A Note on the Noncollapsibility of Rate Differences and Rate Ratios

It is well known that the odds ratio is noncollapsible, in the sense that conditioning on a covariate that is related to the outcome typically changes the size of the odds ratio, even if this covariate is unrelated to the exposure. The risk difference and risk ratio do not have this peculiar property; we say that the risk difference and risk ratio are collapsible. However, noncollapsibility is not unique for the odds ratio; the rate difference and rate ratio are generally noncollapsible as well. This may seem paradoxical, since the rate can be viewed as a risk per unit time, and thus one would naively suspect that the rate difference/ratio should inherit collapsibility from the risk difference/ratio. Adding to the confusion, it was recently shown that the exposure coefficient in the Aalen additive hazards model is collapsible. This may seem to contradict the fact that the rate difference is generally noncollapsible, since the exposure coefficient in the Aalen additive hazards model is a rate difference. In this article, we use graphical arguments to explain why the rate difference/ratio does not inherit collapsibility from the risk difference/ratio. We also explain when and why the exposure coefficient in the Aalen additive hazards model is collapsible.