Multiplicative Interaction and Relative Effects

Why Compare Relative Effects Across Subgroups?

An intervention may change risk by a different proportion in different subgroups. For example, treatment might double risk in one subgroup but increase it by only one-half in another. A multiplicative interaction contrast compares those subgroup-specific relative effects. Its value depends on which relative measure is chosen: a risk ratio and an odds ratio do not generally give the same comparison.

Here, treatment \(A\) is binary, with \(A=1\) denoting treatment and \(A=0\) control. Let \(S\) denote a binary subgroup characteristic, with \(S=0\) and \(S=1\) naming the two subgroups. The outcome \(Y\) is binary, and \(Y=1\) denotes the event of interest. These definitions concern causal effects of treatment within subgroups; a difference in observed outcomes alone does not establish a causal effect.

Definition: Let \(Y(a)\) be the potential outcome under treatment level \(a\), and define the subgroup-specific risk \(r_{as}=P(Y(a)=1\mid S=s)\). The subgroup-specific causal risk ratio is \(RR_s=r_{1s}/r_{0s}\). The multiplicative interaction on the risk-ratio scale is the ratio of risk ratios, \(M_{RR}=RR_1/RR_0\). A value of 1 means the two subgroup-specific risk ratios are equal; a value above or below 1 means they differ in relative magnitude.

The subscript \(s\) identifies the subgroup, not the treatment level: \(RR_0\) compares treatment with control among people with \(S=0\), while \(RR_1\) makes the same comparison among people with \(S=1\). The ratio \(M_{RR}\) compares those two treatment effects. It is not itself a risk ratio comparing two treatment groups.

Calculating and Interpreting a Ratio of Risk Ratios

Suppose treatment is randomized within each subgroup, the treatment and outcome are well defined, and every subgroup has a positive probability of receiving either treatment. With consistency and no interference, the observed risk in each treatment-by-subgroup cell estimates the corresponding potential-outcome risk. Under these conditions, the cell risks can be used to estimate the causal \(RR_s\) values.

Worked Example: Comparing Risk Ratios in Two Subgroups

In a hypothetical randomized study, the event counts are as follows. The treatment was assigned within each subgroup, and each cell contains 200 people.

Subgroup	Control events / total	Treatment events / total
\(S=0\)	20 / 200	40 / 200
\(S=1\)	30 / 200	45 / 200

Among \(S=0\), the control risk is \(20/200=0.10\) and the treatment risk is \(40/200=0.20\). Thus,

RR_0=\frac{0.20}{0.10}=2.00.

Among \(S=1\), the control risk is \(30/200=0.15\) and the treatment risk is \(45/200=0.225\). Therefore,

RR_1=\frac{0.225}{0.15}=1.50, \qquad M_{RR}=\frac{1.50}{2.00}=0.750.

The estimated treatment risk ratio is 25% smaller in \(S=1\) than in \(S=0\), since \(1-0.750=0.250\). In both subgroups treatment is associated with higher risk in these hypothetical data, but the relative increase is smaller in \(S=1\). This describes multiplicative effect modification on the risk-ratio scale; it does not imply that the treatment effect is smaller by the same amount in absolute risk units.

The algebra makes the comparison explicit. By definition,

M_{RR} =\frac{r_{11}/r_{01}}{r_{10}/r_{00}} =\frac{r_{11}r_{00}}{r_{01}r_{10}}.

Here, \(r_{11}\) is the treated risk in \(S=1\), \(r_{01}\) is the control risk in \(S=1\), \(r_{10}\) is the treated risk in \(S=0\), and \(r_{00}\) is the control risk in \(S=0\). This rearrangement shows that the contrast is a cross-product of four risks, not a comparison of the two treated risks alone. Equivalently, taking logarithms gives

\log(M_{RR})=\log(RR_1)-\log(RR_0).

Thus, the multiplicative interaction is zero on the log-ratio scale exactly when \(M_{RR}=1\), or \(RR_1=RR_0\). The ratio’s direction depends on which subgroup is placed in the numerator. Swapping subgroup labels changes \(M_{RR}\) to its reciprocal but does not change whether the subgroup-specific relative effects are equal.

Odds-Ratio Interaction Is a Different Contrast

A second relative-effect measure is the odds ratio. For a risk \(r\), the odds of the event are \(r/(1-r)\). The subgroup-specific causal odds ratio and its multiplicative interaction contrast are

OR_s= \frac{r_{1s}/(1-r_{1s})}{r_{0s}/(1-r_{0s})}, \qquad M_{OR}=\frac{OR_1}{OR_0}.

As with the ratio of risk ratios, \(M_{OR}=1\) means the subgroup-specific odds ratios are equal. But odds and risks are different quantities, so a ratio of odds ratios need not equal a ratio of risk ratios. The chosen measure is part of the definition of the interaction being reported.

Worked Example: Calculating a Ratio of Odds Ratios

Consider another hypothetical randomized study with 100 people in each treatment-by-subgroup cell. The event counts are 20 and 40 for control and treatment, respectively, in \(S=0\), and 50 and 60 in \(S=1\).

For \(S=0\), the control odds are \(0.20/0.80=0.25\), and the treatment odds are \(0.40/0.60=0.6667\). Hence,

OR_0=\frac{0.40/0.60}{0.20/0.80} =\frac{0.6667}{0.25}=2.667.

