Additive Interaction: When “No Interaction” Depends on the Scale
Anas H. Alzahrani, MD PhD MPH
Department of Preventive Medicine and Public Health
Faculty of Medicine, King Abdulaziz University
A paper reports that a treatment works differently in patients with and without a second risk factor. The authors fit a model, inspect the exposure-by-factor product term, see a non-significant p-value, and write: there was no interaction. That sentence may be answering the wrong question.
Interaction is not a single property that lives inside a dataset waiting to be detected. It is a contrast defined on a scale. The same four outcome risks can look additive on one scale and multiplicative on another. A credible analysis therefore starts with the decision: are we asking whether absolute excess risk combines, whether relative risk multiplies, or whether the effect of one intervention changes across strata of another?
The scale is part of the scientific question
Suppose A is a treatment and B is a baseline condition. Let the four risks be p00 for neither exposure, p10 for A only, p01 for B only, and p11 for both.
On the additive scale, ask whether the risk difference for A changes across B:
A positive value means the joint exposure produces more absolute risk than the two individual risk differences would add. A negative value means less. This is often the useful scale when the practical question is how many additional events a combined prevention strategy might avert or create.
On the multiplicative scale, the analogous question is whether the risk ratio for A is the same across levels of B. A regression product term in a log-link, logistic, or proportional-hazards model usually represents a multiplicative contrast unless the model and post-estimation transform say otherwise. A non-significant product term is therefore not evidence that additive interaction is absent.
Interactive additive-interaction explorer
Change the four risks and watch the scale change the story
Enter the outcome risk in each exposure cell. This teaching calculator compares the difference-of-risk-differences with the relative excess risk due to interaction (RERI). It does not estimate uncertainty or remove confounding.
| Contrast | Result | What it means |
|---|---|---|
| A effect when B− vs B+ | +10.0 pp vs +20.0 pp | Absolute risk differences in the two B strata |
| Additive interaction | +10.0 pp | Difference between those two risk differences |
| RERI | +1.00 | Relative excess risk beyond the two individual effects |
| Multiplicative interaction ratio | 1.00× | Approximately multiplicative; 1.00× is the multiplicative null |
RERI is useful, but it is not a magic causal number
When relative risks are available, the relative excess risk due to interaction is:
RERI is zero under additivity of relative risks, positive when the joint effect exceeds that additive expectation, and negative when it falls below it. It is a relative measure, while the difference-of-risk-differences above is expressed in absolute risk units. The two are related, but they are not interchangeable.
RERI also inherits the design and measurement problems that produced the four risks. In an observational study, causal interaction requires a defensible confounding strategy for the relevant exposure effects, adequate overlap in the four cells, and a stable outcome definition. The calculator above uses hypothetical risks. It cannot turn an unadjusted table into a causal result.
How a paper can look convincing and still answer the wrong interaction question
The product-term shortcut
A model reports one interaction p-value, but never states its scale or shows the four joint exposure groups. Readers cannot tell whether “no interaction” means no multiplicative departure, no additive departure, or simply low power.
Odds ratios treated as risk ratios
Plugging odds ratios into an RERI formula derived for risk ratios can mislead when the outcome is not rare. The safer route is to obtain risks or risk ratios compatible with the estimand, then state the approximation if one is being used.
Within-group p-values
“Significant in one subgroup, not significant in the other” is not a test that the effects differ. Compare the effects directly and show the interval for the contrast.
Scale and estimand drift
A paper can compare a baseline risk difference with a hazard-ratio interaction, or combine different follow-up windows, and still use the word “interaction” as though the quantities were aligned.
Effect modification and interaction are related, not identical
Effect modification asks whether the effect of A varies across levels of B. Interaction can ask a broader joint question: what happens when interventions on A and B operate together relative to the reference combination? The language becomes especially important when B is not itself an intervention, but a patient characteristic such as age, genotype, or disease severity.
Calling a baseline characteristic an effect modifier does not prove that it biologically changes treatment action. It may identify a clinically useful stratum, reflect a different target population, or expose a model-scale contrast. The interpretation should follow the estimand and the causal story, not the label attached to a coefficient.
Decision rule
Before interpreting interaction, write one sentence naming the exposure, the modifier or second exposure, the outcome time horizon, and the effect scale. If that sentence is unclear, the interaction result is not ready for a clinical conclusion.
A reviewer’s minimum reporting table
| Question | What to look for | Red flag |
|---|---|---|
| What is being modified? | The effect of A across B strata, or the joint effect of A and B. | “Interaction” appears without a named causal or descriptive question. |
| Which scale? | Risk difference, risk ratio, odds ratio, rate, or hazard, with a reason. | A product term is treated as scale-free. |
| Can the reader reconstruct it? | Four cell risks or effect estimates, reference group, contrast, and interval. | Only the interaction p-value is reported. |
| Is the joint contrast supported? | Adequate events and overlap in all relevant cells, plus confounding and missing-data checks. | A dramatic joint estimate comes from a sparse cell. |
Review the claim in this order
- Name the question. Separate effect modification by a baseline characteristic from a joint intervention question.
- Choose the scale before the model. Use absolute risk when event burden and prevention impact matter; use a relative scale when that is the scientific target. Explain the choice.
- Read the four cells. Check the reference risk, the two single-exposure risks, and the joint risk before reading the interaction label.
- Compare effects directly. Do not infer a difference from two separate significance tests.
- Audit support and uncertainty. Look for sparse cells, overlap, confounding control, missing outcomes, multiplicity, and an interval for the interaction contrast.
- Keep the conclusion proportional. A scale-specific departure is a finding about that estimand. It is not automatically a biological mechanism or a treatment recommendation.
Where Aqrab fits
Interaction claims are a good test of methodological judgment because the model output is usually tidy while the question is not. Aqrab can pressure-test whether the paper names its estimand, keeps the scale stable, reports enough information to reconstruct the contrast, and distinguishes a discovery from a decision-ready subgroup claim.
If you are reviewing a subgroup analysis, a biomarker paper, or a joint-exposure study, try Aqrab for a methods critique pass. If you are building a review workflow, the developer workflows can put the scale, reference group, and reporting checks upstream of publication.
The practical bottom line
“No interaction” is incomplete until the scale is named. A product term can be null while the joint exposures create a meaningful excess of absolute events, or the reverse can happen. Show the four risks, state the estimand, and make the reader’s decision question visible.
The most useful reviewer question is simple: no interaction on which scale, over what time horizon, and for whose decision?
Further reading
- Knol and VanderWeele: Recommendations for presenting analyses of effect modification and interaction — a reporting sequence that includes stratum-specific effects, interaction measures on both scales, and adjusted confounders.
- Interaction in Theory and in Practice — an accessible review of additive interaction, RERI, and the interpretation of joint exposures.
- Li and Chambless: Test for additive interaction in proportional hazards models — a methods paper comparing additive and multiplicative interaction in time-to-event settings.
Keep reading
Don't stop at one method.
Good methods judgment comes from contrast. Read the neighboring guides, see where the assumptions diverge, and avoid treating every observational problem like it needs the same hammer.
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