Interaction and Effect Modification in Clinical and Epidemiological Studies

Part 38 of a Series on the Evaluation of Scientific Publications

Background: A common objective of studies in clinical research is to investigate the effect of an individual factor on the course of a disease. Studying the combined effect of two or more factors on an outcome variable is more complex, as is the question whether an effect varies across subgroups (e.g., men vs. women). The concepts of interaction and effect modification are useful for analyses of this kind.

Methods: This review is based on relevant publications retrieved by a selective search. The concepts of interaction and effect modification are illustrated using examples involving real and fictitious data.

Results: An interaction is present when, for example, taking two drugs reduces the risk of a disease by an amount that differs from the sum of the individual effects. In contrast, an effect modification is present when, for example, the effect of a drug on the risk of disease varies depending on the presence or absence of a particular genotype. In this article, the concepts of interaction, effect modification, synergism, and antagonism are explained. Interactions can occur on either an additive or a multiplicative scale. Moreover, quantitative measures of interactions on the additive scale are explained, and the potential for misinterpreting evaluations of interactions in regression analyses is discussed.

Conclusion: The concepts of interaction and effect modification provide valuable insights for the prevention and treatment of disease.

Cite this as: Kowall B, Stang A, Rathmann W: Interaction and effect changes in clinical and epidemiological studies: Part 38 of a series on the evaluation of scientific publications. Dtsch Arztebl Int 2025; 122: 663–8. DOI: 10.3238/arztebl.m2025.0155

Multiple factors usually interact in the development and treatment of diseases, increasing or decreasing the risk of disease. Many examples can be given. For instance, phenylketonuria is not caused solely by dietary phenylalanine intake or solely by a genetic predisposition to the disease, but by the two working in concert. The effect of two drugs given together is often not simply the sum of their effects when given separately. Driving and drunkenness as individual factors are associated with only a small risk of a traffic accident; only when both factors are present at the same time does the risk increase dramatically. Interaction analysis in epidemiology is the study of such interactions of multiple factors, involving a comparison of their combined effect with the individual effects of each factor acting alone.

Interaction is to be distinguished from effect modification (1, 2), i.e., variation in the strength of an effect across categories of another variable, such as sex, educational level, or the presence or absence of a pre-existing condition. For instance, the risk of sunburn after exposure to sunlight is higher in light-skinned than in dark-skinned people.

In this article, we define, explain, and distinguish among the terms “interaction,” “effect modification,” and “effect measure modification.” We further discuss the important distinction between additive and multiplicative scales in this context (3, 4), as well as the concept of synergisms and antagonisms in interactions. Moreover, we describe methods of quantifying additive interactions as an aid to the interpretation of study results. It will be seen that an approach commonly taken in analyzing interactions, in which interaction terms are inserted into regression models, can generate incorrect conclusions.

Methods

Various aspects of interactions and effect modifications are explained using examples with real and fictitious data based on a selective literature search in PubMed.

Effect modification and interaction

Effect modification

Table 1 contains data from a Dutch multicenter study on mortality after myocardial infarction (5). The effect of diabetes on mortality was examined separately in subjects with and without depression. Among the depressed subjects, diabetes had a markedly stronger effect on mortality than in the non-depressed subjects (25.2% risk difference in mortality with vs. without diabetes in depressed subjects, 9.8% in non-depressed subjects). The relative risk (RR) was also higher in depressed subjects (2.16 vs. 1.73). Thus, an effect modification is evident both in the estimation of risk differences and in the estimation of relative risk, with the effect of the exposure (diabetes yes/no) on the target variable (death yes/no) differing depending on the category of depression (yes/no).

Table 1

The concept of effect modification, illustrated by the relationship between diabetes and mortality: risk differences and relative risks for the effect of diabetes on mortality in the strata of the presence or absence of depression

In Table 2, we provide a fictitious example of how an effect modification may or may not be found depending on the choice of effect measure. When risk differences are used, the effect is identical (5%) in both categories of A; yet, when relative risk is used, it differs between categories (when A = 0, RR = 2.0; when A = 1, RR = 1.33). To emphasize the dependence of effect modification on the choice of effect measure (relative vs. absolute difference), the term effect measure modification is also used as an alternative (3).

