Why a Scaled Criterion Has a Point-Estimate Guardrail
Reference scaling adjusts the FDA bioequivalence criterion for high within-subject variability of the reference product. That adjustment can make the statistical criterion less restrictive as reference variability increases. The FDA procedure therefore also imposes a separate requirement: the test-to-reference point estimate must lie between 80.00% and 125.00%. A result that satisfies the variability-scaled criterion but misses this point-estimate range does not satisfy the combined FDA decision rule.
This requirement is a guardrail on the estimated average treatment difference. It is not another confidence interval, and it does not replace the FDA upper-confidence-bound criterion. Both checks must pass. The analysis here concerns the point-estimate constraint used with the FDA reference-scaled approach; it is not a universal rule for every regulator or every bioequivalence procedure.
Deriving the Equivalent Log-Scale Check
Exposure measures such as AUC and \(C_{\max}\) are commonly analyzed on the natural-log scale. On that scale, the estimated difference \(d\) is the log of the test-to-reference geometric mean ratio. Exponentiating the difference returns the ratio, so the percentage-scale condition can be translated directly to a condition on \(d\).
Because \(0.80=1/1.25\), the two log bounds have equal magnitude and opposite signs:
Thus, on the log scale, the point estimate must lie between approximately \(-0.22314\) and \(0.22314\). This symmetry does not mean the percentage-scale limits are equally distant from 100%: 80% is 20% below 100%, whereas 125% is 25% above it. The limits are reciprocals, which is why their logarithms are symmetric.
The comparison uses the estimated ratio, not the ratio for each participant and not a ratio of arithmetic means calculated without regard to the specified model. The relevant \(d\) must come from the treatment effect estimate in the analysis used for the FDA procedure. If the model includes design terms or other prespecified adjustments, use its estimated test-minus-reference contrast.
Worked Example: A Point Estimate Inside the Range
Suppose a hypothetical analysis of a log-transformed endpoint estimates the test-minus-reference contrast as \(d=0.1655\). The FDA upper-confidence-bound analysis for the same endpoint and analysis yields \(U_{\mathrm{FDA}}=-0.0040\). Treat that upper-bound result as the output of the prescribed FDA analysis; this example focuses on applying the additional point-estimate check.
First convert the log-scale estimate to the test-to-reference geometric mean ratio:
The estimated ratio is 1.180, or 118.0%. Since \(0.80\leq1.180\leq1.25\), the point-estimate constraint passes. Equivalently, \(-0.22314\leq0.1655\leq0.22314\).
The FDA upper-bound check also passes in this hypothetical example because \(-0.0040\leq0\). Therefore, both required checks pass, and the combined FDA reference-scaled criterion is met, subject to the validity of the prescribed model and analysis.
The Combined Decision Is Conjunctive
The point-estimate constraint and the FDA upper-confidence-bound criterion answer related but distinct questions. The upper-bound criterion evaluates the scaled statistical criterion using the estimated treatment effect and reference variability under the prescribed procedure. The point-estimate constraint limits how far the estimated average ratio itself may be from 1.00. A passing result on one cannot compensate for a failing result on the other.
| FDA upper-confidence-bound criterion | Point-estimate range | Combined decision |
|---|---|---|
| Pass | Pass | Both requirements pass |
| Pass | Fail | Does not meet the combined criterion |
| Fail | Pass | Does not meet the combined criterion |
| Fail | Fail | Does not meet the combined criterion |
This is a logical “and”: success requires \(U_{\mathrm{FDA}}\leq0\) and \(0.80\leq\exp(d)\leq1.25\). Passing the point-estimate check alone does not establish bioequivalence, just as passing the scaled criterion alone does not establish it. The upper-bound calculation and its assumptions are addressed in the earlier material on constructing the FDA upper confidence bound; the present check is applied alongside that result.
Worked Example: Scaling Passes but the Point Estimate Fails
Suppose a hypothetical FDA analysis reports \(U_{\mathrm{FDA}}=-0.0100\), which passes the upper-bound criterion. Its estimated log-scale contrast is \(d=0.2390\). Determine whether the combined criterion passes.
Convert the treatment contrast to a ratio:
The point estimate is 127.0%. Since \(1.270>1.25\), it is outside the permitted range. On the log scale, the same failure is \(0.2390>0.22314\). Although \(-0.0100\leq0\) means the reported FDA upper-bound criterion passes, the point-estimate constraint fails. The combined FDA decision therefore does not pass.
This is the essential reason that a variability-scaled result cannot be interpreted on its own. A favorable scaled upper bound does not erase a point estimate outside 80%–125%.
