Borderline NTI Confidence Intervals and Decision Interpretation

Why a Borderline NTI Interval Needs Careful Interpretation

For a narrow-therapeutic-index (NTI) drug, small exposure differences may matter more than they would for many other drugs. The acceptance limits used to assess bioequivalence may therefore be narrower than the conventional 80.00% to 125.00% range. But “NTI” does not, by itself, identify one universal set of limits for every drug, endpoint, or regulatory setting. The decision depends on the criterion specified for the particular assessment.

This tutorial focuses on interpreting a confidence interval near an acceptance boundary. The key distinction is between the estimated treatment ratio and the uncertainty around it. A ratio estimate within the limits is not enough: the required confidence interval must also lie entirely within the applicable limits. Conversely, a confidence interval crossing a boundary means the study has not demonstrated bioequivalence under that criterion; it does not, by itself, prove that the formulations are inequivalent.

Verify the Criterion Before Reading the Interval

A commonly used tighter NTI criterion is 90.00% to 111.11%, often written as 0.90 to \(1/0.90\). This is a useful example, not a universal rule. Product-specific guidance or the applicable regulatory framework may specify different limits, different limits for different endpoints, or a different analysis. The limits and confidence-interval procedure must be prespecified and appropriate to the assessment, not selected after seeing the results.

The examples below use fixed limits of 0.90 and 1.1111 and a 90% confidence interval. If an assessment instead uses an accepted reference-scaled procedure, its decision limits and interval construction must be interpreted according to that procedure; do not substitute the fixed-limit calculation shown here. The earlier material on reference-scaled bioequivalence covers that distinct approach.

For a log-scale estimate \(d\), standard error \(SE\), and degrees of freedom \(df\), a two-sided 90% confidence interval is

Exponentiating both endpoints gives the confidence interval for \(GMR\). The 90% interval corresponds to the two one-sided tests (TOST) equivalence procedure at significance level \(\alpha=0.05\): the null hypothesis is that the ratio is at or beyond at least one acceptance limit, while the alternative is that it lies strictly between both limits. Equivalence is concluded under this procedure only when both one-sided null hypotheses are rejected, which is equivalent to the entire 90% interval being inside the limits.

A Boundary Check, Not a Point-Estimate Check

For the illustrative limits \(L=0.90\) and \(U=1.1111\), the log-scale limits are approximately \(-0.1053605\) and \(0.1053505\). A result is a pass under this criterion only if the lower confidence limit is at least 0.90 and the upper confidence limit is at most 1.1111, before rounding. A point estimate inside the limits does not rescue an interval that extends beyond either boundary.

This is a decision about whether the study demonstrates equivalence to the prespecified standard with the required precision. It is not a test that the true ratio equals 1.00, and a nonpassing result is not proof that the true ratio lies outside the limits.

When the Point Estimate Is Acceptable but the Interval Is Not

An interval that crosses a boundary fails the interval-inclusion requirement, even if the point estimate is comfortably within the range. The correct conclusion is that bioequivalence was not demonstrated under the stated criterion. In regulatory or study reporting, such a result may be described as inconclusive about equivalence, but it is still a nonpassing result for the prespecified decision rule. It should not be reported as proof of inequivalence.

Exact Calculation Matters at the Boundary

A frequent source of errors is rounding the confidence limits too early or applying the wrong acceptance range. Calculations should be made from the unrounded log-scale estimates and confidence limits. Round for presentation only after the decision has been made. If the unrounded endpoint is just outside a limit, displaying a rounded value that appears equal to the limit does not turn it into a pass.

The same log-scale interval can receive different decisions under different standards. For example, the interval \([0.8853,\ 1.1640]\) lies within conventional 80% to 125% limits, but not within the illustrative NTI limits of 90% to 111.11%. It therefore would not pass the latter criterion. This comparison is meaningful only when the applicable standard is known in advance.

Short Justification: Why the Interval Rule Matches TOST

The equivalence alternative on the log scale is \(\log(L)<d<\log(U)\). TOST tests the two parts of the null hypothesis separately: \(H_0:d\leq\log(L)\) against \(H_a:d>\log(L)\), and \(H_0:d\geq\log(U)\) against \(H_a:d<\log(U)\). At \(\alpha=0.05\), each one-sided test uses the corresponding 5% tail of the \(t\) distribution.

The lower test rejects when the estimate is sufficiently above the lower bound:

The upper test rejects when the estimate is sufficiently below the upper bound:

Both tests must reject. Combining the two inequalities gives

which is precisely the condition that the two-sided 90% confidence interval lies wholly within the log limits. Exponentiation preserves the ordering because the exponential function is increasing, so the same inclusion decision holds on the ratio scale.

Check the Analysis and State the Conclusion Precisely

Before interpreting a borderline output, verify that the interval belongs to the intended endpoint and estimand, is on the expected scale, and uses the required confidence level. Check the exact regulatory or protocol criterion, including whether limits differ by endpoint. If multiple endpoints are required to meet their criteria, a pass for one endpoint does not compensate for a nonpass on another.

The interval-based conclusion also depends on the analysis being valid. The model, design, analysis population, and treatment contrast must be appropriate and prespecified; the standard error and degrees of freedom must correspond to that model. Replicate designs may involve variance estimates or criteria not represented by a simple fixed-limit interval. Data-integrity or model diagnostics can identify concerns that warrant investigation, but they do not permit an analyst to change limits after seeing the result.

A useful reporting statement names the endpoint, ratio, interval, limits, and decision. For example: “The estimated test-to-reference ratio was 99.70%, with a 90% confidence interval of 89.94% to 110.52%. Because the lower confidence limit was below the prespecified 90.00% bound, bioequivalence was not demonstrated under the applicable NTI criterion.” This communicates the result without claiming that the formulations have been shown to be inequivalent.

Common Mistakes and Limitations

• Using the point estimate alone. A ratio inside the limits does not establish a pass if its confidence interval crosses a boundary. Apply the interval-inclusion rule.
• Assuming every NTI product has identical limits. Verify the governing product- and endpoint-specific criterion before interpreting the interval.
• Calling a nonpass proof of inequivalence. The result shows that equivalence was not demonstrated with the required criterion and precision; it does not locate the true ratio outside the limits.
• Rounding before deciding. Compare unrounded confidence limits with the exact limits, then round values for reporting.
• Mixing procedures. Do not apply fixed limits to an analysis that uses a different prespecified, accepted procedure, or reinterpret an interval under a more convenient standard.

Even a passing result is a conclusion relative to the chosen endpoints, limits, model, and study precision. It does not guarantee identical exposure in every individual or establish that every clinical outcome is identical. Conversely, an inconclusive or nonpassing interval may reflect limited precision as well as a treatment difference. This tutorial interprets the interval; locating the source of a failed study is a separate analysis.