Additive Interaction and Risk Differences

Why Compare Risk Differences Directly?

A treatment can have different effects in different subgroups. To ask whether those effects differ on the additive scale, compare the subgroup-specific differences in risk. The resulting contrast is called additive interaction: it tells us how much the treatment risk difference in one subgroup differs from the treatment risk difference in another.

This is a comparison of effects, not merely a report of the effect in each subgroup. A large treatment effect in one subgroup does not by itself establish that the effect differs between subgroups; the other subgroup’s effect must also be included in the contrast.

Definition: Let $A$ be a binary treatment, with $A=1$ denoting treatment and $A=0$ denoting control. Let $S$ be a baseline binary subgroup variable, with $S=1$ and $S=0$ identifying the two subgroups, and let $Y$ be a binary outcome. Define $p_{as}=P(Y^a=1\mid S=s)$, the causal risk of the outcome under treatment level $a$ among people in subgroup $s$. The subgroup-specific risk difference is $RD_s=p_{1s}-p_{0s}$. The additive interaction contrast is $$ \Delta_{\mathrm{add}}=RD_1-RD_0=(p_{11}-p_{01})-(p_{10}-p_{00}). $$

The risk difference is measured in risk units. For example, a difference of $-0.10$ is a reduction of 10 percentage points in outcome risk under treatment compared with control. The additive interaction contrast is also in risk units, or equivalently in percentage-point differences when the risks are expressed as percentages. It is not itself a risk.

Sign and Magnitude

The sign is determined by the order of subtraction in the definition: the treatment risk difference for $S=1$, minus the treatment risk difference for $S=0$. If $\Delta_{\mathrm{add}}=0$, the two subgroup-specific risk differences are equal. A positive value means the risk difference is more positive in subgroup $S=1$; a negative value means it is more negative there.

For an adverse outcome that treatment reduces, the subgroup risk differences are often negative. In that setting, a negative additive interaction means the reduction in risk is larger in subgroup $S=1$ than in subgroup $S=0$. A positive interaction could mean that the reduction is smaller in $S=1$, or that treatment raises risk more in $S=1$, depending on the two subgroup effects. Always report the subgroup effects alongside the contrast so the sign has a clear context.

The labels $S=1$ and $S=0$ are also part of the interpretation. Swapping which subgroup is coded 1 reverses the sign of the contrast, while leaving its absolute magnitude unchanged. Similarly, reversing the treatment-versus-control contrast reverses both subgroup risk differences and the interaction contrast. State the coding choices when presenting results.

Worked Example: Interpreting a Negative Additive Interaction

Suppose a hypothetical randomized study considers a treatment intended to prevent an adverse outcome. The population is divided by a baseline risk marker $S$. The causal risks are estimated to be:

Subgroup	Control risk	Treatment risk
$S=0$	0.20	0.10
$S=1$	0.50	0.25

For $S=0$, the risk difference is $RD_0=0.10-0.20=-0.10$, a 10-percentage-point reduction. For $S=1$, it is $RD_1=0.25-0.50=-0.25$, a 25-percentage-point reduction. Therefore,

\Delta_{\mathrm{add}}=RD_1-RD_0=(-0.25)-(-0.10)=-0.15.

The treatment risk difference is 15 percentage points more negative in $S=1$ than in $S=0$. In this example, treatment’s absolute risk reduction is 15 percentage points larger in $S=1$. The interaction contrast does not say that treatment lowers risk by 15 percentage points in $S=1$; that subgroup’s risk reduction is 25 percentage points.

Calculating the Contrast from Hypothetical Data

In a randomized study, subgroup-specific risks can be estimated by the observed proportions with the outcome in each treatment arm and subgroup. To interpret these as estimates of the causal risks, the treatment assignment must be randomized within the relevant subgroup (or otherwise independent of the potential outcomes within it), and both treatment levels must be represented. The treatment and outcome must be well-defined and observed consistently, and the simple individual-level formulation assumes no interference between individuals. Under these conditions, each arm’s observed risk within a subgroup estimates that subgroup’s causal risk under that treatment.

Randomization does not make estimates exact: the observed proportions vary across samples. The calculations below illustrate the point estimates. They do not, by themselves, quantify uncertainty.

Worked Example: Computing Additive Interaction from Event Counts

In a hypothetical randomized trial, treatment and control participants are compared within each of two baseline subgroups. Each cell contains 100 participants, and the table gives the number who experience the adverse outcome.

Subgroup	Control events / participants	Treatment events / participants
$S=0$	18 / 100	10 / 100
$S=1$	40 / 100	24 / 100

The estimated risks in $S=0$ are $18/100=0.18$ under control and $10/100=0.10$ under treatment. Thus,

\widehat{RD}_0=0.10-0.18=-0.08.

The estimated risks in $S=1$ are $40/100=0.40$ under control and $24/100=0.24$ under treatment. Thus,

\widehat{RD}_1=0.24-0.40=-0.16.

The estimated additive interaction is

\widehat{\Delta}_{\mathrm{add}}=(-0.16)-(-0.08)=-0.08.

The estimated treatment risk reduction is 8 percentage points in $S=0$ and 16 percentage points in $S=1$. Their difference is 8 percentage points, with the larger reduction in $S=1$. The arithmetic uses risk differences, not a comparison of event counts across subgroups, whose sizes and baseline risks might differ.

