Interaction Across Risk Difference, Risk Ratio, and Odds Ratio

Why Compare Several Scales?

For a binary outcome, treatment effects can be expressed as differences in risks or as ratios of risks or odds. These scales answer different questions. A risk difference describes the change in outcome probability in percentage points; a risk ratio describes proportional change in risk; and an odds ratio compares the odds of the outcome. When comparing subgroups, the conclusion can depend on which of these contrasts is used.

Baseline risk matters because the same relative change can correspond to different absolute changes. Likewise, a fixed change in risk can represent different relative changes when the starting risks differ. This tutorial makes those relationships explicit and shows how to compare subgroup effects without treating one scale as universally correct.

Definition: Let $A$ denote a binary treatment, with $A=1$ for treatment and $A=0$ for control. Let $Y$ be a binary outcome, where $Y=1$ denotes the outcome of interest, and let $S$ identify a baseline subgroup. Write $Y(a)$ for the potential outcome under treatment level $a$, and define the subgroup-specific causal risk as $p_{as}=P(Y(a)=1\mid S=s)$. The subgroup risk difference, risk ratio, and odds ratio are, respectively, $$ RD_s=p_{1s}-p_{0s}, \qquad RR_s=\frac{p_{1s}}{p_{0s}}, \qquad OR_s=\frac{p_{1s}/(1-p_{1s})}{p_{0s}/(1-p_{0s})}. $$

The risk difference is measured in probability units, often reported as percentage points. A negative risk difference means the treatment reduces the outcome risk when $Y=1$ is adverse. A risk ratio of 1 and an odds ratio of 1 both correspond to equal treatment and control risks within that subgroup; values below 1 indicate a lower treated risk. The numerical sizes of a risk ratio and odds ratio are not directly interchangeable.

To compare effects between two subgroups, label them $S=0$ and $S=1$. The scale-specific interaction contrasts are:

I_{RD}=RD_1-RD_0,\qquad I_{RR}=\frac{RR_1}{RR_0},\qquad I_{OR}=\frac{OR_1}{OR_0}.

For the risk-difference scale, no difference between subgroup effects corresponds to $I_{RD}=0$. On the ratio scales, it corresponds to $I_{RR}=1$ or $I_{OR}=1$. These are comparisons of causal contrasts, not automatically regression coefficients. Whether a particular scale is useful depends on the scientific question and the audience for the result.

How Baseline Risk Connects the Scales

The risk difference and risk ratio are linked by the control risk:

RD_s=p_{0s}(RR_s-1).

Thus, if the risk ratio is the same across subgroups but control risks differ, the risk differences will generally differ. Conversely, equal risk differences need not imply equal risk ratios. The odds ratio also depends on baseline risk, even when the risk ratio is held fixed.

Key relationship: For $p_{0s}>0$, the odds ratio can be written in terms of the risk ratio and the control risk: $$ OR_s=RR_s\frac{1-p_{0s}}{1-RR_s p_{0s}}, $$ provided $RR_s p_{0s}<1$. This condition holds when the treated risk $p_{1s}$ is less than 1. The formula shows that a fixed risk ratio does not generally imply a fixed odds ratio across different baseline risks.

The relationship follows directly from the definitions. Substitute $p_{1s}=RR_s p_{0s}$ into the odds ratio:

OR_s =\frac{p_{1s}}{1-p_{1s}}\frac{1-p_{0s}}{p_{0s}} =\frac{RR_s p_{0s}}{1-RR_s p_{0s}}\frac{1-p_{0s}}{p_{0s}} =RR_s\frac{1-p_{0s}}{1-RR_s p_{0s}}.

This derivation also helps explain why odds ratios can look farther from 1 than risk ratios when outcomes are common. The comparison is especially important when people are accustomed to interpreting an odds ratio as though it were a risk ratio; the two are close when risks are small, but can differ substantially at higher risks.

Worked Comparisons

Worked Example: Equal Risk Ratios, Different Risk Differences

Consider a hypothetical binary adverse outcome. Suppose the control risk is 0.05 in subgroup $S=0$ and 0.30 in subgroup $S=1$. Treatment reduces risk by the same relative proportion in both subgroups, with a risk ratio of 0.80. Assume these subgroup-specific causal risks are identified, for example in a randomized study.

The treated risks are $0.05(0.80)=0.04$ in $S=0$ and $0.30(0.80)=0.24$ in $S=1$. Therefore:

RD_0=0.04-0.05=-0.01,\qquad RD_1=0.24-0.30=-0.06.

The risk-difference interaction is $I_{RD}=-0.06-(-0.01)=-0.05$, or 5 fewer percentage points of treatment-associated risk in subgroup 1 relative to subgroup 0. The risk-ratio interaction is $I_{RR}=0.80/0.80=1$: the relative reduction is the same.

The odds ratios are different. In subgroup 0, $OR_0=(0.04/0.96)/(0.05/0.95)=0.7917$. In subgroup 1, $OR_1=(0.24/0.76)/(0.30/0.70)=0.7368$. Thus $I_{OR}=0.7368/0.7917=0.9307$, not 1. These results do not conflict: the three interaction contrasts answer different scale-specific questions.

