Two Different Sources of Uncertainty
A partially identified effect is not generally a single unknown number. Under the maintained assumptions, the population data distribution may be consistent with a range of causal effects. That range is the identified set. In a sample, the estimated endpoints of the range also vary from sample to sample. Inference must account for this sampling uncertainty without confusing it with the width of the identified set.
For example, an estimated identified interval of \([-2,5]\) does not mean that the effect is known to lie between \(-2\) and \(5\). It is an estimate of the interval’s endpoints. A confidence procedure should allow for the fact that the true lower endpoint might be below \(-2\), and the true upper endpoint might be above \(5\). A confidence interval for a midpoint, or separate 95% intervals for the endpoints, does not automatically provide the desired coverage for the whole identified set.
The inclusion in this definition matters: the target is coverage of every value in the true identified set, not just coverage of one selected effect within it. When a confidence set is itself an interval, a convenient construction is to find simultaneous confidence bounds for \(L(P)\) and \(U(P)\), then take the outer interval extending from the lower confidence limit for \(L(P)\) to the upper confidence limit for \(U(P)\).
Assumptions for Endpoint-Based Inference
The identified set is defined by the substantive restrictions and population quantities in the analysis. Statistical inference does not make those restrictions true; it quantifies sampling uncertainty conditional on the chosen identification analysis. For the procedures below, assume the set is a finite interval and the endpoint estimates are consistent. For the usual normal approximations, also assume that the endpoint estimates have a valid joint large-sample distribution and that their standard errors consistently estimate their sampling variability.
A common regular case is that \((\hat L,\hat U)\) is asymptotically bivariate normal around \((L,U)\). This can arise when endpoints are smooth functions of sample means or other regular estimates. If nuisance quantities are estimated, their contribution must be included in the endpoint standard errors and covariance. The observations must also support the sampling model used—for example, independent and identically distributed observations when that is the assumed design.
These conditions are not automatic. Endpoints defined by a maximum or minimum of competing expressions can behave irregularly when expressions tie or when an active constraint changes with the sample. A routine normal approximation may then be unreliable. The procedures here describe regular endpoint inference; a result near a boundary or other nonregular feature calls for special care rather than an unexamined Wald interval.
Bonferroni Confidence Sets for the Whole Interval
A simple method constructs two endpoint intervals with simultaneous coverage guaranteed by the union bound. Give each endpoint a two-sided confidence interval with confidence level \(1-\alpha/2\). If the two intervals are \(I_L=[a_L,b_L]\) and \(I_U=[a_U,b_U]\), then a confidence set for the entire identified interval is
This construction is valid when \(a_L\leq b_U\). It is intentionally an outer interval: on the event that \(L\in I_L\) and \(U\in I_U\), the true set \([L,U]\) lies inside \([a_L,b_U]\). The endpoint intervals may overlap or be wide; that does not alter the coverage argument.
Compute \(\hat L\), \(\hat U\), and standard errors \(SE(\hat L)\), \(SE(\hat U)\) under the sampling model.
For normal approximations, use the critical value \(z_{1-\alpha/4}\) in each two-sided endpoint interval. Each interval then has error probability \(\alpha/2\).
Set the confidence set’s lower limit to the lower limit of \(I_L\) and its upper limit to the upper limit of \(I_U\).
The justification is short but central. If \(E_L\) is the event that \(I_L\) covers \(L\), and \(E_U\) the event that \(I_U\) covers \(U\), then
Thus \(P(E_L\cap E_U)\geq 1-\alpha\). On \(E_L\cap E_U\), \(a_L\leq L\) and \(U\leq b_U\), so \([L,U]\subseteq[a_L,b_U]\). The guarantee is at least \(1-\alpha\); it can be conservative, especially if the endpoint estimates are strongly dependent.
Worked Example: Bonferroni Bounds for Estimated Endpoints
Suppose a hypothetical analysis estimates an effect’s identified set as \([\hat L,\hat U]=[-2.00,5.00]\), in outcome points. The estimated standard errors are \(SE(\hat L)=0.400\) and \(SE(\hat U)=0.600\). Construct an approximate 95% confidence set for the entire identified interval using Bonferroni.
Here \(\alpha=0.05\), so each two-sided endpoint interval has confidence level \(1-\alpha/2=0.975\), and uses \(z_{1-\alpha/4}=z_{0.9875}\approx2.241\).
For the lower endpoint,
For the upper endpoint,
The outer confidence set is therefore \([-2.896,6.345]\) outcome points. Under the stated large-sample assumptions, this procedure has at least 95% coverage for the entire true identified interval. It is wider than the estimated interval because it includes uncertainty about both endpoints.
Using Joint Endpoint Inference
When a joint covariance estimate is available, a joint confidence region for the endpoint vector can replace the Bonferroni construction. Under the bivariate normal approximation, let \(\widehat\Sigma\) estimate the covariance matrix of \((\hat L,\hat U)\). An approximate \(1-\alpha\) confidence ellipse consists of endpoint pairs \(\theta=(L,U)\) satisfying
Project the ellipse onto each endpoint axis. The projected bounds for \(L\) are \(\hat L\pm\sqrt{\chi^2_{2,1-\alpha}}SE(\hat L)\), and those for \(U\) are \(\hat U\pm\sqrt{\chi^2_{2,1-\alpha}}SE(\hat U)\). Taking the lower projected limit for \(L\) and the upper projected limit for \(U\) gives an outer confidence interval. Since the joint ellipse covers the true endpoint pair with approximate probability \(1-\alpha\), this outer interval covers the entire true identified set whenever the pair is covered.