Table 2

The dependence of effect modification on the chosen effect measure: risk differences and relative risks for the effect of exposure to risk factor B on outcome in the strata of A

Interaction

The term “interaction” refers to a combined effect of two or more factors on a target variable. In the following fictitious example, the risk of developing a certain disease depends on exposure to factors A and B:

• Persons exposed to neither A nor B have a risk of 4% (“baseline risk”), which we will designate here as R₀₀.
• Persons exposed to A, but not B, have a risk of 8% (R₁₀).
• Persons exposed to B, but not A, have a risk 12% (R₀₁).

Stated another way, factor A increases the risk of disease by 4% compared to the baseline risk (8% minus 4%), and factor B by 8% (12%-4%).

As for the risk of disease for individuals exposed to A and B (here designated as R₁₁), three different scenarios can be imagined (Table 3):

Table 3

Estimating an interaction on the additive scale: risks of a fictitious outcome for combinations of two binary risk factors—three scenarios

• No interaction on the additive scale: the increase in risk over baseline for doubly exposed persons is the sum of the increase in risk due to each exposure acting alone. In the example given, R₁₁ in this scenario would be the sum of the baseline risk (4%), the added risk due to A (4%), and the added risk due to B (8%):

R₁₁ = 4% + 4% + 8% = 16%

• Interaction on the additive scale—synergism: If R₁₁ in the example given is, in fact, greater than 16%, i.e., greater than the risk corresponding to addition of the individual effects, then there is an interaction on the additive scale: more precisely, synergism. We refer to the additive scale here because the individual effects are added and subtracted in the relevant calculations, but not multiplied or divided. Synergism means that the effect observed with double exposure is greater than the sum of the effects observed with each kind of single exposure. If, as illustrated in the Figure, the risk for the doubly exposed (R₁₁) equals 20%, there is a synergism with an associated excess risk of 4%.
• Intraction on the additive scale – antagonism: If R₁₁ in the example given is, in fact, less than 16%, then there is also an interaction on the additive scale: an antagonism in this case, because the effect observed with double exposure is less than the sum of the effects observed with each kind of single exposure.

Figure

Disease risks

Interaction contrast (IC) is defined as the deviation of the observed risk with double exposure (R11) from the risk that would result from an effect that was the sum of the two individual effects. IC = 0 means that there is no interaction.

IC = R₁₁ – R₁₀ – R₀₁ + R₀₀ (F1)

One example each of synergism and antagonism are shown in Table 3:

• R₁₁ = 20%, IC = 20% − 8% – 12% + 4% = 4% (synergism: with double exposure, 4% higher risk than with no interaction on the additive scale; scenario 2),
• R₁₁ = 13%, IC = 13% − 8% − 12% + 4% = −3% (antagonism: with double exposure, 3% lower risk than with no interaction on the additive scale; scenario 3).

Interactions can also be calculated on the multiplicative scale, rather than on the additive sale. Estimates of increased risk on the multiplicative scale are given as a multiplicative factor rather than a percentage.

In the example given, A doubles the risk; i.e., the risk with A is twice the risk at baseline without A or B). The multiplicative factor corresponds to the relative risk:

RR₁₀ = R₁₀/R₀₀ = 8%/4% = 2. Similarly, the presence of B trebles the risk, corresponding to a relative risk of RR₀₁ = R₀₁/R₀₀ = 12%/4% = 3.

There is no interaction on the multiplicative scale if the relative risk of the doubly exposed corresponds exactly to the product of the individual relative risks:

RR₁₁ = RR₁₀ · RR₀₁ = 2 · 3 = 6.

RR₁₁ = 6 means that the risk with double exposure is six times the risk with no exposure. This would be true if the risk with double exposure were 24%, which is six times higher than the risk with no exposure (4%). In general, there is no interaction on the multiplicative scale if the relative risk with double exposure exactly equals the product of the two relative risks for single exposure. Expressed as a simple formula, there is an interaction on the multiplicative scale if the following quotient Q deviates from 1:

Table 4 contains examples of synergism (R₁₁ = 32%, Q > 1; scenario 2) and antagonism (R₁₁ = 20%, Q < 1; scenario 3) on the multiplicative scale. In the case of synergism, the risk after a double exposure is eight times the risk after no exposure (RR₁₁ = 8), and the value of Q is 8/(2×3) = 1.33.

