Additive Interaction on the Risk Scale

Why Measure Interaction on the Risk Scale?

When two binary factors are present, the joint risk of an outcome can be compared with the risks associated with each factor separately. On the risk scale, the central question is whether the joint effect differs from the sum of the two separate risk differences. This is a different question from whether a product term is nonzero in a model on another scale: interaction depends on the scale used to define it.

In gene–environment analysis, one factor may be a binary genotype contrast and the other a binary environmental exposure. The risk-scale definitions do not depend on which factor is genetic. They require the same four joint-exposure risks, measured for a clearly defined outcome and time horizon. A positive interaction contrast indicates that the joint risk exceeds the risk expected by adding the two separate risk differences to the baseline risk. It does not, by itself, establish a biological mechanism.

Four Joint-Exposure Risks

Let $G$ and $E$ be binary factors, each coded 0 for its reference category and 1 for its comparison category. Let $Y$ be a binary outcome, such as whether a disease occurs by a specified follow-up time. Define $p_{ge}$ as the risk of the outcome in the joint category $G=g,E=e$. Thus $p_{00}$ is the reference risk, $p_{10}$ and $p_{01}$ are the risks with one factor at a time, and $p_{11}$ is the risk with both factors present.

Definition: The additive interaction contrast (also called the risk-difference interaction) is $$ IC = p_{11}-p_{10}-p_{01}+p_{00}. $$ It is the difference between the risk difference for $G$ when $E=1$ and the risk difference for $G$ when $E=0$: $$ IC = (p_{11}-p_{01})-(p_{10}-p_{00}). $$

Equivalently, $IC$ compares the observed joint risk $p_{11}$ with the risk predicted by adding the separate risk differences to baseline: $$ p_{00}+(p_{10}-p_{00})+(p_{01}-p_{00}). $$ Subtracting this additive prediction from $p_{11}$ gives $p_{11}-p_{10}-p_{01}+p_{00}$. Therefore, $IC=0$ means no interaction on the additive risk scale; it does not mean that the risks are equal or that there is no association. A positive value means the joint risk is above the additive prediction, and a negative value means it is below it.

Relative and Proportional Measures

The interaction contrast is measured in risk units, such as percentage points. Three related measures rescale it or compare excess risks. Their formulas use the same four risks, but their domains and interpretations differ.

Definition: When $p_{00}>0$, define the relative risks $RR_{ge}=p_{ge}/p_{00}$. The relative excess risk due to interaction (RERI) is $$ RERI = RR_{11}-RR_{10}-RR_{01}+1 = \frac{p_{11}-p_{10}-p_{01}+p_{00}}{p_{00}} = \frac{IC}{p_{00}}. $$ Thus RERI is the interaction contrast expressed relative to the baseline risk.

RERI requires $p_{00}>0$, because its relative risks divide by the baseline risk. For a nonzero baseline risk, RERI is positive, zero, or negative exactly when the additive interaction contrast is positive, zero, or negative. RERI is dimensionless. For example, RERI $=1.6$ means the interaction contrast is 1.6 times the baseline risk; it does not mean the joint risk is 1.6 times the baseline risk.

Definition: When $p_{11}>0$, the attributable proportion due to interaction (AP) can be calculated directly as $$ AP = \frac{p_{11}-p_{10}-p_{01}+p_{00}}{p_{11}} = \frac{IC}{p_{11}}. $$ If $p_{00}>0$, this is also $AP=RERI/RR_{11}$. The direct risk formula requires $p_{11}>0$, but does not require $p_{00}>0$.

AP expresses the interaction contrast as a fraction of the joint-exposure risk. It can be positive, zero, or negative, and it is not automatically a probability between 0 and 1. The label “attributable” should be interpreted cautiously: a causal attribution requires causal risks and appropriate identification assumptions, not merely observed risks in four groups.

Definition: When $p_{00}>0$ and $(RR_{10}-1)+(RR_{01}-1)\ne 0$, the synergy index is $$ S = \frac{RR_{11}-1}{(RR_{10}-1)+(RR_{01}-1)}. $$ It compares the joint relative excess risk over baseline with the sum of the two individual relative excess risks.

