On collapsibility and confounding bias in Cox and Aalen regression models

We study the situation where it is of interest to estimate the effect of an exposure variable \(X\) on a survival time response \(T\) in the presence of confounding by measured variables \(Z\). Quantifying the amount of confounding is complicated by the non-collapsibility or non-linearity of typical effect measures in survival analysis: survival analyses with or without adjustment for \(Z\) typically infer different effect estimands of a different magnitude, even when \(Z\) is not associated with the exposure, and henceforth not a confounder of the association between exposure and survival time. We show that, interestingly, the exposure coefficient indexing the Aalen additive hazards model is not subject to such non-collapsibility, unlike the corresponding coefficient indexing the Cox model, so that simple measures of the amount of confounding bias are obtainable for the Aalen hazards model, but not for the Cox model. We argue that various other desirable properties can be ascribed to the Aalen model as a result of this collapsibility. This work generalizes recent work by Janes et al. (Biostatistics 11:572–582, 2010).