Worked FDA Analysis With the Linearized Criterion

From Replicate Data to an FDA Decision

A reference-scaled bioequivalence analysis uses replicate observations to estimate how variable the reference product is within the same subject. The FDA approach combines that estimate with the test–reference treatment contrast through a linearized criterion. Its decision is not based on the variance estimate alone: the uncertainty in the criterion matters, and a separate constraint limits the estimated geometric mean ratio.

The calculations below assume a replicate crossover design and an endpoint analyzed on the natural-log scale, such as AUC or Cmax. The mixed-effects model must represent the design’s treatment, period, and subject structure appropriately, and must estimate the within-subject reference variance from the replicate reference observations. The earlier material on reference-variance estimation and scaled-model coding provides the framework for obtaining those model estimates. Here the focus is how to use the estimates in a complete FDA-style decision.

Definition: Let $d=\mu_T-\mu_R$ be the population mean log-scale contrast between test and reference, and let $\sigma_{WR}^2$ be the within-subject reference variance on the log scale. The FDA linearized criterion is $$ g=d^2-\theta\sigma_{WR}^2, $$ where $\theta=[\ln(1.25)/0.25]^2\approx0.7967$. The criterion supports scaled bioequivalence when its one-sided 95% upper confidence bound is below zero, subject to the additional requirement that the estimated test/reference geometric mean ratio is within 80.00% to 125.0%.

The scale factor $\theta$ is tied to the FDA’s reference-variability threshold. Scaling is used when the within-subject reference standard deviation exceeds approximately 0.294 on the log scale, corresponding to a reference within-subject coefficient of variation of about 30%. The conversion is $CV_{WR}=\sqrt{\exp(\sigma_{WR}^2)-1}$. If the reference variability does not exceed the threshold, the scaled criterion is not the basis for claiming bioequivalence; the applicable unscaled procedure must be used.

For a scaled analysis, the point estimates are $\hat d$ and $\hat\sigma_{WR}^2$, and the estimated criterion is $\hat g=\hat d^2-\theta\hat\sigma_{WR}^2$. A negative $\hat g$ by itself is not sufficient. The upper confidence bound must account for sampling uncertainty in both estimates, including their covariance where relevant. The geometric mean ratio (GMR) estimate is $\exp(\hat d)$; it must independently fall between 0.8000 and 1.250.

Why the Upper Bound Is Needed

The criterion is nonlinear in its inputs: it squares the treatment contrast and subtracts a multiple of the reference variance. Consequently, plugging estimates into $g$ does not show whether the true criterion is convincingly below zero. A confidence bound addresses that uncertainty. One large-sample calculation uses the delta method, which approximates the variance of a smooth function by applying its gradient to the covariance matrix of the estimates.

For $g(d,v)=d^2-\theta v$, where $v=\sigma_{WR}^2$, the gradient is $(2d,-\theta)$. If the fitted model provides $\hat d$, $\hat v$, $SE(\hat d)$, $SE(\hat v)$, and $\operatorname{Cov}(\hat d,\hat v)$, the estimated variance of $\hat g$ is approximated by:

\widehat{\operatorname{Var}}(\hat g) \approx 4\hat d^{\,2}SE(\hat d)^2 +\theta^2SE(\hat v)^2 -4\theta\hat d\operatorname{Cov}(\hat d,\hat v).

The minus sign in the covariance term follows from multiplying the two gradient components, $2\hat d$ and $-\theta$. Omitting covariance is appropriate only when it is zero or when the specified calculation justifiably treats it as zero. A one-sided 95% upper bound is then calculated as $\hat g+t_{0.95,df}SE(\hat g)$. The degrees of freedom must reflect the variance-component analysis; they are not automatically the number of subjects minus one.

Justification: For small estimation errors, a first-order Taylor expansion gives $g(\hat d,\hat v)\approx g(d,v)+2d(\hat d-d)-\theta(\hat v-v)$. Taking the variance of the two error terms gives $4d^2\operatorname{Var}(\hat d)+\theta^2\operatorname{Var}(\hat v)-4\theta d\operatorname{Cov}(\hat d,\hat v)$. Replacing unknown parameters and variances by fitted estimates yields the delta-method variance above. The approximation is most credible when the estimates are reasonably regular and the uncertainty is not dominated by a boundary or severe model misspecification.