If both individual factors increase risk, the denominator of $S$ is positive. In that setting, $S=1$ corresponds to additive risk excesses, $S>1$ indicates that the joint excess exceeds their sum, and $S<1$ indicates a smaller joint excess than their sum. If individual effects are protective or point in different directions, the denominator may be zero or negative; the ratio then does not support the usual “synergy” interpretation. Report the risks and $IC$ rather than forcing a directional label.

Worked Calculations

Worked Example: Positive Additive Interaction

Suppose a hypothetical cohort has the following risks of a binary outcome over the same follow-up period: $p_{00}=0.05$, $p_{10}=0.10$, $p_{01}=0.12$, and $p_{11}=0.25$. Assume these are risks in a common target population and that all four exposure categories are represented.

Interaction contrast. Substituting the four risks gives $$ IC=0.25-0.10-0.12+0.05=0.08. $$ The corresponding separate risk differences are $0.10-0.05=0.05$ and $0.12-0.05=0.07$, whose sum is $0.12$. The additive prediction for joint exposure is therefore $0.05+0.12=0.17$, and the observed joint risk $0.25$ exceeds it by $0.25-0.17=0.08$, consistent with the contrast.

RERI. The relative risks are $RR_{10}=0.10/0.05=2$, $RR_{01}=0.12/0.05=2.4$, and $RR_{11}=0.25/0.05=5$. Therefore $$ RERI=5-2-2.4+1=1.6. $$ This also equals $IC/p_{00}=0.08/0.05=1.6$. The interaction contrast is 1.6 times the baseline risk.

AP and synergy index. The attributable proportion is $AP=0.08/0.25=0.32$, or 32%. The synergy index is $$ S=\frac{5-1}{(2-1)+(2.4-1)} =\frac{4}{2.4}=1.667. $$ Both individual relative excess risks are positive, so $S>1$ indicates that the joint relative excess risk is greater than their sum. These calculations describe risk-scale contrasts; a causal interpretation would additionally require that the risks represent the relevant causal risks.

Worked Example: Negative Additive Interaction Despite Elevated Joint Risk

Consider another hypothetical set of risks: $p_{00}=0.10$, $p_{10}=0.20$, $p_{01}=0.25$, and $p_{11}=0.30$. The joint-exposure risk is still higher than baseline, but that fact alone does not determine whether additive interaction is positive.

Interaction contrast. $$ IC=0.30-0.20-0.25+0.10=-0.05. $$ The individual risk differences are $0.20-0.10=0.10$ and $0.25-0.10=0.15$. Their sum predicts a joint risk of $0.10+0.10+0.15=0.35$, which is 0.05 above the actual joint risk of 0.30. The interaction is negative on the additive risk scale.

Relative measures. The relative risks are $RR_{10}=2$, $RR_{01}=2.5$, and $RR_{11}=3$. Hence $$ RERI=3-2-2.5+1=-0.5, \qquad AP=\frac{-0.05}{0.30}=-0.1667. $$ The synergy index is $$ S=\frac{3-1}{(2-1)+(2.5-1)} =\frac{2}{2.5}=0.800. $$ Both individual effects increase risk, so the usual interpretation applies: the joint excess is smaller than the sum of the individual excesses. The negative AP is not a negative percentage of people; it is the interaction contrast divided by the joint risk.

Worked Example: Zero Baseline Risk and the Domain of AP

Suppose hypothetical risks are $p_{00}=0$, $p_{10}=0.10$, $p_{01}=0.20$, and $p_{11}=0.40$. This example shows why the direct AP formula has a broader domain than its expression using RERI.

Interaction contrast. $$ IC=0.40-0.10-0.20+0=0.10. $$ The additive prediction is $0+(0.10-0)+(0.20-0)=0.30$; the joint risk exceeds that prediction by 0.10.