Worked Example: Projecting a Joint Confidence Ellipse
In a hypothetical analysis, \(\hat L=-1.00\) and \(\hat U=4.00\) outcome points, with standard errors \(0.300\) and \(0.500\). Suppose a valid estimated covariance matrix for the two endpoint estimates is available, and use the 95% bivariate-normal ellipse, for which \(\chi^2_{2,0.95}=5.991\).
The projection multiplier is \(\sqrt{5.991}\approx2.448\). The lower endpoint’s projected limits are
The upper endpoint’s projected limits are
The resulting outer confidence set is \([-1.734,5.224]\) outcome points. The covariance affects the shape and orientation of the ellipse, although these projection widths use the marginal standard errors. This joint construction is not necessarily narrower than Bonferroni; its usefulness depends on the joint approximation and the procedure’s critical value.
What the Confidence Set Can Establish
A confidence set describes sampling uncertainty about the identified set under the maintained assumptions. If its lower limit is above zero, the procedure supports the conclusion that the entire true identified set is positive at the stated confidence level. If its upper limit is below zero, it supports the corresponding negative conclusion. If it contains zero, the data and assumptions do not establish that every compatible effect has the same sign.
This is not a probability statement that the fixed population set has a 95% chance of lying inside the computed interval. In repeated samples, the procedure is designed to cover the fixed true set at least 95% of the time (or asymptotically so). Nor does a confidence set repair weak or implausible identifying assumptions: the guarantee is conditional on the assumptions that define the population identified set.
Worked Example: Assessing Whether Every Compatible Effect Is Positive
A hypothetical study estimates an identified interval \([1.40,4.20]\) outcome points, with endpoint standard errors \(0.250\) and \(0.400\). Use the 95% Bonferroni confidence-set method to assess whether all effects in the true identified set are positive.
Using \(z_{0.9875}=2.241\), the lower endpoint’s interval is
The upper endpoint’s interval is
The outer confidence set is \([0.8398,5.096]\) outcome points. Its lower limit is above zero, so the procedure supports the conclusion that the whole true identified set lies above zero, at least 95% confidence coverage under the stated assumptions. It does not identify one particular effect value, and it does not establish that the effect is exactly the estimated interval \([1.40,4.20]\).
Diagnostics, Alternatives, and Common Mistakes
Before interpreting the result, check that the estimated endpoints are ordered, the standard errors correspond to the endpoint estimators actually used, and the sampling method is compatible with the data design. Inspect whether the endpoints arise from smooth, stable formulas or from competing constraints whose active identity changes across samples. Very wide outer limits may signal substantial sampling uncertainty even if the estimated identified interval is narrow; conversely, a wide estimated set can remain wide even with precise endpoints. These are different issues and should be reported separately.
Bonferroni is simple and protects simultaneous coverage without requiring endpoint independence, but it can be conservative. A joint confidence region uses dependence information and can be more tailored, provided its joint distributional approximation is valid. Separate pointwise 95% intervals are not a substitute: the probability that both endpoints are covered can be below 95%. A confidence interval for a midpoint answers a different question and cannot ensure that the full identified set is covered.
- Mistake: treating \([\hat L,\hat U]\) as a confidence interval. It estimates the population identified set but does not, by itself, account for endpoint sampling error. Use a confidence construction for the endpoints.
- Mistake: reporting two pointwise 95% endpoint intervals as 95% joint coverage. Their joint coverage is not guaranteed to be 95%. Use a simultaneous procedure such as Bonferroni or a valid joint region.
- Mistake: interpreting a confidence set as a point estimate. The outer set represents sampling uncertainty about the range of compatible effects; it does not select one effect from that range.
These methods rely on their sampling approximations and can be poor when endpoint estimators are nonregular, especially near boundary configurations. A routine standard-error calculation should not be treated as a universal solution in those cases. Report the identifying assumptions, the estimated identified interval, the confidence construction, and the confidence set as distinct components of the analysis.
Check Your Understanding
Answer each question, then compare with the solutions below.
- An estimated identified interval is \([-0.50,2.00]\), with endpoint standard errors \(0.200\) and \(0.300\). Using 95% Bonferroni coverage and \(z_{0.9875}=2.241\), calculate the outer confidence set.
- Why do two separate 95% endpoint intervals not automatically give 95% coverage for the entire true identified set?
- What does a confidence set whose lower limit is above zero support, and what does it not claim?
- Name one condition needed for the usual endpoint normal approximation and one reason it might fail.
Solutions. 1. The lower endpoint interval has lower limit \(-0.50-2.241(0.200)=-0.9482\). The upper endpoint interval has upper limit \(2.00+2.241(0.300)=2.672\). Thus the outer confidence set is \([-0.9482,2.672]\). 2. Each interval has its own 95% coverage, but the chance both cover can be less than 95%; without a simultaneous procedure the union-bound guarantee is not at the requested level. 3. It supports, at the procedure’s stated coverage level and under its assumptions, that the entire true identified set is positive. It does not assign a probability to a fixed set after observing the data or identify one exact effect. 4. The endpoint estimators need a valid joint large-sample distribution with consistently estimated standard errors; a maximum or minimum with tied competing constraints can make the endpoint nonregular and invalidate the routine approximation.