What Happens When an Interval Endpoint Equals a Margin

Why Exact Contact With a Margin Matters

An equivalence conclusion depends on more than whether an estimate is between two margins. The uncertainty around that estimate must also be sufficiently small: the confidence interval must lie strictly inside the equivalence region. That strictness matters at the boundary. If a confidence limit is exactly equal to a margin, the corresponding one-sided test is exactly at its critical value, so the test does not pass under the strict rejection rule used here.

This is a boundary case, not a practically likely event in continuously distributed data. But it is important for interpreting software output, checking calculations, and writing a reproducible decision rule. A displayed endpoint may appear equal to a margin because of rounding even when the underlying values differ. Conversely, calculations that round an intermediate value can turn an intended equality into a small inequality.

For a difference-scale analysis, the margins and parameter are expressed in the units of the difference. For a ratio analysis, TOST is commonly performed on the log-ratio scale, where ratio limits become additive limits. The same boundary logic applies on either scale: a test statistic exactly at its critical value does not satisfy a strict rejection inequality.

Strict Tests and Their Confidence-Interval Equivalent

Suppose the estimate is \(\hat{\theta}\), its standard error is \(SE\), and inference uses a critical value \(c\) for a one-sided test at level \(\alpha\). For a large-sample normal procedure, \(c=z_{1-\alpha}\); for a \(t\)-procedure, \(c=t_{1-\alpha,df}\). The model must justify the chosen reference distribution: for example, a normal approximation for the estimate with a valid standard error, or a \(t\)-statistic with the stated degrees of freedom under the fitted model's assumptions.

The lower-margin test rejects when the estimate is sufficiently above \(L\). The upper-margin test rejects when the estimate is sufficiently below \(U\). Written in terms of the component statistics, the decision rules are:

Both inequalities must hold. Equality in either one is not rejection under these strict rules. Equivalently, the one-sided p-value for that component is equal to \(\alpha\) when its statistic is exactly \(c\); it has not fallen below the significance threshold.

The two rules can be rearranged directly, without relying on a verbal interpretation. Since \(SE>0\), the lower test rejects exactly when \(\hat{\theta}-cSE>L\). The upper test rejects exactly when \(\hat{\theta}+cSE<U\). Thus both reject exactly when the interval formed with those endpoints is strictly inside the margins.

For \(\alpha=0.05\), the matching interval is a 90% two-sided confidence interval, since \(1-2\alpha=0.90\). This relationship only holds when the interval and tests use the same estimate, standard error, reference distribution, degrees of freedom where applicable, and margins. Comparing a 90% interval from one model with TOST statistics from another does not establish this duality.

Worked Boundary Cases

Why the Derivation Determines the Boundary Rule

The confidence-interval rule follows from the test rules, rather than being an additional convention. For the lower test, the rejection region is \(T_L>c\). Multiplying by the positive \(SE\) and rearranging gives \(\hat{\theta}-cSE>L\), which says that the interval's lower limit must exceed the lower margin. If the lower limit equals \(L\), the rearrangement gives \(T_L=c\), not \(T_L>c\). The same algebra for the upper test requires \(\hat{\theta}+cSE<U\). Equality at \(U\) likewise produces a statistic equal to \(c\), not greater than \(c\).

A closed confidence interval that merely shares an endpoint with the closed interval \([L,U]\) can be described informally as “within or on the margins.” That wording is not the strict containment condition corresponding to both tests passing. For TOST, the acceptance region for the parameter is open, \(L<\theta<U\), and the confidence interval must be contained in that open region. Exact contact leaves a boundary point in the confidence interval, so strict containment fails.

Precision, Diagnostics, and Common Mistakes

Exact equality should be assessed using the unrounded values used in the test. Printed estimates, standard errors, confidence limits, and p-values are often rounded for display. A limit printed as 0.800 may be 0.7996 or 0.8004 internally; those values lie on opposite sides of a 0.800 margin. The display alone cannot resolve the decision.

For a deliberate boundary check, retain the defining quantities at full available precision and calculate the limit and test statistic from the same inputs. If a hypothetical example defines \(SE=5/c\), using \(SE=3.039\) in subsequent arithmetic does not preserve exact equality. The rounded standard error can make the computed statistic slightly larger or smaller than \(c\). Likewise, do not round a p-value to 0.050 and infer that its unrounded value equals \(\alpha=0.05\).

• Check the matching procedure. Verify that the confidence interval and both tests use the same estimate, \(SE\), critical distribution, and degrees of freedom.
• Inspect unrounded limits. Compare the stored limits with the prespecified margins before applying display formatting.
• Check the direction. The lower limit must be greater than \(L\); the upper limit must be less than \(U\). Reversing either inequality changes the conclusion.
• Check the comparison operator. Document whether the decision rule uses strict inequalities. In exact continuous models, equality has probability zero, so alternative conventions at the single point can yield the same nominal level; nevertheless, implementations and reports need a consistent rule for computed boundary cases.

A common mistake is to say, “The confidence interval reaches the margin but does not cross it, so equivalence is established.” That reasoning treats touching as strict containment. The correction is to test each endpoint using the defined inequalities: \(CI_L>L\) and \(CI_U<U\). If either comparison is equality, the corresponding component test is at the critical value and equivalence is not concluded under the strict rule.

The interval is the most direct presentation for seeing endpoint contact, while the two test statistics show which one-sided condition fails. Neither replaces the need to state the decision rule. A rounded table is convenient for communication; a full-precision computation is necessary for resolving a boundary comparison. If finite numerical precision makes equality indistinguishable from a tiny difference, retain greater precision or use a numerically stable comparison rather than silently treating the displayed values as exact.