AP. Since $p_{11}>0$, the direct risk formula is defined: $$ AP=\frac{0.40-0.10-0.20+0}{0.40} =\frac{0.10}{0.40}=0.25. $$ Thus the interaction contrast is one quarter of the joint risk. However, $RR_{ge}=p_{ge}/p_{00}$ cannot be calculated because $p_{00}=0$. Therefore RERI and the representation $AP=RERI/RR_{11}$ are undefined here. The synergy index, which also uses relative risks based on $p_{00}$, is undefined as well. One must not infer that AP is undefined merely because RERI is undefined.

Derivation and Conditions for Interpretation

The contrast follows directly from comparing the effect of one factor across the two levels of the other. At $E=1$, the risk difference for $G$ is $p_{11}-p_{01}$; at $E=0$, it is $p_{10}-p_{00}$. Subtracting the latter from the former gives $$ (p_{11}-p_{01})-(p_{10}-p_{00}) =p_{11}-p_{01}-p_{10}+p_{00} =p_{11}-p_{10}-p_{01}+p_{00}=IC. $$ Thus additive interaction is a difference of risk differences. If this contrast is zero, the risk difference associated with $G$ is the same at both levels of $E$, and equivalently the risk difference associated with $E$ is the same at both levels of $G$.

This algebra does not require a causal model. If $p_{ge}$ denotes observed risk among people with $G=g,E=e$, $IC$ describes an association contrast in those groups. To interpret it causally, define the risks as $p_{ge}=P(Y^{g,e}=1)$, where $Y^{g,e}$ is the potential outcome under joint exposure setting $(g,e)$. Identifying those causal risks from observed data requires assumptions such as consistency, positivity for all four exposure categories in the relevant covariate strata, and exchangeability of exposure assignment given the measured covariates. Without adequate control of confounding or other failures of identification, a calculated contrast is not necessarily causal.

The four risks must also refer to the same outcome definition, follow-up horizon, and target population. If estimates come from different populations or time horizons, their contrast may not answer a coherent interaction question. A numerical point estimate alone does not quantify sampling uncertainty; estimates near zero may be imprecise. The formulas define the estimand and its calculation, not an uncertainty interval or a test.

Common Mistakes and Practical Interpretation

Calling any elevated joint risk positive interaction. A joint risk can exceed baseline while $IC$ is negative, as in the second example. Compare it with the additive prediction, not baseline alone.
Confusing risk difference with relative risk. $IC$ is in risk units; RERI and $S$ are dimensionless ratios. State which measure is being reported and retain the risk units for $IC$.
Using RERI when the baseline risk is zero. RERI requires $p_{00}>0$. AP can still be computed by its direct formula if $p_{11}>0$, even though the RERI-based expression for AP is unavailable.
Interpreting AP as a guaranteed fraction of cases caused by interaction. AP is an algebraic proportion of the joint risk represented by the interaction contrast. Causal language requires causal risks and the assumptions needed to identify them; even then, a negative AP is possible.
Calling $S>1$ synergy without checking directions. The usual interpretation assumes positive individual excess risks, so the denominator is positive. If it is zero or negative, the index may be undefined or misleading for that label.

Key takeaway: Start with $IC=p_{11}-p_{10}-p_{01}+p_{00}$, which directly measures departure from additivity in risk units. RERI scales that contrast by $p_{00}$, while direct AP scales it by $p_{11}$. Check each denominator before calculating or interpreting a ratio.

Check Your Understanding

Use the four-risk definitions and show the calculation for each numerical answer.

If $p_{00}=0.08$, $p_{10}=0.14$, $p_{01}=0.16$, and $p_{11}=0.28$, calculate $IC$. Is the joint risk above or below its additive prediction?
For the risks in question 1, calculate RERI and AP. State what each denominator represents.
Can AP be calculated when $p_{00}=0$ and $p_{11}>0$? Explain why the answer differs for RERI.
Suppose both individual relative risks exceed 1 and $S=0.7$. What does this indicate about the joint relative excess risk compared with the sum of the individual relative excess risks?
What additional distinction must be made before describing an additive interaction contrast as causal rather than associational?