The FDA Linearized Reference-Scaled Criterion

Why Linearize a Reference-Scaled Boundary?

Conventional average bioequivalence compares the treatment difference on the log scale with fixed limits. That approach does not widen its boundary as within-subject variability increases. Reference-scaled methods instead use the within-subject variability of the reference product to define an expanding boundary. The FDA linearized criterion expresses that idea in a form that compares a squared treatment difference with a reference-scaled variance term.

This tutorial focuses on the criterion itself: its terms, derivation, assumptions, and interpretation. It is a population-level condition, not by itself a complete statistical decision procedure. In particular, using sample estimates in the formula does not account for their uncertainty. The later construction of an upper confidence bound addresses that inferential step.

The difference \(d\) is expressed in log units, and both \(d^2\) and \(\sigma_{WR}^2\) are in squared log units. The constant \(\theta\) is dimensionless, so the two terms being compared have compatible units. The reference variance is specifically within-subject reference variance; it is not the total variance of all observations, nor the variance of the treatment difference.

From a Scaled Boundary to the Criterion

The criterion can be understood by first expressing the treatment difference as an absolute distance from zero. With \(k=\ln(1.25)/\sigma_{W0}\), the corresponding population boundary is \(|d|\leq k\sigma_{WR}\). The distance is allowed to grow in proportion to the reference within-subject standard deviation. Squaring both nonnegative sides gives \(d^2\leq k^2\sigma_{WR}^2\), which is equivalent to the linearized form after setting \(\theta=k^2\).

The word “linearized” refers to the way the criterion is written as a squared treatment difference minus a constant times the reference variance. It does not mean that the treatment difference itself is unsquared, or that the criterion is linear in every parameter. It is a convenient algebraic representation of a variance-scaled population boundary.

Using \(\sigma_{W0}=0.25\), the scaling constant is

The intermediate ratio is approximately \(0.8926\), using the unrounded value of \(\ln(1.25)\); its square is approximately \(0.7967\). This scaling constant is fixed by the specified reference standard-deviation scale and the conventional upper limit. It is not recalculated for each study. The study-specific reference variance enters separately through \(\sigma_{WR}^2\).

Worked Calculations

Assumptions and Scope of Interpretation

The criterion has a meaningful interpretation only when its inputs refer to the intended bioequivalence estimand and are on a consistent log scale. In the usual formulation, \(\mu_T\) and \(\mu_R\) are the population log-scale means for test and reference, and \(\sigma_{WR}^2\) is the within-subject reference variance for the relevant endpoint and design. The data must come from a design that can identify reference within-subject variability, such as an appropriate replicate design. A simple comparison that observes each subject on only one formulation cannot, by itself, provide the required within-subject reference variance.

The population statement also presumes that the reference variance is a well-defined feature of the target population under the study conditions and that the model used to estimate it is appropriate. The criterion does not specify how to estimate variance components, handle incomplete periods, or construct inferential bounds. Those tasks depend on design and statistical analysis choices. A calculated value based on estimates is therefore a descriptive plug-in quantity, not automatically a valid regulatory conclusion.

The assumptions should be checked in context: confirm the analysis scale, endpoint, reference formulation, and variance definition; confirm the replicate observations support estimation of \(\sigma_{WR}^2\); and inspect whether the selected model and data are suitable for that estimation. A mislabeled total variance or an incorrectly transformed CV cannot be repaired by applying the criterion correctly afterward.

Relation to Other Equivalence Boundaries

For comparison, conventional average bioequivalence uses a fixed log-scale interval determined by the 80%–125% limits. The FDA scaled boundary instead has half-width \(k\sigma_{WR}\). At the reference scale \(\sigma_{WR}=0.25\), this half-width is \(\ln(1.25)\), so the boundary matches the conventional log-scale half-width. Below or above that reference standard deviation, the scaled half-width contracts or expands in direct proportion to \(\sigma_{WR}\).

This is distinct in form from the EMA expanded-limit approach and its cap. Those approaches express the expanded limits and cap directly; the FDA linearized criterion expresses a squared difference minus a reference-scaled variance term. The formulas should not be interchanged just because both approaches use reference variability. They use different regulatory formulations and associated decision rules.

The main benefit of the FDA expression is algebraic clarity: the squared treatment difference and scaled variance term can be combined into one quantity whose population value must be nonpositive. Its limitation is equally important: this population inequality alone does not quantify sampling uncertainty. A point estimate of \(d^2-\theta\sigma_{WR}^2\) at or below zero is not, by itself, the complete FDA statistical test. The upper-confidence-bound procedure is needed to make an inferential decision.

Diagnostics, Failure Modes, and a Common Mistake

Several checks help detect errors before interpreting a result. First, verify that \(\sigma_{W0}=0.25\) is a standard deviation and that it has not been confused with a variance of \(0.25\). Second, verify that the variance inserted is the within-subject reference variance, not the reference standard deviation and not a pooled or total variance. Third, check the sign and arithmetic: the criterion is \(d^2-\theta\sigma_{WR}^2\), with the variance term subtracted. Fourth, calculate the equivalent boundary \(k\sigma_{WR}\) as an independent check. The sign of the criterion should agree with whether \(|d|\) is below or above that boundary.

A warning sign is a reported boundary that does not increase when \(\sigma_{WR}\) increases, or a calculation with squared-log units in one term and unsquared log units in the other. Another warning sign is a conclusion based only on rounded quantities when the criterion is very close to zero. Keep adequate precision through the calculation and round only the reported result.

Common mistake: “The reference standard deviation is \(0.25\), so the variance term is \(0.25\), and the scaling constant is \(0.25\).” This confuses the reference standard-deviation scale, the study’s reference variance, and the fixed scaling constant. The correction is to calculate \(\theta=[\ln(1.25)/0.25]^2\approx0.7967\), then multiply that constant by the study-specific variance \(\sigma_{WR}^2\). At \(\sigma_{WR}=0.25\), that variance is \(0.0625\), not \(0.25\).