Why the Same Relative Effect Can Mean Different Benefits
A treatment can have the same proportional effect in two populations yet prevent different numbers of events. The reason is that the populations may start at different levels of untreated risk. If an intervention reduces risk by the same fraction in both groups, a population with more events to prevent can have a larger absolute reduction. In drug development, confusing these two features can lead to an incorrect expectation of clinical benefit when results are applied to a new population.
The distinction is especially useful when an effect estimate appears stable across populations but the expected event rate differs. The treatment effect and baseline risk should be described separately, then combined to obtain the absolute outcome expected in the target population. The introductory material on standardization establishes how to average conditional outcomes over a target covariate distribution; here the focus is what those averages reveal about baseline risk versus treatment effect.
These are population-level quantities for the stated outcome, time horizon, treatment strategy, and target population. The risk ratio compares treated and control risks; the absolute reduction subtracts those risks. Neither should be replaced by a conditional estimate without saying so. For example, a risk ratio within each age stratum is not automatically the same quantity as the risk ratio after averaging risks over the whole target population.
Conditions for Separating the Components
A calculation is only meaningful if the source and target quantities refer to comparable questions. The endpoint definition, follow-up window, treatment strategies, and handling of intercurrent events must align. The earlier material on aligning estimands emphasizes this comparability: a difference in outcome definition cannot be repaired by arithmetic standardization.
For a causal interpretation, treatment assignment in the source evidence must support estimation of the conditional risks under both strategies. In a randomized trial, randomization supports this within the trial, subject to appropriate conduct and outcome ascertainment. In nonrandomized evidence, conditional exchangeability requires that, after accounting for the relevant measured covariates, treatment assignment is independent of the potential outcomes. Transport to the target additionally requires that the conditional outcome relationships used for prediction apply there, conditional on the variables represented in the model.
- Consistency and a well-defined intervention: the observed outcome under the assigned strategy corresponds to the outcome under that strategy, and the strategy has a sufficiently clear meaning.
- Conditional exchangeability for treatment effects: within levels of measured covariates, the source data identify the risks under treatment and control. Randomization is one way to support this condition.
- Transportability conditional on measured covariates: after conditioning on the covariates used for standardization, relevant conditional risks are applicable to the target.
- Positivity and overlap: the source evidence must provide support for the covariate patterns that occur in the target, and both treatment strategies must be represented where needed.
- Aligned outcome and time horizon: baseline and treated risks must refer to the same binary endpoint and period in both populations.
These conditions are substantive, not merely computational. A fitted model can produce a target risk even when the target contains covariate patterns absent from the source, but that number depends on extrapolation rather than supported transport. Likewise, a stable relative effect does not rule out unmeasured effect modification or different treatment implementation in the target.
An Exact Decomposition
The following identity separates the difference in absolute benefit between target and source into a component associated with baseline risk and a component associated with relative effect. It is an algebraic decomposition, not by itself evidence that either component is causal.
To verify it, use \(B_g=R_{0g}-R_{1g}=R_{0g}(1-Q_g)\), which follows from \(Q_g=R_{1g}/R_{0g}\). Subtract the source benefit from the target benefit and add and subtract \(R_{0T}(1-Q_S)\):
Under this ordering, the first term is the baseline-risk component: it asks how much absolute benefit would differ if the source relative risk applied at the target untreated risk. The second term is the relative-effect component, evaluated at the target untreated risk. The allocation depends on the chosen reference population and order of replacement; another valid decomposition can allocate the interaction differently. The total \(B_T-B_S\), however, does not depend on that choice.
In the special case \(Q_S=Q_T=q\), the relative-effect component is zero and the identity simplifies to \(B_T-B_S=(R_{0T}-R_{0S})(1-q)\). Thus a common relative risk can coexist with different absolute reductions whenever untreated risks differ and \(q\ne1\). Conversely, equal untreated risks do not guarantee equal benefits if the relative effects differ.
Worked Example: A Common Risk Ratio, Different Absolute Benefits
Suppose a hypothetical source population has 70% low-risk and 30% high-risk participants, while the target population has 40% low-risk and 60% high-risk participants. The untreated event risks are 10% in the low-risk stratum and 40% in the high-risk stratum. Assume treatment halves risk in each stratum, so the treated risks are 5% and 20%, respectively. The outcome and follow-up period are identical in both populations, and assume the conditional risks are valid for the target.
