Regulatory Criteria

The regulatory criterion is the optional second positional argument to pumas_be (or the first argument to assess_be). When it is omitted, the default is StandardBioequivalenceCriterion (standard average bioequivalence). Each criterion is a singleton type implementing a specific regulatory agency's rule. This page states what each criterion checks and names the document it implements, which is cited in full under Regulatory references; for the mathematical derivations see the Pumas course on bioequivalence.

Standard average bioequivalence (ABE) is established by the two one-sided tests (TOST) procedure at the 5% level, or equivalently by requiring the 90% confidence interval for the geometric mean ratio (GMR) of test to reference to lie within [0.80, 1.25].

The other criteria modify these fixed limits for two drug classes: highly variable drugs (HVD), whose within-subject coefficient of variation CVᵣ exceeds 30% and for which fixed limits are unduly strict, and narrow therapeutic index drugs (NTID), for which fixed limits are unduly permissive. Both modifications scale the acceptance region with the reference within-subject standard deviation σwᵣ (on the log scale) through a regulatory constant θ (alt. k = √θ).

Standard ABE methodology is harmonized internationally by the ICH M13A guideline (2024); the harmonization of the highly variable and narrow therapeutic index approaches is deferred to ICH M13C, under development at the time of writing, so the HVD and NTID criteria below remain agency-specific.

Summary

* Not every replicate design qualifies for the FDA RSABE criteria. Howe's statistic is only defined on the RSABE-supporting subset: the designs the package classifies under the "Replicated crossover (supports ABE, ABEL, and RSABE)" paradigm, namely the partially-replicated RRT|RTR|TRR or one of the fully-replicated RTRT|TRTR, RTTR|TRRT, RRTT|TTRR (see Study Designs). A replicate design outside this subset still supports EMA ABEL, but not the FDA RSABE statistic. The FDA_HighlyVariable criterion reverts to standard ABE at or below cv_min, but reference scaling itself requires one of these designs; FDA_NarrowTherapeuticIndex additionally requires a fully replicated one of them so that σ̂wₜ (and hence the variability ratio) is estimable. EMA ABEL, by contrast, runs on any replicated design (partial or full), since it needs only σ̂wᵣ.

The constant is calibrated as θ = (ln Δ / σ_W0)², where Δ is the limit multiplier and σ_W0 the baseline within-subject standard deviation at which the scaled and standard limits coincide. Its square root, k = √θ, is the equivalent factor on the standard-deviation scale: θ multiplies the reference variance σwᵣ² in the linearized criterion, while k multiplies the standard deviation σwᵣ in the implied acceptance limits exp(±k σwᵣ). So, for example, the EMA's θ = 0.5776 corresponds to the commonly cited k = 0.76. The reference_scale_default function returns θ for a criterion; passing reference_scale to pumas_be or run_be overrides it directly.

Details of the FDA criteria

The FDA frames reference scaling as a hypothesis test on the linearized criterion

\[(\mu_T - \mu_R)^2 - \theta\, \sigma_{wR}^2 < 0,\]

decided through Howe's Approximation I.^[howe] The criterion is decomposed as x = β̂² − SE² (estimating (μ_T − μ_R)², bias-corrected) and y = −θ σ̂wᵣ², each with its own upper bound Uₓ = max(|ℓ|, |u|)² (from the 90% CI [ℓ, u] of the log-GMR) and U_y = y·d / χ²_{d,0.95}. The Howe statistic is

\[\hat{h} = (x + y) + \sqrt{(U_x - x)^2 + (U_y - y)^2},\]

and reference-scaled bioequivalence is concluded when ĥ ≤ 0 (a non-positive 95% upper confidence bound for the linearized criterion). The auxiliary quantities β̂, SE, and σ̂wᵣ come from a within-subject-contrast linear model, not from the mixed-model variance components (see Within-subject variability).

The FDA_HighlyVariable criterion (RSABE for HVD). The scaling constant is θ = (ln 1.25 / 0.25)² ≈ 0.7967 (σ_W0 = 0.25, i.e. the scaled limits meet [0.80, 1.25] at CVᵣ ≈ 25.4%). When the estimated CVᵣ exceeds cv_min (default 0.30, i.e. σ̂wᵣ > 0.294), bioequivalence requires ĥ ≤ 0 and the GMR point estimate in [0.80, 1.25]; otherwise the analysis reverts to standard ABE. Implements the FDA progesterone product-specific guidance.

The FDA_NarrowTherapeuticIndex criterion (RSABE for NTID). Three criteria must hold simultaneously: standard ABE (CI ⊆ [0.80, 1.25]); the RSABE test ĥ ≤ 0 with θ = (ln(1/0.9) / 0.1)² ≈ 1.1101 (σ_W0 = 0.10); and an upper bound on the within-subject standard-deviation ratio,

\[\frac{\hat{\sigma}_{wT}/\hat{\sigma}_{wR}}{\sqrt{F_{d_T, d_R,\, 0.05}}} \le 2.5,\]

constructed from the F-distribution. A fully replicated design is required so that σ̂wₜ is estimable. Implements the FDA warfarin sodium product-specific guidance.