Worked Example: Equal Risk Differences, Different Relative Effects

In another hypothetical setting, suppose control risks are 0.05 and 0.30 in subgroups 0 and 1, respectively, and treatment lowers each risk by 0.02. The treated risks are therefore 0.03 and 0.28. The risk differences are:

RD_0=0.03-0.05=-0.02,\qquad RD_1=0.28-0.30=-0.02.

There is no risk-difference interaction: $I_{RD}=-0.02-(-0.02)=0$. But the risk ratios are $RR_0=0.03/0.05=0.6000$ and $RR_1=0.28/0.30=0.9333$. Their interaction contrast is $I_{RR}=0.9333/0.6000=1.556$. The same 2-percentage-point risk reduction represents a larger proportional reduction against the lower baseline risk.

The odds ratios also differ. In subgroup 0, $OR_0=(0.03/0.97)/(0.05/0.95)=0.5876$. In subgroup 1, $OR_1=(0.28/0.72)/(0.30/0.70)=0.9074$. Hence $I_{OR}=0.9074/0.5876=1.544$. Equal absolute changes do not imply equal relative changes on either ratio scale.

Worked Example: Equal Odds Ratios, Different Risks and Risk Ratios

Suppose a hypothetical treatment has an odds ratio of 0.50 in each of two subgroups. Let the control risk be 0.10 in subgroup 0 and 0.50 in subgroup 1. To find the treated risk $p_1$ from a control risk $p_0$ and odds ratio $OR$, solve $OR=(p_1/(1-p_1))/(p_0/(1-p_0))$ for $p_1$:

p_1=\frac{OR\,p_0}{1-p_0+OR\,p_0}.

For subgroup 0, $p_{10}=0.50(0.10)/(0.90+0.50(0.10))=0.05/0.95=1/19$, approximately 0.05263. For subgroup 1, $p_{11}=0.50(0.50)/(0.50+0.50(0.50))=0.25/0.75=1/3$, approximately 0.3333. In both subgroups the odds ratio is 0.50 by construction.

The risk differences are $RD_0=1/19-0.10=-0.04737$ and $RD_1=1/3-0.50=-0.1667$. The risk ratios are $RR_0=(1/19)/0.10=0.5263$ and $RR_1=(1/3)/0.50=0.6667$. Thus a common odds ratio does not mean equal absolute or relative risk changes. Here the higher-baseline-risk subgroup has the larger absolute reduction, while its risk ratio is closer to 1.

Interpreting and Reporting the Comparison

A useful report gives subgroup risks as well as the contrasts. Risks show the starting points and make it possible to understand what a ratio means in absolute terms. For example, “the risk ratio was 0.80 in both subgroups” communicates equal proportional risk reductions, but does not communicate equal absolute reductions if the control risks differ. Conversely, reporting only risk differences can conceal differences in proportional effect.

These causal interpretations require a design or assumptions that identify the subgroup potential-outcome risks. In a randomized trial, treatment assignment must be randomized within the relevant subgroups, with positive probability of each treatment level. In an observational setting, a simple identification argument requires consistency, conditional exchangeability of treatment and potential outcomes within subgroup, and positivity of treatment assignment within subgroup. If measured covariates are needed to make exchangeability plausible, subgroup risks must be adjusted or standardized over those covariates rather than treated as raw risks.

In every case, the outcome, treatment, subgroup definition, and scale should be stated. For an adverse outcome, a risk difference of $-0.02$ means a 2-percentage-point lower risk under treatment than control in that subgroup. A risk ratio of 0.80 means the treated risk is 80% of the control risk, or 20% lower relative to control. Neither statement alone establishes that the treatment caused the observed contrast; causal language depends on the identification conditions.

Common Mistake: Treating “No Interaction” as Scale-Free

A common mistake is to say that effects are “the same across subgroups” after finding the same risk ratio, and then assume the risk differences and odds ratios must also be the same. That conclusion does not follow. The first worked example has equal risk ratios but unequal risk differences and odds ratios. The correction is to name the scale: effects can be equal on one scale and unequal on another, because each scale compares risks differently and depends differently on baseline risk.

There is no single scale that answers every practical question. If the decision concerns how many adverse outcomes might be prevented, the risk difference is often directly relevant. If the question concerns proportional change, the risk ratio may be more informative. Odds ratios may be appropriate for particular analyses, but they should be interpreted as odds contrasts, not casually relabeled as risk ratios.

Key takeaway: Always show or consider subgroup-specific baseline risks alongside treatment contrasts. State whether the comparison is on the risk-difference, risk-ratio, or odds-ratio scale, and interpret the interaction contrast on that same scale.

Check Your Understanding

Use the definitions and examples above to answer the following questions.

If two subgroups have the same risk ratio but different control risks, what relationship determines whether their risk differences are also equal?
For a risk-ratio interaction contrast, what value represents equal risk ratios across two subgroups? How does that value differ for a risk-difference interaction contrast?
A treatment lowers risk from 0.20 to 0.10 in one subgroup and from 0.60 to 0.50 in another. Calculate both risk differences and risk ratios, and describe which scale shows equal effects.
Why can the odds ratio and risk ratio differ more when the outcome is common than when it is rare?
What assumptions or study-design features are needed before observed subgroup risks can be interpreted as causal risks?