Worked FDA Analyses

Each example uses hypothetical model output from a replicate study. To make the arithmetic transparent, the examples state the estimated covariance and use a one-sided $t$ critical value with the stated degrees of freedom. In an actual analysis, the model, variance estimator, covariance, and degrees of freedom must be obtained using a method appropriate to the study design and the applicable FDA implementation.

Worked Example: Both FDA Requirements Are Met

Suppose the fitted model gives $\hat d=0.03922$, $\hat v=0.1200$, $SE(\hat d)=0.0400$, and $SE(\hat v)=0.0600$, with covariance estimated as zero for this calculation. Let $df=20$, so $t_{0.95,20}=1.725$. The reference variance exceeds the scaling threshold $0.294^2=0.08644$, so evaluate the scaled criterion.

Estimated reference variability. The within-subject reference standard deviation is $\sqrt{0.1200}=0.3464$. Its coefficient of variation is $\sqrt{\exp(0.1200)-1}=0.3571$, or 35.71%, above the approximate 30% threshold.

Estimated criterion.

\hat g=(0.03922)^2-(0.7967)(0.1200) =0.001538-0.09560=-0.09406.

Uncertainty and upper bound.

SE(\hat g) \approx\sqrt{4(0.03922)^2(0.0400)^2+(0.7967)^2(0.0600)^2} =\sqrt{0.002295}=0.04791.

$$ U=-0.09406+(1.725)(0.04791)=-0.01140. $$

Because $U<0$, the linearized criterion passes. The GMR estimate is $\exp(0.03922)=1.040$, or 104.0%, within 80.00% to 125.0%. Both requirements are met, so this hypothetical result supports bioequivalence under the stated FDA scaled analysis.

Worked Example: A Negative Estimate Does Not Guarantee a Pass

Consider a second hypothetical analysis with $\hat d=0.08000$, $\hat v=0.09000$, $SE(\hat d)=0.0400$, $SE(\hat v)=0.0600$, and zero estimated covariance. Again take $df=20$ and $t_{0.95,20}=1.725$. Since $0.09000>0.08644$, the estimated reference variance exceeds the scaling threshold.

The estimated criterion is negative:

\hat g=(0.08000)^2-(0.7967)(0.09000) =0.006400-0.07170=-0.06530.

But its standard error is approximately:

SE(\hat g) \approx\sqrt{4(0.08000)^2(0.0400)^2+(0.7967)^2(0.0600)^2} =\sqrt{0.002326}=0.04823.

The upper bound is $U=-0.06530+(1.725)(0.04823)=0.01791$, which is above zero. The linearized criterion therefore does not pass. The GMR estimate is $\exp(0.08000)=1.083$, or 108.3%, inside the required point-estimate range, but that does not rescue the failed upper-bound requirement. This example shows why checking only the sign of $\hat g$ is insufficient.

Worked Example: A Passing Criterion With a Failing Point Estimate

Suppose a third hypothetical analysis gives $\hat d=0.2390$, $\hat v=0.2000$, $SE(\hat d)=0.0200$, and $SE(\hat v)=0.0500$, with zero estimated covariance. Take $df=20$ and $t_{0.95,20}=1.725$. The reference variance exceeds $0.08644$, so evaluate the scaled criterion.

The estimated criterion and its standard error are:

\hat g=(0.2390)^2-(0.7967)(0.2000) =0.05712-0.1593=-0.1022,

SE(\hat g)\approx \sqrt{4(0.2390)^2(0.0200)^2+(0.7967)^2(0.0500)^2} =\sqrt{0.001678}=0.04096.

Thus $U=-0.1022+(1.725)(0.04096)=-0.03156$, below zero, so the linearized criterion passes. However, the GMR estimate is $\exp(0.2390)=1.270$, or 127.0%, above 125.0%. The point-estimate constraint fails; the overall FDA bioequivalence conclusion therefore fails despite the passing upper bound.

