Diagnosing Confidence Intervals That Cross an Equivalence Limit

Why a Crossed Limit Matters

In an equivalence analysis, the question is not simply whether an estimated treatment difference is small or whether its confidence interval overlaps the acceptable range. The evidence must support both required claims: the difference is greater than the lower equivalence limit and less than the upper equivalence limit. A confidence interval that crosses either limit leaves one of those claims unsupported.

This tutorial focuses on diagnosing that failure. An interval may cross only the lower limit, only the upper limit, or both. These are different patterns: each identifies which one-sided component of the two one-sided tests (TOST) does not pass. In every pattern, however, partial overlap with the equivalence region is insufficient to conclude equivalence.

Model and Interval Criterion

Suppose an analysis provides an estimate \(\hat{\theta}\) and standard error \(SE\). For a normal approximation, assume the estimate is approximately normally distributed around \(\theta\), with the reported \(SE\) appropriate for that estimator. For a t-based analysis, use the corresponding t distribution and \(df\). The analysis must also target the intended treatment contrast, and the limits \(L\) and \(U\) must have been specified for that contrast before interpreting the result.

The standard error must reflect the study design and model. For example, if the contrast is estimated from paired or crossover data, the uncertainty calculation must account for that pairing. The examples below use a normal approximation and a significance level of \(\alpha=0.05\); they are hypothetical calculations, not study results.

For a normal approximation, the lower-bound test statistic is \(z_L=(\hat{\theta}-L)/SE\), and the upper-bound test statistic is \(z_U=(\hat{\theta}-U)/SE\). At \(\alpha=0.05\), each one-sided test uses the critical value \(z_{0.95}=1.645\). The lower component passes when \(z_L>1.645\); the upper component passes when \(z_U<-1.645\).

These two conditions can also be read from the corresponding central 90% confidence interval. In the normal case, that interval is formed using \(1.645\), not \(1.96\), because a 90% two-sided interval has 5% in each tail.

The equivalence follows directly from the endpoint inequalities. The lower endpoint exceeds \(L\) exactly when \(\hat{\theta}-1.645SE>L\); rearranging gives \((\hat{\theta}-L)/SE>1.645\), which is the lower-test condition. The upper endpoint is below \(U\) exactly when \(\hat{\theta}+1.645SE<U\); rearranging gives \((\hat{\theta}-U)/SE<-1.645\), which is the upper-test condition. Thus, if the interval crosses a limit, the corresponding endpoint inequality fails, and so does that component of TOST.

For a t-based analysis, replace 1.645 by the one-sided t critical value for the stated \(df\). The matching interval is then the central 90% t interval. The diagnostic principle is unchanged: compare each endpoint with its corresponding margin, rather than judging the interval by its overlap alone.

Worked Diagnoses

For a concrete set of examples, suppose the contrast is a log treatment/reference ratio, \(\theta=\log(T/R)\), and the prespecified ratio limits are 0.80 and 1.25. On the log scale these are approximately \(L=\log(0.80)=-0.2231\) and \(U=\log(1.25)=0.2231\). Exponentiating a log-scale interval gives the corresponding ratio-scale interval; because the exponential function is increasing, it preserves the ordering of the endpoints.

How to Read Each Crossing Pattern

The table assumes that the interval is the one corresponding to the specified TOST level and that no endpoint lies exactly on a margin. The crossing labels are about the endpoints: an interval crosses the lower limit when its lower endpoint is below \(L\), and crosses the upper limit when its upper endpoint is above \(U\). Do not label a result as crossing both limits merely because it crosses one and equivalence fails.

“Fails” here means that the relevant one-sided test did not reject its null hypothesis at the chosen \(\alpha\). It does not prove that the true contrast is outside the equivalence range. The result may be inconclusive: the available data have not established the required inequality. A different analysis or more precise data could lead to a different decision, but the equivalence limits themselves should not be changed to make the observed interval fit.

Common Misreadings and Diagnostic Checks

• Calling any overlap with the equivalence region a success. An interval can partly overlap \((L,U)\) and still include parameter values beyond a limit. Equivalence requires the full corresponding interval to lie within the limits.
• Assuming both components fail whenever equivalence fails. That is not necessarily true. An interval crossing only the lower limit fails the lower-bound component; the upper-bound component can pass. Check both endpoints separately.
• Using the wrong interval level. At \(\alpha=0.05\), the usual matching interval is a 90% two-sided interval, not a 95% interval. A mismatched level can make the interval diagnosis disagree with the TOST decision.
• Relying on the point estimate alone. A point estimate inside the margins does not ensure that the uncertainty interval is inside them. In the first example, the estimate is within the margins but the lower component still fails.
• Confusing failure to establish equivalence with proof of non-equivalence. A failed component means the required evidence was not obtained at the chosen level. It is not, by itself, proof that the true contrast lies outside the margins.

As a practical check, record the interval’s lower endpoint, upper endpoint, and both margins before making a conclusion. Then state the lower-component result and upper-component result separately. If the analysis is on a log scale, make the comparison on that scale using log-transformed margins, or exponentiate both interval endpoints and compare them with the ratio limits. Do not compare log-scale endpoints directly with ratio-scale limits.

Why the Diagnosis Is Useful

Naming the failed component makes the result more informative than reporting only “not equivalent.” It tells a reader whether the uncertainty extends below the acceptable range, above it, or beyond both sides. This distinction can guide review of the estimate, its uncertainty, the model, and the study design, while keeping the inferential conclusion appropriately limited.

The confidence-interval check and the two one-sided tests are alternative ways to express the same decision when they use matching assumptions, contrast, margins, and significance level. The interval is often easier to diagnose visually; the test statistics make explicit which inequality has not been supported. Neither method permits a conclusion of equivalence from partial overlap.