Residual carryover and the ICH M13A 5% pre-dose rule

Question

In a single-dose 2x2 crossover, ICH M13A excludes a period whose pre-dose concentration is above 5% of that period's Cmax. A 2x2 analysis needs both periods, so the participant leaves the analysis. M13A also asks for at least 12 evaluable participants and does not ask for a test of carryover.

Two questions follow, for washouts between 2 and 10 mean half-lives:

Model

The study is reported in the ADEMP structure (aims, data-generating mechanisms, estimand, methods, performance measures; Morris, White and Crowther 2019). All data are simulated.

With a constant residual fraction r = AUC[c_i1(· + W)] / AUC[c_i1] and no within-participant noise, the naive estimate has a closed form (the algebra of Ocaña, Sánchez and Carrasco 2015 and earlier carryover work):

At θ = 1.25 the bias is about −0.225 r, and at θ = 0.80 about +0.225 r: in both cases toward 1, which is toward equivalence at the boundary that defines consumer risk. Cmax has no closed form and is reported from the simulation.

Replicates and random numbers. The replicate count is sized from a target MCSE of 0.005 for a rate of 0.05: 0.05 × 0.95 / 0.005² = 1,900, rounded up to 2,000. The generator is L'Ecuyer-CMRG with seed 20261025 and one stream per (θ, disposition) cell. Within a cell every washout, arm and configuration reuses the same simulated participants (common random numbers), so differences between arms are paired within study. Committed run: 2000 replicates per cell (MCSE 0.0049 at a rate of 0.05 and at most 0.0112, at a rate of 0.5), 8 cells, seed 20261025, R version 4.6.0 (2026-04-24); the grid took 9.7 s on 4 cores (2026-10-25).

Analyses compared

Every arm uses the same estimator: with d_i = log Y(period 1) − log Y(period 2), log GMR-hat = (mean d_TR − mean d_RT) / 2, SE = ½ s_p √(1/n_TR + 1/n_RT) with s_p the pooled within-sequence SD of d and n − 2 degrees of freedom, and the 90% CI from the t distribution. This equals the participant + period + formulation model fitted by least squares (check T3). BE is concluded if and only if the 90% CI lies within 0.80 to 1.25 on unrounded limits. A study with fewer than 12 evaluable participants or an empty sequence is not evaluable and does not conclude BE.

The six analysis arms.

The rule's leftover bias splits exactly, study by study: A2 − A0 = (A2 − A2s), the residual left in the retained participants, + (A2s − A0), the selection.

Known-answer checks

The test script checks the model, the NCA and the estimator against answers known in advance, runs negative controls, checks the 5% rule on hand-built profiles, recomputes every arm from its definition and checks the MCSE formulas, before any operating characteristic is read. KA1 to KA4 and NC1 and NC2 are the known answers and controls of the study plan.

Known-answer checks and controls, written by tests/test_carryover.R (21 PASS, 0 FAIL, 2 INFO). INFO rows compare against the unvalidated pilot and are reported, not enforced.

Results

Base configuration (no assay error, sampling to 72 h), AUC(0-t), monoexponential disposition unless stated. Every configuration, disposition and endpoint is in out/oc_results.csv.