Boundary Values, Rounding, and Reporting
The stated limits are inclusive: a point estimate exactly equal to 0.80 or 1.25 satisfies the point-estimate condition. In practice, however, calculations should use the unrounded model estimate and ratio. Display rounding can make a result close to a limit look equal to it when the underlying value is just outside, or make it appear outside when the underlying value is just inside. Report sufficient digits to make the decision reproducible, while also presenting a readable percentage.
Worked Example: Checking a Lower-Boundary Estimate
Suppose a hypothetical analysis gives \(d=-0.22314\) and \(U_{\mathrm{FDA}}=-0.0006\). Use the log-scale and ratio-scale checks to evaluate the point estimate and the combined result.
The lower log bound is \(\ln(0.80)\approx-0.22314\). Thus the reported log estimate is at the lower boundary to the shown precision. Transforming it gives:
At the exact boundary, \(G=0.80\), so the inclusive point-estimate condition passes. The reported upper bound also passes because \(-0.0006\leq0\). On these hypothetical values, both conditions pass. For an actual analysis, the decision must be based on the unrounded estimate: if its full-precision value were below \(\ln(0.80)\), the point-estimate check would fail even if a rounded display showed 0.8000.
Assumptions, Diagnostics, and Common Pitfalls
The arithmetic of the point-estimate constraint is simple, but its interpretation depends on a valid estimate. Use the contrast from the prespecified FDA analysis, on the log scale, for the endpoint and analysis population under evaluation. The underlying study and model must support that analysis: treatment, period, sequence, and replicate information should be handled as prescribed; the required within-subject reference variability must be estimated from suitable replicate data; and missing observations or model failures must not be ignored. A ratio computed from a different model, a different analysis population, or an unadjusted summary should not be substituted merely because it is available.
Useful checks include confirming the direction of the contrast, confirming that the ratio is test divided by reference, and independently checking that exponentiating \(d\) reproduces the reported ratio. A sign reversal changes \(G\) to \(1/G\); a numerator-reference reversal can therefore create an apparent discrepancy. Also inspect the full-precision value when the ratio is near 0.80 or 1.25. The point-estimate check itself does not diagnose a poor model, invalid variance estimate, protocol deviations, or data errors; those need their own review.
A frequent mistake is to reason, “The FDA scaled upper bound is below zero, so bioequivalence is established.” The correction is to evaluate both conditions explicitly. Another mistake is to treat 80%–125% as a requirement that the entire confidence interval lie within those limits. That is not what this point-estimate constraint says: it restricts the point estimate, while the separate scaled upper-bound calculation supplies its own inferential criterion. Do not convert the point-estimate check into an unrequested confidence-interval test.
The principal alternative is conventional average bioequivalence using two one-sided tests and the corresponding confidence-interval limits. That approach uses fixed limits and does not replace the FDA reference-scaled decision rule for a study analyzed under that procedure. EMA expanded limits are another regulatory approach, with their own conditions and rules; they should not be mixed with the FDA point-estimate constraint. The correct procedure is determined by the applicable regulatory framework and protocol, not selected after seeing which gives a favorable result.
Finally, passing these statistical checks is evidence under the specified analysis and regulatory decision rule, not proof that two products are identical in every respect or that every patient will have the same exposure. The point-estimate restriction is one component of a defined bioequivalence assessment. Its usefulness is clearest when interpreted together with the scaled criterion, the study design, and the quality of the underlying data.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A hypothetical analysis gives \(d=0.1000\). Calculate the test-to-reference geometric mean ratio and decide whether the point-estimate constraint passes.
- The FDA upper-confidence-bound result is \(-0.0030\), but the estimated ratio is 1.26. Does the combined criterion pass? Explain.
- Why is it incorrect to say that the 80%–125% point-estimate constraint requires the full confidence interval to fall inside those limits?
- A report gives a rounded ratio of 0.800. What additional information should be checked before concluding that the point-estimate constraint passes?
Solutions. (1) \(G=\exp(0.1000)=1.105\), approximately 110.5%; it lies between 0.80 and 1.25, so the point-estimate check passes. (2) No. The upper-bound criterion passes because \(-0.0030\leq0\), but 1.26 exceeds 1.25, so the point-estimate check fails and the combined criterion is not met. (3) The constraint applies to the point estimate. The separate FDA upper-bound procedure supplies the inferential check; imposing an entire-interval requirement would change the stated rule. (4) Check the unrounded model estimate and full-precision ratio. A displayed 0.800 could represent a value below 0.80, which would fail the inclusive bound.