RERI: Relative Excess Risk Due to Interaction

RERI (relative excess risk due to interaction) is one of three standard measures researchers use to test whether two exposures combine additively or interact — alongside AP (attributable proportion due to interaction) and S (the synergy index). All three are computed from the same underlying relative risks and answer a question a multiplicative regression model’s interaction coefficient does not: not “is the joint effect a different multiple of the separate effects,” but “does exposure to both factors together produce more, less, or exactly as much excess risk as adding the two separate effects.”

Why additive interaction is the scale public-health decisions actually need

Standard logistic, Poisson, and Cox regression estimate effects on a multiplicative scale — a product term’s coefficient tests whether the joint relative risk equals the product of the two separate relative risks. Kenneth Rothman argued, in the paper that introduced the synergy index (Rothman KJ, “The estimation of synergy or antagonism,” American Journal of Epidemiology, 1976;103(5):506-511), that this is the wrong reference model for most public-health questions. Departure from additivity, not multiplicativity, is what corresponds to biological synergism and to the number of excess cases a joint-exposure reduction would actually prevent. A statistically significant multiplicative interaction term and a meaningful additive interaction are two different findings, and a model can show one without the other — a non-significant product-term p-value does not mean RERI is zero, and vice versa.

The three measures: RERI, AP, and S

Let RR₁₁ be the relative risk for people exposed to both factors, RR₁₀ exposed to only the first, RR₀₁ exposed to only the second, and RR₀₀ = 1 the doubly-unexposed reference group. The three additive-interaction measures (formalized by Tyler VanderWeele and Mirjam Knol, “A Tutorial on Interaction,” Epidemiologic Methods, 2014;3(1):33-72) are:

• RERI = RR₁₁ − RR₁₀ − RR₀₁ + 1. Zero means the joint effect is exactly the sum of the two separate effects (pure additivity, no interaction). Positive means superadditivity — the combination does more harm than the parts added together. Negative means subadditivity.
• AP (attributable proportion due to interaction) = RERI / RR₁₁. This rescales RERI as the share of risk in the doubly-exposed group that is attributable to the interaction itself, rather than to either exposure alone — useful for communicating impact, since “36% of the excess risk in doubly-exposed people is due to the interaction” reads more directly than a raw RERI value.
• S (synergy index) = (RR₁₁ − 1) / [(RR₁₀ − 1) + (RR₀₁ − 1)]. S = 1 means no interaction, S > 1 means positive interaction, S < 1 means negative interaction. S expresses the same underlying comparison as a ratio rather than a difference, and is undefined when RR₁₀ = RR₀₁ = 1 (dividing by zero).

All three are internally consistent — they are different transformations of the same three relative risks, not three independent pieces of evidence. Most methods papers recommend reporting RERI and AP together (AP for interpretability, RERI because it is the quantity the confidence interval is usually built around) and adding S when a bounded, symmetric-around-1 statistic is more convenient for the audience.

Computing RERI from a regression model

You rarely observe RR₁₁/RR₁₀/RR₀₁ directly. In practice they come from a regression model fit with both exposures and their product term, then evaluated as linear combinations of the fitted coefficients against the doubly-unexposed reference:

• RR₁₀ = exp(β₁) — the coefficient on exposure A alone.
• RR₀₁ = exp(β₂) — the coefficient on exposure B alone.
• RR₁₁ = exp(β₁ + β₂ + β₁₂) — both main-effect coefficients plus the interaction coefficient.

One caveat that’s easy to skip: if the model is logistic regression, its exponentiated coefficients are odds ratios, not risk ratios. RERI/AP/S are additive-risk-scale measures, so an odds-ratio-based calculation is only a good approximation when the outcome is rare across every exposure stratum (the standard rare-disease assumption). For a common outcome, use a model that estimates risk ratios directly — Poisson regression with robust variance, a log-binomial model, or marginal standardization from the fitted logistic model — rather than treating odds ratios as risk ratios by default.

A worked example (illustrative, not from a real study)

The numbers below are a constructed illustration to show the mechanics, not a finding from any published dataset. Suppose a case-control study estimates, relative to people exposed to neither factor: RR₁₀ = 1.8 for exposure A alone, RR₀₁ = 1.5 for exposure B alone, and RR₁₁ = 3.6 for both together.

• RERI = 3.6 − 1.8 − 1.5 + 1 = 1.3 — the joint effect is 1.3 units of relative risk larger than additivity alone would predict.
• AP = 1.3 / 3.6 ≈ 0.36 — about 36% of the risk in the doubly-exposed group is attributable to the interaction itself, over and above the two exposures’ independent contributions.
• S = (3.6 − 1) / [(1.8 − 1) + (1.5 − 1)] = 2.6 / 1.3 = 2.0 — the excess risk from joint exposure is twice what the sum of the two separate excess risks alone would produce.

All three point to the same conclusion (meaningful positive additive interaction) because they’re the same underlying comparison expressed three ways — which is exactly why reviewers rarely need more than RERI plus one of AP or S.

Confidence intervals for RERI, AP, and S

Because all three measures are nonlinear combinations of model coefficients (sums and ratios of exponentiated betas, not a single coefficient), their standard errors are not something a regression model reports directly — you cannot just read a CI off the interaction term’s own output. The standard approach is the delta method applied to the covariance matrix of β₁, β₂, and β₁₂ (the variance formula most commonly cited traces to Hosmer and Lemeshow’s 1992 treatment of interaction contrasts), or a bootstrap over the fitted model. Confidence intervals on RERI and S are often wide relative to the point estimate specifically because of this nonlinear propagation — a wide interval is not automatically a sign of a coding error. Several tools compute this without a manual delta-method derivation: Stata’s user-written and now built-in reri command, R’s epiR::epi.interaction() and the dedicated interactionR package, and published SAS macros built around the same Hosmer-Lemeshow-derived variance formula.

Reporting RERI in a manuscript — what reviewers expect

A few things reviewers in epidemiology and clinical journals consistently look for, per the STROBE reporting guidance’s treatment of interaction and effect modification and the VanderWeele/Knol tutorial cited above:

• Report the point estimate and confidence interval for RERI (or AP/S), not just the interaction coefficient’s p-value from the regression table — they answer different questions.
• State the reference category explicitly (doubly-unexposed) and confirm both exposures were coded so the reference group has the lowest risk when considered jointly.
• If either exposure is protective rather than a risk factor (RR < 1), the standard AP/S formulas can give misleading results without recoding — Knol and colleagues (“Estimating measures of interaction on an additive scale for preventive exposures,” European Journal of Epidemiology, 2011;26(6):433-438) set out the recoding rule: redefine the reference categories so the jointly-lowest-risk combination is the comparator, before computing RERI/AP/S, rather than applying the risk-factor formulas unmodified to a preventive exposure.
• Name the software/package and variance method used — delta method versus bootstrap CIs can differ meaningfully at small cell counts, and reviewers increasingly expect that noted rather than assumed.

Frequently asked questions

For the distinction this all sits on top of, see effect modification vs. confounding. For the underlying regression models RERI is usually computed from, see logistic regression, fitting logistic regression in R, and the Cox proportional hazards model. Related CASRAI Dictionary terms: confounding variable and confidence interval. See the Research Methods hub for the full quantitative-analysis sub-cluster.