For \(S=1\), the control odds are \(0.50/0.50=1.00\), and the treatment odds are \(0.60/0.40=1.50\). Therefore,

OR_1=\frac{1.50}{1.00}=1.50, \qquad M_{OR}=\frac{1.50}{2.667}=0.5625.

The treatment odds ratio is 43.75% smaller in \(S=1\) than in \(S=0\), because \(1-0.5625=0.4375\). For these same risks, the risk ratios are \(0.40/0.20=2.00\) in \(S=0\) and \(0.60/0.50=1.20\) in \(S=1\), giving \(M_{RR}=1.20/2.00=0.600\). The two interaction contrasts differ. Both calculations are correct for their stated measures; neither should be silently substituted for the other.

Obtaining Causal Risks With Covariates

In an observational study, treatment groups may differ in factors that predict the outcome. Within each subgroup, a causal interpretation requires that measured covariates \(C\) suffice to control confounding, along with consistency and positivity. One way to estimate the subgroup-specific risks is to standardize the treatment-specific outcome risks over the covariate distribution in that subgroup:

r_{as} =\sum_c P(Y=1\mid A=a,S=s,C=c)\,P(C=c\mid S=s).

This identifies the potential-outcome risk under conditional exchangeability, \(Y(a)\perp A\mid (S,C)\), for each treatment level and subgroup, provided that each relevant covariate stratum has a positive probability of receiving both treatment levels. The standardization weights \(P(C=c\mid S=s)\) define the covariate distribution for the subgroup whose effect is being estimated.

Worked Example: Standardizing Before Comparing Risk Ratios

Suppose an observational study measures a binary prognostic covariate \(C\), classified as low or high. Assume, for this hypothetical example, that within each \(S,C\) combination treatment is conditionally exchangeable, both treatment levels occur, and the recorded conditional risks are consistent with the treatment received. The proportions and conditional risks are:

Subgroup	Covariate distribution	Control risk by \(C\)	Treatment risk by \(C\)
\(S=0\)	50% low, 50% high	0.10 low, 0.30 high	0.20 low, 0.45 high
\(S=1\)	80% low, 20% high	0.20 low, 0.40 high	0.30 low, 0.48 high

For \(S=0\), standardize both treatment risks to the 50%/50% covariate distribution:

r_{00}=0.50(0.10)+0.50(0.30)=0.200, \qquad r_{10}=0.50(0.20)+0.50(0.45)=0.325.

The subgroup risk ratio is \(RR_0=0.325/0.200=1.625\). For \(S=1\), use that subgroup’s 80%/20% covariate distribution:

r_{01}=0.80(0.20)+0.20(0.40)=0.240, \qquad r_{11}=0.80(0.30)+0.20(0.48)=0.336.

Thus \(RR_1=0.336/0.240=1.400\), and the standardized ratio of risk ratios is

M_{RR}=\frac{1.400}{1.625}=0.8615.

Under the stated assumptions, the estimated treatment risk ratio is about 13.85% smaller in \(S=1\) than in \(S=0\). The standardization is important: it compares treatment and control risks using the same covariate distribution within each subgroup, rather than attributing differences in covariate composition to treatment.

Assumptions, Scope, and Common Mistakes

A ratio of subgroup-specific relative effects is a causal contrast only when its component risks are causal risks. In a randomized study, randomization within subgroups supports identification; in an observational study, identification depends on the relevant exchangeability, positivity, and consistency conditions. If people’s outcomes also depend on other people’s treatment, the potential outcomes must specify the relevant exposure mapping or spillover conditions. The examples here assume no interference, or an exposure regime held fixed.

The relative-effect measure also matters. A ratio of risk ratios compares proportional changes in risk; a ratio of odds ratios compares proportional changes in odds. These are not interchangeable, particularly when events are common. A result for one measure does not, by itself, establish the same interaction conclusion for another measure.

Mistake: “The treated risk is higher in one subgroup, so there is multiplicative interaction.” The interaction contrast compares the treatment-to-control relative effect in each subgroup; it is not a comparison of treated risks alone.
Mistake: “A ratio of odds ratios is a ratio of risk ratios.” Odds and risks have different denominators. Calculate and label the chosen measure explicitly.
Mistake: “An interaction ratio of 1 means treatment has no effect.” It means the relative treatment effects are equal across the two subgroups. Both subgroup-specific ratios could be greater than 1, less than 1, or equal to 1.
Mistake: “Observed subgroup ratios are automatically causal.” In observational data, confounding control and positivity are needed to identify the causal risks used in those ratios.

Key takeaway: Define the subgroup-specific relative effects first, then divide one by the other. A ratio of risk ratios equals 1 when subgroup risk ratios are equal; a ratio of odds ratios answers the analogous question for odds. Always state the measure and the assumptions that support a causal interpretation.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

In \(S=0\), treatment risk is 0.12 and control risk is 0.06. In \(S=1\), treatment risk is 0.18 and control risk is 0.12. Calculate both risk ratios and their ratio.
What does a ratio of risk ratios of 1 mean? What does it not imply about the treatment effect?
Why can a ratio of odds ratios differ from a ratio of risk ratios even when calculated from the same four risks?
In an observational study, name the assumptions needed to interpret standardized subgroup risks as causal risks.
If the subgroup labels are reversed, what happens to the ratio of risk ratios, and does the equality-of-effects conclusion change?