how rare the time-to-event outcome must be for the Cox proportional hazards model–based difference method to avoid substantial non-collapsibility and produce reliable mediation-effect estimates in longitudinal settings with time-dependent mediators and confounders.

Background

The paper’s simulations indicate that the Cox proportional hazards model–based difference method performs well when the event is rare, but produces biased estimates when the baseline event probability is increased. The authors emphasize that the simulation results do not constitute a mathematical proof and that the required degree of rarity has not been established.

Identifying a quantitative or theoretically justified rarity threshold would clarify when applied researchers can use the Cox-based difference method for causal mediation analysis with time-dependent mediators, time-dependent confounders, and time-to-event outcomes.

References

For example, although our simulations showed negligible non-collapsibility issues for the Cox model based difference method in most scenarios with a rare outcome, exactly how rare the outcome needs to be remains unclear.

— Causal Mediation Analysis with a Time-Dependent Mediator, Time-Dependent Confounders and a Time-to-Event Outcome: Revisiting the Difference Method  (2608.13094 - Denz et al., 13 Aug 2026) in Discussion, Section 6