# Relative excess risk due to interaction (RERI) from a multiplicative model

Co-authors and reviewers sometimes argue, that we should calculate the RERI even though we use a multiplicative model. This is easily done in both Cox (Redirecting) and logistic models (Rothman KJ. Modern epidemiology. 1986?), but has also been criticized (https://doi.org/10.1093/aje/kwg113 and Commentary: Much Ado About Interactions on JSTOR) especially because the RERI from a multiplicative model will depend on other covariates.

With an environment (E) and gene (G) dummy and a Cox model, λ(t)=λ0(t) e^(β1 E + β2 G + β3 EG), where β3 is the (multiplicative) interaction term. An estimate for the relative excess risk due to interaction is RERI = e^(β1 + β2 + β3 ) - e^β1 - e^β2 + 1 and confidence intervals are typically obtained via the delta method Redirecting .

As I understand it, the argument for using the RERI is, that it measures a biological interaction, whereas the multiplicative interaction merely measures a statistical interaction, which has been debated A competing risks approach to “biologic” interaction | SpringerLink and Invariants and noninvariants in the concept of interdependent effects - PubMed . I usually refuse to do that, because of the above and because I don’t want to do post-hoc analyses. Also the RERI is sometimes suggested because the multiplicative interaction does not suggest an interaction.

For instance, this paper, Redirecting, was rejected in other journals, because reviewers did insist, that we should estimate the RERI for every single nucleotide polymorphism. The reviewer argued, that the RERI is a measure of a biological interaction and not only a statistical interaction, as the RERI is an interaction on the additive scale.

I argue – if you want an additive interaction, you should start with an additive model. To me, an interaction term on the additive scale, but not on the multiplicative scale suggests, that the multiplicative model is preferable, as it has a parameter less. I don’t even understand interactions as more than a model departure.

My questions to you – am I wrong? Should I plan to estimate the RERI? Is the RERI from a multiplicative model any good?

My goal here is to know what you think about RERI and what you would do if a reviewer asked to calculate it. Does interaction on the additive scale have anything to do with biology.

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