For the source population, standardize the untreated and treated risks using source weights:
Therefore \(B_S=0.19-0.095=0.095\), or 9.5 percentage points, and \(Q_S=0.095/0.19=0.50\).
For the target population, use its own covariate weights:
Thus \(B_T=0.28-0.14=0.14\), or 14 percentage points, and \(Q_T=0.14/0.28=0.50\). Both strata have a risk ratio of 0.50, and the standardized risk ratio is also 0.50 because the same conditional multiplier applies in each stratum. The target absolute reduction is 4.5 percentage points larger than the source reduction; the target has more high-risk participants, not a stronger relative treatment effect.
The decomposition gives the same result. The baseline-risk component is \((0.28-0.19)(1-0.50)=0.09(0.50)=0.045\). The relative-effect component is \(0.28(0.50-0.50)=0\). Their sum, 0.045, equals \(B_T-B_S=0.14-0.095\).
When Relative Effects Also Differ
The preceding example depends on a common conditional risk ratio. If the relative effect varies with covariates, changing the covariate mix can change the standardized risk ratio as well as the absolute benefit. Even if the conditional effects are known, the population-level ratio is a ratio of standardized risks, not generally a weighted average of the stratum-specific ratios.
Worked Example: Decomposing a Difference With Effect Modification
Use the same hypothetical population proportions and untreated stratum risks as above. Now suppose the treatment risk ratio is 0.60 in the low-risk stratum and 0.40 in the high-risk stratum. The resulting treated risks are \(0.10(0.60)=0.06\) and \(0.40(0.40)=0.16\). The covariate-specific effects are assumed applicable to both populations.
In the source population:
So \(B_S=0.19-0.09=0.10\), or 10 percentage points, and \(Q_S=0.09/0.19\approx0.4737\).
In the target population:
Thus \(B_T=0.28-0.12=0.16\), or 16 percentage points, and \(Q_T=0.12/0.28\approx0.4286\). The high-risk stratum, which is more common in the target, also has the stronger relative reduction.
The target-source benefit difference is \(0.16-0.10=0.06\). Using the stated decomposition, the baseline-risk component is \((0.28-0.19)(1-0.4737)\approx0.04737\), and the relative-effect component is \(0.28(0.4737-0.4286)\approx0.01263\). Together they equal \(0.06000\), up to rounding. This example shows why a population-level relative effect may change when treatment-effect modifiers are distributed differently.
Using a Transported Relative Effect to Predict Absolute Benefit
When evidence supports a common relative risk \(q\) in the target, a target untreated risk can be combined with it to predict the target treated risk and absolute reduction:
This calculation requires more than knowing a source relative effect. It requires that the relative effect applies to the target under the specified treatment and endpoint, and that the target baseline risk is appropriately estimated for the same time horizon and covariate distribution. If \(q\) is transported only conditionally, estimate the relevant conditional risks and standardize them; do not assume that a conditional ratio equals the marginal target ratio in the presence of effect modification.
Worked Example: Converting a Transported Relative Effect Into Benefit
Suppose a hypothetical source analysis supports a risk ratio of 0.60, and substantive evidence supports applying that same relative risk to a target population. An independently estimated target untreated risk over the aligned follow-up period is 25%. Assume the endpoint, treatment strategy, and time horizon match, and that the target baseline-risk estimate is suitable for the population of interest.
The predicted treated risk is \(R_{1T}=0.60(0.25)=0.15\), or 15%. The predicted absolute reduction is \(B_T=0.25-0.15=0.10\), or 10 percentage points. The interpretation is that, under the stated transport assumptions, the treatment is predicted to reduce the event risk from 25% to 15% in this target population.
For comparison, if the source untreated risk were 10% with the same ratio of 0.60, the source treated risk would be \(0.60(0.10)=0.06\), and its absolute reduction would be \(0.10-0.06=0.04\), or 4 percentage points. Carrying the 4-point source reduction directly into the target would therefore not match the 10-point reduction implied by the target risk and transported ratio. These are hypothetical population risks, not a claim about any particular medicine or study.