Why This Is a Difference of Differences

The additive interaction can be expanded into four risks. Starting from the definition, subtract the control-to-treatment risk difference in $S=0$ from the corresponding risk difference in $S=1$:

\begin{aligned} \Delta_{\mathrm{add}} &=(p_{11}-p_{01})-(p_{10}-p_{00})\\ &=p_{11}-p_{01}-p_{10}+p_{00}. \end{aligned}

This is a difference of differences: first take the treatment-minus-control risk difference within each subgroup, then subtract the subgroup-specific effects. Equivalently, it compares the change in risk from control to treatment across the two subgroups. The four-term expression makes an important point explicit: all four treatment-by-subgroup risks contribute to the contrast.

The contrast is zero exactly when the two risk differences are equal:

\Delta_{\mathrm{add}}=0 \quad\Longleftrightarrow\quad p_{11}-p_{01}=p_{10}-p_{00}.

This is equality of effects on the additive scale. It does not require the risks themselves to be equal across subgroups. For instance, control risks can differ substantially while the treatment changes risk by the same number of percentage points in each subgroup.

Worked Example: Equal Additive Effects Despite Different Risks

Consider hypothetical causal risks for an outcome in two subgroups:

Subgroup	Control risk	Treatment risk
$S=0$	0.10	0.05
$S=1$	0.40	0.35

In $S=0$, the risk difference is $0.05-0.10=-0.05$, or a 5-percentage-point reduction. In $S=1$, it is $0.35-0.40=-0.05$, also a 5-percentage-point reduction. Hence,

\Delta_{\mathrm{add}}=(-0.05)-(-0.05)=0.

The baseline risks differ by 30 percentage points, but the treatment risk differences are equal. Thus there is no additive interaction for these risks, even though the outcome risks are not the same in the two subgroups.

Keep the Interaction Contrast Separate from Each Subgroup Effect

A subgroup-specific treatment effect answers, “What is the treatment risk difference within this subgroup?” Additive interaction answers a different question: “How different are the two subgroup-specific treatment risk differences?” Reporting both effects is useful, but neither one alone is the interaction.

Worked Example: Two Effects and Their Contrast

Suppose a hypothetical randomized trial has the following event counts, with 100 participants in each treatment-by-subgroup cell.

Subgroup	Control events / participants	Treatment events / participants
$S=0$	12 / 100	6 / 100
$S=1$	30 / 100	16 / 100

For $S=0$, the estimated risk difference is $6/100-12/100=0.06-0.12=-0.06$, a 6-percentage-point reduction. For $S=1$, it is $16/100-30/100=0.16-0.30=-0.14$, a 14-percentage-point reduction. The additive interaction is

\widehat{\Delta}_{\mathrm{add}}=(-0.14)-(-0.06)=-0.08.

The separate subgroup estimates describe reductions of 6 and 14 percentage points. The contrast describes an 8-percentage-point difference between those reductions. A report that gives only the two effects leaves the reader to calculate that comparison; a report that gives only the interaction does not show either subgroup’s absolute effect.

Likewise, whether a statistical test for one subgroup’s effect is significant and whether a test for the other subgroup’s effect is significant do not answer whether the effects differ. The interaction is a contrast between estimates; its uncertainty must be assessed for that contrast itself. A difference between “significant” and “not significant” is not automatically a statistically significant difference. Point estimates alone also do not establish how precisely the interaction has been estimated.

Assumptions, Interpretation, and Common Pitfalls

The causal interpretation depends on how the risks are identified. In a randomized trial, random assignment within subgroup supports comparison of the treatment arms for that subgroup, provided treatment is well-defined, outcomes are consistently measured, both arms are possible in each subgroup, and there is no interference under the individual-level setup. In an observational study, observed risks cannot automatically be treated as causal risks: identification would require appropriate control of confounding and other conditions for the target population. The formula defines the estimand; design and assumptions determine whether data identify it.

The risk difference is often useful when the question concerns absolute changes in event probability. Its units are directly interpretable as percentage points, and the additive interaction directly compares those absolute changes. But a zero additive interaction does not mean that treatment has no effect in either subgroup: both effects could be equal and nonzero, as in the preceding example. Conversely, a nonzero additive interaction does not say which subgroup has the larger treatment effect until the treatment direction, subgroup coding, and outcome are stated.

Common mistake: “The treatment effect is statistically significant in $S=1$ but not in $S=0$, so there must be additive interaction.” This compares two separate tests, not the treatment risk differences. The correction is to calculate $RD_1-RD_0$ and assess uncertainty for that contrast. The two subgroup-specific estimates and their uncertainty may be informative, but the difference in significance labels is not itself evidence that the effects differ.

When reporting additive interaction, give the four risks or the two subgroup-specific risk differences, the contrast, its units, and the coding of treatment and subgroup. This makes it possible to check the calculation and see what the sign means. For example: “The estimated treatment risk differences were $-8$ percentage points in $S=0$ and $-16$ percentage points in $S=1$; the additive interaction, $RD_1-RD_0$, was $-8$ percentage points.”

Key takeaway: Additive interaction is the difference between subgroup-specific risk differences. Its sign and magnitude describe how much the absolute treatment effect differs between the coded subgroups; they do not replace the subgroup effects themselves.

Check Your Understanding

Use the definition and keep the treatment and subgroup coding explicit.

If $RD_0=-0.04$ and $RD_1=-0.12$, calculate the additive interaction and interpret its sign in percentage points.
Can two subgroups have different control risks and zero additive interaction? Explain using the definition.
For an adverse outcome, what does a positive additive interaction mean in general? Why should you inspect both subgroup risk differences before giving it a more specific interpretation?
Why does finding a statistically significant treatment effect in one subgroup and a nonsignificant effect in another not establish additive interaction?
If the subgroup labels $S=0$ and $S=1$ are exchanged, what happens to the sign and absolute magnitude of the additive interaction?