From Weighted Risks to Population Contrasts
A weighted analysis of anticoagulation may report a stroke risk of 6% under one treatment strategy and 9% under another. Those two risks answer an absolute question about each strategy; their difference and ratio answer two distinct comparative questions. The risk difference measures the change in probability on an absolute scale, while the risk ratio measures the proportional change. Both are marginal when they compare risks averaged over the same target population rather than risks conditional on particular covariate values.
The interpretation depends on what population the weighted risks represent. In a causal analysis, the target is commonly the study population’s baseline covariate distribution, with treatment assignment or treatment history reweighted to support comparison of strategies in that population. Stroke and bleeding are separate outcomes: a favorable stroke contrast does not by itself determine the bleeding contrast or establish an overall clinical benefit.
For stroke, \(Y^a=1\) means a stroke occurs over the outcome period under strategy \(a\); for bleeding, define a separate binary outcome and risks. A negative risk difference means lower risk under strategy 1, whereas a risk ratio below 1 means its risk is a smaller fraction of the comparator risk. Neither measure is inherently more important: the risk difference conveys absolute frequency, and the risk ratio conveys relative frequency.
Worked Example: Reporting Stroke and Bleeding Contrasts
Suppose a weighted analysis estimates a 6% stroke risk under anticoagulation and a 9% risk under the comparator. For bleeding, suppose the corresponding risks are 4% and 2.5%. Calculate and interpret each marginal contrast.
Stroke risk difference. The risks as probabilities are \(R_1=0.06\) and \(R_0=0.09\), so
This is a 3-percentage-point lower stroke risk under anticoagulation in the target population. Per 1,000 people represented by that population, the difference corresponds to 30 fewer strokes over the specified period, if the causal interpretation and the time horizon are appropriate.
Stroke risk ratio.
The estimated stroke risk under anticoagulation is about two-thirds of the comparator risk, or about 33.3% lower on a relative scale.
Bleeding risk difference and ratio.
The estimated bleeding risk is 1.5 percentage points higher, or 15 additional events per 1,000 people over the stated period, and is 1.6 times the comparator risk. The two outcomes therefore show different directions of contrast. These calculations alone do not combine them into a single benefit-harm measure.
What Makes the Risks Marginal?
Let \(L\) denote baseline covariates, such as age or clinical history. A conditional risk \(P(Y^a=1\mid L=l)\) describes risk among people with covariates \(l\). The marginal risk \(R_a\) averages that conditional risk over the distribution of \(L\) in the chosen target population. Thus a marginal contrast reflects both the stratum-specific risks and the population’s composition.
For a baseline treatment \(A\), inverse-probability weighting can represent this target through a weighted pseudo-population. Write \(e_a(L)=P(A=a\mid L)\). A normalized weighted risk in arm \(a\) has the form
Here \(I(A_i=a)\) is 1 when person \(i\) received strategy \(a\) and 0 otherwise. In longitudinal settings, the weights may instead represent treatment histories or strategies; the interpretation still requires that the weighted risks refer to the same target population. The numbers in the following examples illustrate contrasts, not a prescription for constructing longitudinal weights.
Proof. Conditional on \(L\), the denominator’s integrand has expectation
Therefore its unconditional expectation is 1. For the numerator, use consistency on the event \(A=a\), then conditional exchangeability:
Taking expectations over \(L\) gives \(E[I(A=a)Y/e_a(L)]=E[Y^a]\). Dividing by the denominator expectation of 1 proves the identity. \(\square\)
The result explains the population target of the weighted risks; it does not guarantee that a finite-sample estimate is accurate. With the stated conditions, independent observations, and true weights, the law of large numbers makes the sample averages in the normalized estimator converge to their expectations. The denominator converges to 1, so the ratio converges in probability to \(E[Y^a]\). Estimated weights require additional regularity, including adequate estimation of treatment probabilities.
Worked Example: How Covariate-Stratum Risks Produce Marginal Contrasts
Suppose 40% of the target population is in a higher-risk stratum and 60% is in a lower-risk stratum. Weighted stroke risks are as follows:
| Baseline stratum | Population proportion | Risk under anticoagulation | Risk under comparator |
|---|---|---|---|
| Higher-risk | 0.40 | 0.20 | 0.30 |
| Lower-risk | 0.60 | 0.02 | 0.04 |
Average each strategy’s risks over the same target-population proportions:
Thus \(RD=0.092-0.144=-0.052\), a 5.2-percentage-point lower marginal stroke risk. The marginal risk ratio is
The stratum-specific risk differences are \(-0.10\) and \(-0.02\). Averaging those differences with the target-population proportions gives
which agrees with the marginal risk difference. The conditional risk ratios are \(0.20/0.30=2/3\) and \(0.02/0.04=1/2\); neither alone equals the marginal ratio. The marginal ratio can be written as a weighted average of these conditional ratios, using each stratum’s contribution to comparator risk:
The weights \(5/6\) and \(1/6\) are the two strata’s shares of the total comparator risk, not their population proportions. This distinction matters when explaining why marginal and conditional risk ratios differ.
