Why the Scale Changes the Question
A treatment can have different subgroup effects when effects are measured as differences in risk, yet the same effects when measured as ratios of risk. This is not a contradiction. The two measures ask different questions: a risk difference compares absolute changes in outcome probability, while a risk ratio compares relative changes. Consequently, asking whether treatment effect varies by subgroup is incomplete unless the effect scale is stated.
The distinction matters for interpretation. An absolute change can describe how many additional or fewer outcomes occur per person treated, while a relative change describes how large the treated risk is compared with the untreated risk. Subgroups can have different baseline risks, so the same relative change can correspond to different absolute changes.
The sign of a risk difference indicates direction: a negative value means treatment lowers the outcome risk compared with control, and a positive value means it raises that risk. A risk ratio below 1 indicates a lower risk under treatment; a ratio above 1 indicates a higher risk. In all cases, the outcome and subgroup need to be described in context.
A Link Between Relative and Absolute Effects
For any subgroup with a positive control risk, the risk difference and risk ratio are linked by a simple identity. Starting with \(RR_s=p_{1s}/p_{0s}\), multiply both sides by \(p_{0s}\) to obtain \(p_{1s}=p_{0s}RR_s\). Subtract \(p_{0s}\) from both sides:
Thus, for the same risk ratio, the risk difference depends on the control risk. If the relative effect is a doubling of risk, for example, the absolute increase is twice the control risk: a control risk of 0.10 gives an increase of 0.10, while a control risk of 0.30 gives an increase of 0.30. Conversely, the same absolute increase can yield different risk ratios when the control risks differ.
This identity explains the scale dependence without implying that one scale is correct and the other is wrong. Each is a valid summary, provided its meaning fits the question. It also shows why a subgroup comparison of treatment effects cannot be interpreted without specifying which contrast is being compared.
Worked Example: Same Risk Ratio, Different Risk Differences
Suppose a hypothetical randomized trial compares a preventive treatment with control in two prespecified subgroups. In each subgroup and treatment arm, 1,000 people are observed. Assume treatment was randomized within each subgroup, each person’s outcome depends only on their own assigned treatment, and the observed outcome is the outcome under the assigned treatment. These conditions allow the arm-specific subgroup proportions to estimate the subgroup causal risks; the calculations below are point estimates, not a claim that the risks are known without sampling error.
| Subgroup | Control outcomes | Treatment outcomes | Control risk | Treatment risk |
|---|---|---|---|---|
| A | 100 of 1,000 | 200 of 1,000 | 0.10 | 0.20 |
| B | 300 of 1,000 | 600 of 1,000 | 0.30 | 0.60 |
For subgroup A, the estimated risk difference is \(200/1000-100/1000=0.20-0.10=0.10\), or 10 percentage points. Its estimated risk ratio is \((200/1000)/(100/1000)=0.20/0.10=2.00\).
For subgroup B, the estimated risk difference is \(600/1000-300/1000=0.60-0.30=0.30\), or 30 percentage points. Its estimated risk ratio is \((600/1000)/(300/1000)=0.60/0.30=2.00\).
The estimated treatment effects differ on the risk-difference scale: the increase is 10 percentage points in A and 30 percentage points in B. They do not differ on the risk-ratio scale: each subgroup has a treatment risk twice its control risk. The same subgroup risks therefore support effect modification on one scale but not the other.
The interpretation is scale-specific. These data suggest a larger absolute increase in subgroup B, while the relative increase is the same in both subgroups. Neither statement should be substituted for the other.
The Reverse Pattern Is Also Possible
The pattern can reverse. If two subgroups have the same risk difference but different control risks, their risk ratios will generally differ. This follows from \(RR_s=1+RD_s/p_{0s}\): holding \(RD_s\) fixed while changing \(p_{0s}\) changes the ratio.
Worked Example: Same Risk Difference, Different Risk Ratios
Consider another hypothetical randomized comparison, with 1,000 people in each subgroup-treatment combination. Suppose the observed outcome counts are as follows.
| Subgroup | Control outcomes | Treatment outcomes | Control risk | Treatment risk |
|---|---|---|---|---|
| A | 100 of 1,000 | 200 of 1,000 | 0.10 | 0.20 |
| B | 400 of 1,000 | 500 of 1,000 | 0.40 | 0.50 |
For subgroup A, the estimated risk difference is \(0.20-0.10=0.10\), or 10 percentage points, and the estimated risk ratio is \(0.20/0.10=2.00\).
For subgroup B, the estimated risk difference is \(0.50-0.40=0.10\), also 10 percentage points, and the estimated risk ratio is \(0.50/0.40=1.25\).