Diagnostics, Alternatives, and Limitations
Before interpreting the decomposition, examine whether its ingredients are supported by the data and assumptions. Useful checks include comparing source and target covariate distributions, checking whether target patterns lie within source support, and inspecting calibration of the conditional untreated-risk model where suitable outcome data are available. Examine treatment-effect estimates across covariate strata or clinically justified subgroups when the sample size and design permit. Large differences may indicate effect modification, sparse data, or model misspecification; they should not be resolved by simply declaring the relative effect constant.
A practical diagnostic sequence is:
Confirm that populations, treatment strategies, outcome, and time horizon define the same estimand.
Determine whether the target covariate patterns are represented in the source evidence and whether each treatment strategy is supported where required.
Review the target untreated-risk estimate and the conditional treatment-effect model rather than checking only the final absolute benefit.
Standardize both treatment and control risks over each population's covariate distribution, then form the absolute and relative contrasts.
The main alternative to transporting a relative effect and combining it with target baseline risk is to transport an absolute treatment effect directly. That can be appropriate when the absolute effect is plausibly stable for the target and the source and target baseline risks are similar. It is risky when baseline risk differs substantially: under a common risk ratio, the absolute reduction necessarily scales with untreated risk. Direct standardization of both conditional risks is often more transparent when covariate-specific outcome predictions are credible. Neither approach is a substitute for checking transport assumptions.
The decomposition is scale-dependent. A constant risk ratio and a constant risk difference are different assumptions, and neither implies the other in general. A constant risk difference would yield the same absolute benefit across populations if it truly applied, while the resulting relative risk could vary with baseline risk. Results can also depend on the outcome frequency, follow-up period, competing events, missing data, and unmeasured effect modifiers. Uncertainty in the baseline-risk estimate and relative-effect estimate should be carried into the predicted absolute benefit; a point calculation alone does not show how precise that prediction is.
Common Mistake
- Wrong reasoning: “The risk ratio is 0.50 in both populations, so the treatment must prevent the same percentage-point risk in both.”
- Correction: A risk ratio describes proportional change, not the number of events prevented. If the control risks are 10% and 28%, a common ratio of 0.50 implies absolute reductions of \(0.10(1-0.50)=0.05\), or 5 percentage points, and \(0.28(1-0.50)=0.14\), or 14 percentage points.
A related arithmetic error is to average stratum-specific treated risks using the wrong population weights. Always calculate \(R_{1g}\) using the covariate distribution of population \(g\); then calculate \(B_g=R_{0g}-R_{1g}\) and \(Q_g=R_{1g}/R_{0g}\) from those same standardized risks. Mixing weights or contrasts can produce internally inconsistent conclusions.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- A hypothetical target population has untreated risk 20% and a transported risk ratio of 0.75. What are its predicted treated risk and absolute reduction?
- Why can equal conditional risk ratios across strata yield different absolute reductions in source and target populations?
- For a source with \(R_{0S}=0.12\), \(Q_S=0.50\), and a target with \(R_{0T}=0.24\), \(Q_T=0.50\), calculate both absolute benefits and identify the decomposition components.
- If relative effects differ across covariate strata, why is it generally incorrect to average the stratum-specific risk ratios and call the result the standardized risk ratio?
- Name two conditions or diagnostics that should be checked before interpreting a transported target absolute benefit.
Solutions. (1) \(R_{1T}=0.75(0.20)=0.15\), or 15%, and \(B_T=0.20-0.15=0.05\), or 5 percentage points. (2) With a common ratio \(q\), \(B_g=R_{0g}(1-q)\); different untreated risks therefore give different absolute reductions. (3) \(B_S=0.12(1-0.50)=0.06\), or 6 percentage points, and \(B_T=0.24(1-0.50)=0.12\), or 12 percentage points. The difference is 6 points, entirely the baseline-risk component: \((0.24-0.12)(1-0.50)=0.06\); the relative-effect component is zero. (4) The standardized ratio is the ratio of standardized treated and untreated risks, \(E_g[m_1(X)]/E_g[m_0(X)]\), not generally the average of stratum ratios. The stratum weights in that ratio are influenced by baseline risks. (5) Examples include aligned endpoint and time horizon, support or positivity in the target covariate patterns, credible conditional transportability, adequate calibration of target untreated risk, and assessment of covariate-specific effect variation.