Table 4

Estimating an interaction on the multiplicative scale: variation of Table 3 with estimation of relative risks instead of risk differences

In scenario 1 in Table 4, lthere is no interaction on the multiplicative scale, but the interaction contrast is 8% (IC = 24% − 12% − 8% + 4% = 8%). This means that there is interaction on the additive scale. This leads to an important conclusion: the presence or absence of an interaction depends on the choice of scale (additive or multiplicative). It can be shown, in fact, that whenever there is no interaction on the additive scale, there will be an interaction on the multiplicative scale; likewise, whenever there is no interaction on the multiplicative scale, there will be an interaction on the additive scale (4).

Differences between interaction and effect modification

The analyses of interactions and effect modifications have different purposes (6). In effect modification, a relationship is examined in two or more subgroups, and an effect estimator is obtained for each subgroup. The analysis of effect modifications can identify subpopulations in which effects are particularly strong or weak. For example, one may want to know whether a drug has an especially strong effect in a subgroup, or whether the exposure in question increases the risk of disease especially strongly in certain groups of people. Such findings are important for both the treatment and the prevention of disease. The analysis of interactions (but not of effect modifications) also enables risk estimation for persons exposed to both factors. The goal of interaction analysis is to determine whether the combined effect of two exposures differs from the sum or product of the effects of the two individual exposures: e.g., whether persons who smoke and drink are at greater risk for a disease than would be predicted from the individual risks associated with smoking and drinking. Another example is the question whether two drugs taken simultaneously cause adverse side effects more often than would be expected from a combination of the individual risks.

The choice of scale—additive or multiplicative?

At first glance, the dependence of the findings on the choice of scale may make them seem arbitrary. One argument in favor of using the additive scale is that absolute changes are more relevant to clinical practice than relative changes, e.g., when there is a question of the increase or decrease in the number of disease events after exposure to a risk factor. The additive scale is more relevant in the context of medical care and public health than the multiplicative scale (7, 8, 9).

This is illustrated by a fictional example from VanderWeele and Knol, in which ethical considerations are set aside for the purpose of discussion (Table 5) (10). Assume that the probability that a particular drug (D) will cure a disease is found to depend on the patient’s genotype (G). Among patients with genotype G0, the spontaneous cure rate is 2%, and the cure rate if D is given is 5%; among those with genotype G1, the spontaneous cure rate is 4%, and the cure rate if D is given is 10%. Thus, 3 of 100 patients with genotype G0 and 6 of 100 with genotype G1 will benefit from the drug.

Table 5

Probability of remission depending on drug (D) and genotype (G)

One might conclude that it would be better to restrict the scarce drug D to patients with genotype G1 to help as many patients as possible.

The observed difference of 6 versus 3 cures brought about by the drug depending on the genotype corresponds to an interaction contrast (IC) of 0.03, or 3%, on the additive scale. Yet, on the multiplicative scale, no interaction is seen. When the appropriate values are entered into equation F2, the result is 1 (RR₁₁ = 0.10/0.02 = 5; RR₁₀ = 0.04/0.02 = 2; RR₀₁ = 0.05/0.02 = 2.5; und thus 5 / (2 · 2.5) = 1). Simply put, the drug D always multiplies an individual’s probability of cure by 2.5, regardless of genotype. The benefit of the drug seems larger among G1 patients only on the additive scale, and not on the multiplicative scale.

In the public health setting, the additive scale is preferred, as already mentioned. It is nonetheless recommended in the existing guidelines for the publication of results from randomized clinical trials (the CONSORT statement) and in the STROBE guidelines for the publication of results from observational studies that both absolute and relative effect measures should be reported; in other words, that both the additive and the multiplicative scale should be considered (CONSORT: Item 17b, STROBE: Item 16c) (11, 12).

Measures of interaction on the additive scale

We will mention two measures that are used to quantify the strength of interactions on the additive scale. Confidence intervals can also be calculated for these measures.

One such measure is the relative excess risk due to the interaction (RERI)—alternatively designated the interaction contrast ratio (ICR)—and the other is the attributable proportion (AP) (13). A value of 0 for either measure indicates that there is no interaction on the additive scale. The formulae for both are given in the Box. These measures can be interpreted as follows:

RERI is the ratio of the excess risk, expressed as the interaction contrast (IC, equation F1), to the baseline risk in the doubly unexposed. For the synergism in Table 3, the RERI is 1.0. This means that the increase in risk due to the interaction in doubly exposed persons (IC = 4%) is equal, in this case, to the baseline risk (4%).

