Additive and multiplicative interaction
- Measure
Two factors together can add or multiply their effects. Whether there is interaction depends on the scale you look at.
What it is
When two exposures act together, their joint effect is compared with what would be expected if they did not interact. On the additive scale the expected value is the sum of the excess risks; on the multiplicative scale, the product of the relative risks. The same table can show interaction on one scale and not on the other, so the scale must always be stated. Public health prefers the additive scale because it speaks in preventable cases.
Formula and reading
- R00 = risk with neither factor; R10 = A only; R01 = B only; R11 = both.
- Additive expectation: R11 = R10 + R01 − R00. Higher = additive synergy (superadditive).
- Multiplicative expectation: RR11 = RR10 × RR01. Ratio RR11 / (RR10 × RR01) > 1 = supramultiplicative.
- RERI = RR11 − RR10 − RR01 + 1. RERI > 0 superadditive; < 0 subadditive.
Worked example
- Cohort, 10-year risk of an outcome per 1,000: neither factor 5; A only 20; B only 10; A and B 40.
- RR against the group with neither: A 20/5 = 4; B 10/5 = 2; both 40/5 = 8.
- Multiplicative scale: expected 4 × 2 = 8; observed 8. No multiplicative interaction.
- Additive scale: expected 5 + 15 + 5 = 25 per 1,000; observed 40: an excess of 15 per 1,000.
- RERI = 8 − 4 − 2 + 1 = 3, above 0: positive additive interaction.
- Removing B prevents 40 − 20 = 20 cases per 1,000 among those with A, and only 10 − 5 = 5 among those without.
- Reading: with no multiplicative interaction, preventing B pays off 4 times more in people who have A.
The two scales of interaction
| Aspect | Additive | Multiplicative |
|---|---|---|
| Base measure | Risk difference | Relative risk or OR |
| No interaction if… | Excess risks add up | RRs multiply |
| Index | RERI (0 = none) | Ratio of RRs (1 = none) |
| What model assesses it | Additive risk models or the RERI | Product term in logistic or Cox regression |
| What it is for | Preventable cases and public health priorities | Mechanisms and statistical models |
Markers
- RERI
- RR11
- Synergy
- Additive scale
How to read it
- Report the risks of all four groups (neither, A only, B only, both) so readers can see both scales.
- A positive RERI shows where to concentrate prevention: in the group where the other factor is present.
- In case-control studies the RERI is calculated with ORs as approximations of the RRs.
Common pitfalls
- Saying ‘there is no interaction’ without naming the scale.
- Drawing conclusions about additive interaction from the product term of a logistic regression.
- Reading statistical interaction as proof of a joint biological mechanism.
Test yourself
Think of the answer, then tap the question to see it.
Risks per 1,000: neither 5, A only 20, B only 10, both 40. Is there multiplicative interaction?
No: RR of A = 4, of B = 2, expected 4 × 2 = 8, equal to the observed 40/5 = 8.
With the same data, what is the RERI and what does it mean?
RERI = 8 − 4 − 2 + 1 = 3: positive additive interaction; together they cause more cases than the sum of their excesses.
Why does public health prefer the additive scale?
Because it measures preventable cases: it shows in which group removing a factor prevents the most disease.
Which scale does a product term in logistic regression assess?
The multiplicative one (on the OR scale).
Connections
Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.
Sources: Genetic Epidemiology (Centers for Disease Control and Prevention, 1997); Principles of Epidemiology | Lesson 3 - Section 6 (Centers for Disease Control and Prevention, 2012); Chapter 10: Analysing data and undertaking meta-analyses (Cochrane, 2024).
Spotted a mistake?We are always improving for the community. Tell us what is wrong and our team will review it.Report a mistake
Representative model, for educational purposes only.