Checks, Alternatives, and Limitations

Before interpreting a result, verify that the inputs actually correspond to the FDA criterion. The variance must be the within-subject reference variance on the log scale, not a total variance or a pooled variability estimate that combines different sources. The contrast must be test minus reference in the same model and on the same endpoint scale. The model must match the replicate design, and the variance estimate and its degrees of freedom must be supported by that analysis.

Diagnostics matter because a precise-looking upper bound can be misleading if the model is not credible. Examine residual patterns, convergence, influential observations, and the stability of the reference-variance estimate. Small replicate studies can produce highly uncertain variance estimates; observations with unusual reference replicates may have substantial influence. If model assumptions or variance structure are doubtful, investigate the cause and consider justified sensitivity analyses rather than selecting the result that produces the preferred conclusion.

The FDA linearized criterion is not the same decision rule as the EMA approach with expanded limits. The FDA calculation tests a confidence bound on a variance-scaled criterion and applies a separate GMR point-estimate constraint. The expanded-limit approach expresses its requirement through limits that depend on reference variability. The approaches may therefore produce different outcomes for the same study; one rule should not be substituted for the other or treated as a mere change of notation.

The delta-method upper bound is an approximation. It can be less reliable when estimates are far from regular conditions, when the variance estimate is unstable, or when uncertainty is strongly affected by a boundary. The covariance term can also matter; setting it to zero without justification may change the calculated bound. Accordingly, the worked arithmetic illustrates the reasoning and the required checks, not a replacement for the prespecified analysis method or applicable regulatory software.

A common mistake is to conclude bioequivalence as soon as $\hat g<0$, or to conclude it from a GMR inside 80% to 125% alone. The first ignores uncertainty in the criterion; the second ignores the scaled criterion altogether. The correct conclusion requires the reference-variability condition, an upper confidence bound below zero, and the point-estimate constraint. A failed requirement means the analysis has not established bioequivalence under this rule; it does not prove that the products are clinically different.

Confirm the analysis inputs.
Check the endpoint, log scale, replicate design, fitted model, and that the variance estimate is the within-subject reference variance.

Check the scaling condition.
Compare $\hat v$ with $0.294^2$, or express the estimated reference variability as $CV_{WR}=\sqrt{\exp(\hat v)-1}$.

Evaluate the criterion with uncertainty.
Compute $\hat g$, its standard error, and the one-sided 95% upper confidence bound using an appropriate covariance estimate and degrees of freedom.

Apply the separate point constraint.
Calculate $\exp(\hat d)$ and verify that it is between 0.8000 and 1.250.

State the conclusion conditionally and completely.
Conclude that the analysis supports bioequivalence only when the applicable scaled analysis is used, the upper bound is below zero, and the GMR constraint is met.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

If $\hat v=0.0700$, is the estimated reference variance above the FDA scaling threshold? What does this imply about using the scaled criterion?
With $\hat d=0.0500$ and $\hat v=0.1500$, calculate $\hat g$ using $\theta=0.7967$.
A calculated upper bound is $-0.0040$, but the GMR estimate is 126.0%. Does the analysis meet both requirements?
Why is the estimated criterion $\hat g<0$ alone insufficient to establish bioequivalence?

Solutions. (1) The threshold is $0.294^2=0.08644$, and $0.0700<0.08644$. The estimated reference variability does not exceed the scaling threshold, so the scaled criterion is not the basis for the decision; use the applicable unscaled procedure. (2) $\hat g=(0.0500)^2-(0.7967)(0.1500)=0.002500-0.1195=-0.1170$. (3) No. The upper bound is below zero, but 126.0% exceeds the 125.0% upper point-estimate limit. (4) It is only a point estimate and does not account for sampling uncertainty. The one-sided 95% upper confidence bound must be below zero, and the separate GMR constraint must also be met.

Worked FDA Analysis With the Linearized Criterion

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