Details of the EMA criteria

The EMA_HighlyVariable criterion (ABEL: average bioequivalence with expanding limits). Unlike the FDA's hypothesis-test formulation, ABEL retains the confidence-interval-within-bounds paradigm and merely widens the bounds. When CVᵣ exceeds cv_min (default 0.30) the acceptance limits expand to

\[\left[100\, e^{-k\, \hat{\sigma}_{wR}},\; 100\, e^{k\, \hat{\sigma}_{wR}}\right], \qquad k = \sqrt{\theta} = 0.76\ \ (\theta = 0.5776),\]

where the CVᵣ entering σ̂wᵣ is clamped at cv_max (default 0.50), so the bounds stop widening beyond CVᵣ = 50% and plateau at ≈ [69.84, 143.19]%; at or below cv_min, standard ABE applies instead. The GMR point estimate must also lie in [0.80, 1.25]. ABEL applies to Cmax; the EMA requires AUC to meet the standard limits regardless of variability. The estimate σ̂wᵣ follows the EMA Q&A's reference-only fixed-effects ANOVA (its SAS scheme; see Within-subject variability), distinct from the Q&A's Methods A/B/C, which are the treatment-comparison models (see Models). Implements the EMA Guideline on the Investigation of Bioequivalence.

The EMA_NarrowTherapeuticIndex criterion (tightened ABE). The acceptance limits are simply narrowed to [0.90, 1.11]; the model and inference are otherwise identical to standard ABE, and no replicate design is needed.

Multiplicity

No correction to the nominal level is applied for multiple tests. The TOST is an intersection–union test (both one-sided tests must reject for bioequivalence to be concluded), so it controls the type I error at the nominal level without adjustment. The same argument applies to co-primary endpoints (e.g. AUC and Cmax), which must all pass: no α correction is applied across them, and each endpoint is run with a separate pumas_be call. In multi-formulation designs each test formulation is compared to the reference separately and reported with its own confidence interval and assessment, again at the nominal level.

Confidence levels

Three arguments to pumas_be / run_be set the confidence levels behind the criteria's interval and reference-scaling quantities:

The level_rsabe_stat and level_var_ratio arguments only affect the reference-scaling criteria: level_rsabe_stat the FDA RSABE bound, level_var_ratio the FDA NTID variability ratio.

A note on the last one: level_var_ratio is a two-sided level, so its default 0.9 reads as (90%) in the Variability Ratio Quantile output row, but the figure reported is the upper limit of that interval, evaluated at 1 − (1 − level_var_ratio)/2, so the default yields a one-sided 95% bound, the F quantile written F_{d_T, d_R, 0.05} in the FDA criteria above.

Overriding cv_min and cv_max

These override the switching/cap thresholds on CVᵣ. When nothing (default), the criterion's regulatory defaults apply. The cv_min override is consumed only by FDA_HighlyVariable, FDA_NarrowTherapeuticIndex, and EMA_HighlyVariable, and cv_max only by EMA_HighlyVariable. Passing one to a criterion that does not use it raises an ArgumentError (a no-op value equal to the default is accepted).

Example: Health Canada expanding limits

Health Canada's standard for highly variable drug products is the same expanding-limits procedure as the EMA's: the acceptance limits widen as exp(±k·σ̂wᵣ) with the identical constant k = 0.76 (θ = 0.5776). The two differ only in where the expansion is capped.

Its rule is stated directly on σ̂wᵣ: the standard [80, 125]% when σ̂wᵣ ≤ 0.294 (CVᵣ ≤ 30%), the scaled limits when 0.294 < σ̂wᵣ ≤ 0.534 (30% < CVᵣ ≤ 57.4%), and a fixed [66.7, 150]% above that, whereas the EMA instead stops expanding at CVᵣ = 50%, giving [69.84, 143.19]%. That final clause is exactly the clamping EMA_HighlyVariable already performs, so the criterion reproduces the Health Canada standard once the cap is moved through cv_max:

using Bioequivalence, PharmaDatasets

be_data = dataset(joinpath("bioequivalence", "RTRT_TRTR", "PJ2017_4_4"))
pumas_be(be_data, EMA_HighlyVariable; endpoint = :AUC, cv_max = 0.574)

Only the cap changes: the Maximal CVᵣ for Reference Scaling row now reads 57.4 | 0.534. Since this reference has CVᵣ ≈ 35% (below both agencies' caps), the acceptance interval here is identical to the one the EMA default would give. The two sets of limits coincide for every CVᵣ up to 50% and differ above it: past its own cap Health Canada plateaus at [66.7, 150]%, whereas the EMA has already plateaued at [69.84, 143.19]%.

Everything else carries over from EMA_HighlyVariable: at or below cv_min (30%) the analysis reverts to standard ABE, and the GMR point estimate must lie within [80, 125]%, the same bound Health Canada places on the relative mean.

The two agencies differ in which endpoint is scaled: Health Canada expands the interval for AUC (its Cmax standard asks only that the relative mean fall in [80, 125]%, with no confidence interval), whereas the EMA expands Cmax and holds AUC to the standard limits. The package is endpoint-agnostic and scales whichever endpoint is analyzed, so the choice is made through the endpoint argument. Health Canada also excludes critical-dose drugs from this approach, a restriction the package does not encode.

Helpers

• The reference_scale_default function returns the criterion's θ (or nothing).
• The reference_scaled_acceptance_bounds function returns the implied lower/upper GMR bounds for a given σwᵣ and θ (the EMA ABEL expanded limits).

The agency documents these criteria implement are listed under Regulatory references on the Overview page.