If \(R_0>0\), then
In the common case where \(r_0(l)>0\) in every stratum with \(p(l)>0\), the second sum vanishes and the first expression is a weighted average of conditional risk ratios. Derivation. By averaging conditional risks, \(R_a=\sum_l p(l)r_a(l)\). Subtracting the two identities yields the risk-difference formula. For the ratio, use \(r_1(l)=r_0(l)\{r_1(l)/r_0(l)\}\) when \(r_0(l)>0\), divide the sum for \(R_1\) by \(R_0\), and retain any strata with \(r_0(l)=0\) separately. Their contribution is shown explicitly because a conditional risk ratio is undefined there. \(\square\)
This result shows precisely why a marginal risk ratio should not be casually described as the average of conditional risk ratios using population proportions. Even when all conditional ratios exist, its averaging weights depend on the comparator risks as well as on the population proportions.
Worked Example: Calculating Risks From Weighted Totals
Consider a small weighted analysis of stroke. In the anticoagulation arm, four observations have weights 2, 1, 3, and 4, with stroke outcomes 1, 0, 0, and 0. In the comparator arm, four observations have weights 1, 2, 1, and 4, with outcomes 0, 1, 0, and 0. Calculate the normalized weighted risks and contrasts.
For anticoagulation, the total weight is \(2+1+3+4=10\). The weighted event total is \((2)(1)+(1)(0)+(3)(0)+(4)(0)=2\), so \(\widehat R_1=2/10=0.20\).
For the comparator, the total weight is \(1+2+1+4=8\). The weighted event total is \((1)(0)+(2)(1)+(1)(0)+(4)(0)=2\), so \(\widehat R_0=2/8=0.25\).
The estimated marginal risk difference is \(0.20-0.25=-0.05\), and the estimated risk ratio is \(0.20/0.25=0.80\). In this weighted sample, anticoagulation is associated with a 5-percentage-point lower risk and a risk estimated at 80% of the comparator risk. These are descriptive calculations; a causal interpretation still depends on identification assumptions and appropriate weighting.
Interpretation, Diagnostics, and Alternatives
State the outcome, time horizon, target population, and scale when reporting a contrast. For example: “In the target population represented by the weighted analysis, the estimated risk of stroke by 12 months was 6% under anticoagulation and 9% under the comparator, a risk difference of −3 percentage points and a risk ratio of 0.67.” This wording prevents a relative reduction from being mistaken for an absolute change and makes clear that risks refer to a defined period.
A weighted risk difference is often more directly useful for discussing expected numbers of events, while a risk ratio can make relative comparisons easier across settings with different baseline risks. Neither scale eliminates uncertainty: estimates should be accompanied by appropriate confidence intervals, and an interval for a ratio is typically interpreted on the ratio scale. A risk ratio is undefined if the comparator marginal risk is zero; it can also be unstable when that risk is very small.
Several checks help reveal when an interpretation may be unreliable:
- Inspect weighted risks and denominators. Confirm that the numerator counts weighted events and the denominator is the total weight in the corresponding strategy group.
- Check covariate balance and positivity. Severe remaining imbalance or very large weights can indicate that the weighted groups do not adequately represent comparable people, or that some treatment probabilities are near zero.
- Compare crude and weighted summaries carefully. A difference between them may reflect confounding adjustment, but it is not itself proof that the weighted estimate is correct.
- Check outcome and time definitions. A risk requires a clearly defined event and follow-up period; contrasts from different periods or differently defined outcomes cannot be compared as if they were the same estimand.
A common mistake is to call \(RR=0.67\) a 33-percentage-point reduction. That reasoning confuses relative and absolute scales: \(1-0.67\) is approximately a 33% relative reduction, while the absolute reduction is \(R_1-R_0\). At risks of 6% and 9%, the absolute reduction is 3 percentage points. Another mistake is to treat a weighted marginal risk as the risk for every person or every covariate stratum. It is an average over the target population; effects can differ across strata.
The main alternatives answer related but different questions. A crude risk contrast compares the observed groups without adjustment and may reflect baseline differences. A conditional model may report covariate-specific comparisons, whose meaning depends on the model and covariate values. Standardization instead predicts each strategy’s risk across the target covariate distribution and averages those predictions. Properly constructed weighting and standardization can target the same marginal risks under suitable assumptions, but their finite-sample behavior and modeling requirements can differ. An odds ratio is not a risk ratio; even absent confounding, conditional and marginal odds ratios can differ because odds ratios are non-collapsible.
Check Your Understanding
Answer each question, then check the solutions below.
- Weighted stroke risks are 8% under anticoagulation and 10% under the comparator. Calculate the risk difference and risk ratio, and interpret both.
- Why is a marginal risk ratio not generally the population-proportion-weighted average of stratum-specific risk ratios?
- In the two-stratum example above, what population quantity determines the weights for averaging the conditional risk ratios?
- What does it mean for weighted risks to be marginal, and name one assumption needed for their causal interpretation.
- Why is it incorrect to describe a risk ratio of 0.80 as an absolute reduction of 20 percentage points?
Solutions. (1) \(RD=0.08-0.10=-0.02\), or 2 percentage points lower; \(RR=0.08/0.10=0.80\), or 20% lower on a relative scale. (2) The marginal ratio divides averaged risks, and the conditional ratios are weighted by each stratum’s contribution to comparator risk, not generally by population proportions. (3) Each stratum’s share of the total comparator risk determines its ratio weight, provided the conditional comparator risk is positive. (4) The risks average over the target population’s covariate distribution; consistency, conditional exchangeability, and positivity are examples of assumptions needed for causal identification. (5) The ratio describes a proportional comparison. An absolute reduction requires subtracting risks and depends on their levels.