AP is the ratio of the excess risk, expressed as the interaction contrast (IC), to the risk in the doubly exposed; in other words, it is the fraction of the risk in the doubly exposed that can be attributed to the interaction. For the synergism in Table 3, the AP equals IC/R₁₁ = 4%/20% = 0.2. This means that 20% of the risk in the doubly exposed can be attributed to the interaction of the two exposures.

Box

Measures of interaction on the additive scale

eBox

The interpretation of interaction analyses

Evaluations of interactions in regression analyses

Interaction analyses in clinical studies are often performed as part of a regression analysis. For readers with a basic understanding of regression analyses (see [14]), interaction analyses in regression models are described in more detail in the eMethods section, which also includes a discussion of a possible misinterpretation of evaluations of interactions in regression analyses.

What readers of scientific publications should bear in mind

The mere finding of an interaction does not imply anything about its direction, its strength, or the scale used. Readers should closely consider the following:

• On what scale was the interaction examined, and why? Were interactions evaluated with absolute or relative effect sizes?
• In which direction does the interaction go? In the case of synergism, the effect is stronger than would be expected from the sum or product of the individual effects; in the case of antagonism, it is weaker.
• Is the strength of the interaction quantified, e.g., by specification of the RERI or AP?
• Has an effect modification possibly been neglected (e.g., a difference in the effect of a drug depending on the sex of the patient)?

Conflict of interest statement
The authors state that they have no conflict of interest.

Manuscript received on 9 April 2025, revised version accepted on 27 August 2025.

Translated from the original German by Ethan Taub, M.D.

Corresponding author
Prof. Dr. rer. nat. Dr. rer. san. Bernd Kowall

References

1. Corraini P, Olsen M, Pedersen L, Dekkers OM, Vandenbroucke JP: Effect modification, interaction and mediation: An overview of theoretical insights for clinical investigators. Clin Epidemiol 2017; 9: 331–8

2. Bours MJL: A nontechnical explanation of the counterfactual definition of effect modification and interaction. J Clin Epidemiol 2021; 134: 113–24

3. Rothman KJ: Epidemiology. An Introduction. 2 nd edition. Oxford University Press; 2012.

4. Rothman KJ, Greenland S, Lash TL: Modern Epidemiology. 3 rd edition. Lippincott Williams & Wilkins; 2008.

5. Bot M, Pouwer F, Zuidersma M, van Melle JP, de Jonge P: Association of coexisting diabetes and depression with mortality after myocardial infarction. Diabetes Care 2012; 35: 503–9

6. Knol MJ, Groenwold RHH: [Effect modification and interaction]. Ned Tijdschr Geneeskd 2015; 159: A8499.

7. Rothman KJ, Greenland S, Walker AM: Concepts of interaction. Am J Epidemiol 1980; 112: 467–70

8. Knol MJ, VanderWeele TJ: Recommendations for presenting analyses of effect modification and interaction. Int J Epidemiol 2012; 41: 514–20

9. Weinberg CR: Interaction and exposure modification: Are we asking the right questions? Am J Epidemiol 2012; 175: 602–5

10. VanderWeele TJ, Knol MJ: A Tutorial on interaction. Epidemiol Methods 2014; 3: 33–72

11. Schulz KF, Altman DG, Moher D , CONSORT Group: CONSORT 2010 Statement: Updated guidelines for reporting parallel group randomised trials. J Clin Epidemiol 2010; 63: 834–40

12. STROBE Statement—checklist of items that should be included in reports of observational studies: www.strobe-statement.org (last accessed on 6 April 2025).

13. Andersson T, Alfredsson L, Källberg H, Zdravkovic S, Ahlbom A: Calculating measures of biological interaction. Eur J Epidemiol 2005; 20: 575–9

14. Zapf A, Wiessner C, König IR: Regression analyses and their particularities in observational studies—part 32 of a series on evaluation of scientific publications. Dtsch Arztebl Int 2024; 121: 128–34

e1. Rothman KJ: Six persistent research misconceptions. J Gen Res Med 2014; 29: 1060–4