From Stratum-Specific Bounds to a Population Bound
When a population is divided into covariate strata, a bound on the causal effect in each stratum is not yet a bound for the whole target population. The population effect depends on how common each stratum is in that population. Consequently, aggregation requires weighting each stratum’s lower and upper endpoints by its target-population proportion.
The distinction is substantive, not just arithmetic. Giving every stratum equal weight generally describes an artificial population in which all strata are equally common. If the scientific target is a particular population, its stratum proportions determine the aggregation.
This setup assumes that the stratum-specific intervals apply to the same target population and effect scale used in the aggregation. The strata must cover that population, and each \(q_z\) must be its proportion in the target population. The bounds \(L_z\) and \(U_z\) must be valid for the conditional effect \(\tau_z\), under the assumptions used to obtain them. Aggregation does not make invalid conditional bounds valid.
Why the Endpoints Are Weighted
The population effect is a weighted average of the stratum-specific effects. Since \(L_z\leq\tau_z\leq U_z\) and \(q_z\geq0\), multiplying each inequality by \(q_z\) preserves its direction. Adding the resulting inequalities gives
The middle term is \(\tau\), so the left and right terms are valid lower and upper bounds for the population effect. This argument establishes validity even when we do not know whether the endpoints can actually be attained.
For the aggregated interval to be sharp, the endpoints must also be feasible. In particular, it must be possible for the stratum effects to attain their lower endpoints together, and likewise their upper endpoints together, under the information and assumptions defining the identified set. When the feasible choices in different strata are unrestricted relative to one another and each conditional endpoint is attainable, the aggregate lower and upper bounds are sharp. If additional restrictions link the strata, the weighted interval remains valid but may be wider than the sharp aggregate interval.
Worked Example: Two Strata With Unequal Population Shares
Suppose a hypothetical target population has two strata. Stratum 1 makes up 70% of the population and has a conditional average treatment effect between \(-0.20\) and \(0.40\). Stratum 2 makes up 30% and has a conditional effect between \(-0.10\) and \(0.50\). The outcome is binary, and effects are expressed as risk differences.
The weights are \(q_1=0.70\) and \(q_2=0.30\); they are nonnegative and sum to one. Assuming the supplied conditional bounds are valid for these strata and this target, the population lower bound is
The population upper bound is
Thus the population risk difference is bounded by \([-0.17,\,0.43]\), or from a 17-percentage-point decrease to a 43-percentage-point increase. The calculation does not establish that either endpoint is attainable; that depends on whether the stratum-specific endpoint choices are jointly feasible.
An unweighted average of the lower endpoints would be \((-0.20-0.10)/2=-0.15\), and an unweighted average of the upper endpoints would be \((0.40+0.50)/2=0.45\). Those numbers give equal influence to the two strata, contrary to the stated target population’s 70%/30% composition.
Choosing Weights for the Intended Target
The weights are part of the estimand. They are not automatically the proportions in a study sample, nor are they automatically the proportions among treated units or untreated units. For the population average treatment effect defined above, use the distribution of \(Z\) in the target population. If the scientific question instead concerns a different target group, its stratum distribution may call for different weights and a different aggregate effect.
This distinction matters when a study’s composition differs from the population of interest. It also matters when some strata are rare: their endpoints may be wide, but their contribution to this particular population bound is scaled by their population share. A stratum with \(q_z=0\) contributes nothing to the target effect. Its conditional bound is irrelevant to this aggregation, though it may matter for a different target population.
Worked Example: Reweighting to the Target Population
Suppose three hypothetical strata have conditional risk-difference bounds \([-0.04,\,0.08]\), \([-0.10,\,0.20]\), and \([-0.02,\,0.06]\). A study sample contains 20%, 50%, and 30% of its observations in the three strata, but the specified target population contains 50%, 30%, and 20%, respectively. Assume the conditional bounds apply to the target strata, and the target weights are known.
Use the target proportions, which are nonnegative and sum to one. The lower endpoint is
The upper endpoint is
The target population’s average risk difference is therefore bounded by \([-0.054,\,0.112]\), or \([-5.4,\,11.2]\) percentage points. Using the study’s proportions instead would calculate a different weighted interval. That calculation might be relevant to the study-sample population, but it would not answer the stated target-population question.
Validity, Sharpness, and Cross-Stratum Restrictions
The weighted-inequality argument always gives a valid outer interval when the conditional bounds are valid and the weights are correct. It does not, by itself, show that every value inside that interval is feasible. Sharpness requires attention to what combinations of stratum effects are allowed.
For example, if the feasible set is the Cartesian product of the conditional intervals—meaning any permitted value in one stratum can occur alongside any permitted value in another—then the minimum weighted sum is achieved by selecting every lower endpoint, and the maximum by selecting every upper endpoint. With nonnegative weights, those choices minimize and maximize each term, respectively. If a cross-stratum assumption rules out one of those combinations, the calculated endpoints need not be attainable.