The estimated absolute increase is the same in both subgroups, so these point estimates show no effect modification on the risk-difference scale. The relative increase differs: the risk is 100% higher in A and 25% higher in B. Thus the estimates show effect modification on the risk-ratio scale. As in the first example, these are sample estimates; uncertainty would matter when drawing conclusions beyond the observed data.
Interpreting Subgroup Risks Causally
The scale comparison concerns causal subgroup risks, not merely observed associations. In a randomized trial, treatment assignment must be randomized within the subgroups being compared, or otherwise be independent of the potential outcomes within each subgroup. Each treatment level must have a positive probability of assignment in each subgroup, so both arms can be compared there. Consistency requires that the observed outcome under the assigned treatment equals the corresponding potential outcome. The examples also assume no interference: one person’s outcome is not changed by another person’s treatment.
Under these conditions, the observed risk in treatment arm \(a\) within subgroup \(s\) estimates \(P(Y(a)=1\mid S=s)\). In an observational study, randomization is absent, so causal interpretation requires additional identification assumptions, such as conditional exchangeability given measured covariates, as well as consistency and positivity. Merely calculating subgroup proportions does not establish those conditions.
The subgroup definition also matters. A contrast among people classified by a characteristic measured before treatment answers a different question from one defined by a characteristic that treatment itself could change. For the basic comparisons here, treat subgroup membership as fixed and defined independently of treatment assignment.
Worked Example: Equal Relative Reductions With Different Absolute Benefits
Suppose a hypothetical randomized treatment is evaluated for a harmful binary outcome in two subgroups. In each subgroup and arm, 1,000 people are observed. Assume randomized assignment within subgroups, consistency, positivity, and no interference, so the observed risks can be used as estimates of the corresponding subgroup causal risks.
| Subgroup | Control outcomes | Treatment outcomes | Control risk | Treatment risk |
|---|---|---|---|---|
| A | 400 of 1,000 | 200 of 1,000 | 0.40 | 0.20 |
| B | 100 of 1,000 | 50 of 1,000 | 0.10 | 0.05 |
For subgroup A, the estimated risk difference is \(0.20-0.40=-0.20\), a reduction of 20 percentage points, and the estimated risk ratio is \(0.20/0.40=0.50\).
For subgroup B, the estimated risk difference is \(0.05-0.10=-0.05\), a reduction of 5 percentage points, and the estimated risk ratio is \(0.05/0.10=0.50\).
The relative reduction is 50% in each subgroup, since a risk ratio of 0.50 means the treatment risk is half the control risk. The absolute reduction is four times larger in A than in B: 20 rather than 5 percentage points. A report that says only “the treatment effect is the same” would omit which of these distinct comparisons it means.
Choosing and Reporting the Scale
There is no scale-free definition of whether subgroup effects differ. The risk difference is expressed in probability points, so it directly describes an absolute contrast. The risk ratio is dimensionless and describes a relative contrast. Which is most useful depends on the scientific and decision context. If the concern is the number of outcomes prevented or caused per treated person, an absolute measure is often directly relevant. If the question concerns proportional change in risk, a relative measure may be more informative.
When reporting subgroup comparisons, state the subgroup risks and the scale-specific contrasts, or make the chosen scale explicit. “There is no effect modification” should be qualified: for example, “the estimated risk ratios are equal across these subgroups” is precise, while an unqualified statement may incorrectly suggest equality on every scale. Likewise, a difference between subgroup estimates should not automatically be treated as reliable evidence of a population difference; sampling uncertainty is separate from the definition of effect modification.
Common Mistake
- Wrong reasoning: “The relative effect is the same in both subgroups, so the treatment effect does not vary.”
- Correction: Equal risk ratios establish equality only on the risk-ratio scale. If baseline risks differ, the risk differences can still differ, as in the first worked example.
- Wrong reasoning: “The risk difference is the same, so the relative effects must also be the same.”
- Correction: Equal risk differences do not establish equal risk ratios when control risks differ, as in the second worked example.
Check Your Understanding
Use the definitions and examples above to answer the following questions.
- In subgroup A, the control risk is 0.15 and the treatment risk is 0.30. In subgroup B, the control risk is 0.25 and the treatment risk is 0.50. Compare the risk differences and risk ratios across the subgroups.
- Two subgroups each have an estimated risk difference of 0.08, but their control risks are 0.10 and 0.40. Calculate both risk ratios and explain what they imply about scale-specific effect modification.
- Explain why a risk ratio of 0.70 represents the same relative reduction in two subgroups but need not represent the same absolute reduction.
- Name conditions that allow observed randomized-trial risks within a subgroup to estimate its causal risks.
- Rewrite the statement “there is no effect modification” so that it clearly specifies a scale.