Worked Example: A Valid Interval That Is Not Sharp
Consider a hypothetical population split equally between two strata, so \(q_1=q_2=0.50\). Suppose each stratum’s conditional effect is bounded by \([-1,\,1]\), but an additional, known cross-stratum restriction requires \(\tau_2=-\tau_1\). The outcome effect is measured on a scale where these values are feasible within the stated conditional intervals.
Using only the stratum-specific intervals gives the outer lower and upper bounds
However, the cross-stratum restriction implies
Thus the actual population effect is exactly zero under the joint restriction. The interval \([-1,\,1]\) is valid but not sharp in this example: its endpoints would require both stratum effects to equal \(-1\), or both to equal \(1\), neither of which satisfies \(\tau_2=-\tau_1\). The lesson is to distinguish adding conditional interval information from characterizing all joint restrictions.
Diagnostics and Common Mistakes
Before reporting an aggregate interval, verify that the weights and endpoints match the intended estimand. A small arithmetic error can be consequential, but a correct calculation with the wrong weights still answers the wrong question.
- Check the weights. Confirm that every \(q_z\geq0\) and that \(\sum_z q_z=1\). If they do not sum to one, determine whether strata are missing, the weights use a different scale, or the target population has been specified incorrectly.
- Check the ordering. Each stratum should satisfy \(L_z\leq U_z\). The aggregate endpoints should also satisfy \(L\leq U\).
- Check the target. Label whether the weights describe the target population, the sample, or another group. Do not switch among them without changing the estimand.
- Check the scale and units. All stratum effects must be expressed on a compatible scale. A weighted average of risk differences is interpretable in percentage points; one cannot directly average a risk difference with an odds ratio.
- Check sharpness separately. Do not call the endpoints sharp merely because each stratum has sharp bounds. Confirm that the endpoint choices are jointly feasible under all restrictions.
A common mistake is to average the stratum endpoints equally because there are \(K\) strata. That reasoning treats each stratum as equally prevalent, regardless of its size. The correction is to use the target proportions \(q_z\) in both endpoint calculations. Another mistake is to infer that the interval’s midpoint is the population effect; a midpoint is not identified by the bounds alone.
| Approach | Calculation | What it represents |
|---|---|---|
| Target-weighted aggregation | \(\sum_z q_zL_z\) to \(\sum_z q_zU_z\) | Valid population bounds when conditional bounds apply to the target strata |
| Unweighted endpoint average | \(\frac{1}{K}\sum_z L_z\) to \(\frac{1}{K}\sum_z U_z\) | Bounds for an equally weighted mixture of strata, not generally the target population |
| Pooled worst-case calculation | Bound the pooled effect directly from pooled information | A different calculation; it need not equal aggregation of conditional bounds |
The comparison is about the quantity each calculation targets, not a guarantee that one interval is always narrower. Aggregation also has limitations: uncertainty in estimated stratum proportions or conditional endpoints is not represented by the arithmetic alone, and finite samples may leave some strata poorly informed. An interval computed from estimated inputs should not be described as a confidence interval unless a separate inferential procedure accounts for sampling uncertainty.
Check Your Understanding
Use the target-population weights specified in each question. Solutions are included below the questions.
- Two strata have target weights \(0.25\) and \(0.75\), with effect bounds \([-0.40,\,0.20]\) and \([-0.10,\,0.30]\). Calculate the aggregate interval.
- Why are weights of \(0.50\) and \(0.50\) inappropriate for a target population whose stratum shares are \(0.20\) and \(0.80\), unless that equal-weight population is the intended target?
- Suppose every stratum-specific interval is valid, but a known joint restriction prevents all lower endpoints from occurring together. What can be concluded about the weighted lower endpoint?
- A target has three strata with weights \(0.40\), \(0.35\), and \(0.25\). The first stratum has bounds \([-0.05,\,0.10]\), the second \([-0.02,\,0.08]\), and the third \([-0.10,\,0.20]\). Calculate the lower and upper aggregate endpoints.
Solutions. (1) The lower endpoint is \(0.25(-0.40)+0.75(-0.10)=-0.10-0.075=-0.175\). The upper endpoint is \(0.25(0.20)+0.75(0.30)=0.05+0.225=0.275\), so the interval is \([-0.175,\,0.275]\). (2) Equal weights describe an equally weighted mixture, not the stated target; use \(0.20\) and \(0.80\) to represent that target. (3) The weighted lower endpoint remains a valid lower bound, but it may not be attainable and therefore need not be sharp. (4) The lower endpoint is \(0.40(-0.05)+0.35(-0.02)+0.25(-0.10)=-0.020-0.007-0.025=-0.052\). The upper endpoint is \(0.40(0.10)+0.35(0.08)+0.25(0.20)=0.040+0.028+0.050=0.118\), giving \([-0.052,\,0.118]\).