• Naive analysis. At a true GMR of 1.25 the consumer risk of including everyone is 0.318 (MCSE 0.010) at 2 mean half-lives, 0.159 (0.008) at 3, 0.073 (0.006) at 5 and 0.052 (0.005) at 10, against 0.051 (0.005) with no residual. The bias is toward 1, as the closed form predicts.
• Interval coverage. With no residual the 90% CI covers the true GMR in 0.900 (MCSE 0.007) of studies. Including everyone, coverage falls to 0.678 (0.010) at 2 mean half-lives and 0.832 (0.008) at 3; under the M13A rule it is 0.878 (0.019) at 4 half-lives and 0.903 (0.007) at 5.
• M13A rule, short washouts. At 2 and 3 mean half-lives the rule excludes 100% and 90% of participants and no study is evaluable, so consumer risk is 0 and so is power. At 4 half-lives 87% of studies are not evaluable and power at a true GMR of 0.95 is 0.115 (MCSE 0.007), against 0.987 (0.003) with no residual.
• M13A rule, borderline washouts. At 5 mean half-lives the rule drops 34% of participants (43% in sequence TR, 25% in RT at a true GMR of 1.25) and brings consumer risk to 0.051 (MCSE 0.005); power at 0.95 falls to 0.888 (0.007). At 6 half-lives it drops 16% and consumer risk is 0.059 (0.005), close to the naive 0.065 (0.005).
• Where the rule's bias comes from. At a true GMR of 1.25 the M13A arm's bias is -0.0054 (MCSE 0.0008) at 5 half-lives; selection contributes -0.0043 (0.0008) and retained residual -0.0010 (0.0001). Selection is 80% (MCSE 3%) of the total at 5 half-lives and 74% (3%) at 6; at 8 it is 44% (5%), where the two parts are comparable, and at 10 it is 28% (6%), where the retained residual is the larger part (delta-method MCSEs).
• Subtraction (sensitivity analysis). With a monoexponential residual and no assay error, subtracting the extrapolated pre-dose concentration matches the oracle almost exactly, because the model it extrapolates is the model that generated the data: over the base grid the largest per-study difference in log GMR-hat from the oracle is 2.1e-09 (λ_z is fitted to period-1 points that still carry a trace of absorption; 1.7e-11 with the true k), and every study reaches the oracle's BE decision. Under the biexponential disposition its consumer risk at 3 half-lives is 0.049 (MCSE 0.005) against 0.045 (0.005) with no residual; with 8% assay error and an LLOQ it is 0.045 (0.005) against 0.048 (0.005).
• Sampling. With sampling to 72 h and a mean half-life of 24 h, 20% of profiles have AUC(0-t) below 80% of AUC(0-inf), and 56% of studies cross the M13A 20% trigger even with no residual. Sampling to 120 h brings these to 0.6% and 0.0%.
• Endpoints. Without assay error, Cmax and the joint decision (AUC(0-t) and Cmax both within the limits) give the same picture as AUC(0-t): their BE rates differ from it by at most 0.0015 and 0.0015 in the base configuration with the monoexponential disposition, and by at most 0.015 anywhere without assay error, because within-participant variation then largely scales each profile as a whole. With 8% assay error each sample has its own error and the endpoints separate: the joint rate is up to 0.076 lower than the AUC(0-t) rate (A1 arm, 2 mean half-lives, true GMR 1.25, biexponential disposition).

Consumer risk

T2. Rate of concluding BE on AUC(0-t), true GMR 1.25 (consumer risk). Studies that are not evaluable count as not concluding BE. Columns: washout as a multiple of the mean half-life (24 h); KA1 = 10 times the largest individual half-life in each replicate. Cells: estimate (MCSE). n/e: no evaluable study.

T2. Rate of concluding BE on AUC(0-t), true GMR 0.80 (consumer risk). Studies that are not evaluable count as not concluding BE. Columns: washout as a multiple of the mean half-life (24 h); KA1 = 10 times the largest individual half-life in each replicate. Cells: estimate (MCSE). n/e: no evaluable study.

Power

T2. Rate of concluding BE on AUC(0-t), true GMR 0.95 (power). Studies that are not evaluable count as not concluding BE. Columns: washout as a multiple of the mean half-life (24 h); KA1 = 10 times the largest individual half-life in each replicate. Cells: estimate (MCSE). n/e: no evaluable study.

T2. Rate of concluding BE on AUC(0-t), true GMR 1.00 (power). Studies that are not evaluable count as not concluding BE. Columns: washout as a multiple of the mean half-life (24 h); KA1 = 10 times the largest individual half-life in each replicate. Cells: estimate (MCSE). n/e: no evaluable study.

Bias and interval coverage

T1a. Bias of log GMR-hat, AUC(0-t), true GMR 1.25. Columns: washout as a multiple of the mean half-life (24 h); KA1 = 10 times the largest individual half-life in each replicate. Cells: estimate (MCSE). n/e: no evaluable study.

T1b. Bias of log GMR-hat, AUC(0-t), true GMR 0.80. Columns: washout as a multiple of the mean half-life (24 h); KA1 = 10 times the largest individual half-life in each replicate. Cells: estimate (MCSE). n/e: no evaluable study.

T1c. Coverage of the 90% CI for the log GMR, AUC(0-t), true GMR 1.25 (studies evaluable in that arm). Columns: washout as a multiple of the mean half-life (24 h); KA1 = 10 times the largest individual half-life in each replicate. Cells: estimate (MCSE). n/e: no evaluable study.

Exclusions and studies that are not evaluable

T3. M13A pre-dose rule: share of participants excluded (overall and by sequence) and share of studies not evaluable (fewer than 12 participants or an empty sequence). Columns: washout as a multiple of the mean half-life (24 h); KA1 = 10 times the largest individual half-life in each replicate. Cells: estimate (MCSE). n/e: no evaluable study.

The AUC coverage condition

T4. M13A coverage condition AUC(0-t) / AUC(0-inf) ≥ 0.80 at a true GMR of 1.25: mean share of analyzed profiles below 0.80, and share of studies in which more than 20% of profiles fall below it (M13A then asks for a discussion of validity). Columns: washout as a multiple of the mean half-life (24 h); KA1 = 10 times the largest individual half-life in each replicate. Cells: estimate (MCSE). n/e: no evaluable study.

Selection versus retained residual

T5. The M13A arm's bias split into retained residual (A2 - A2s) and selection (A2s - A0), and the rule against including everyone (A2 - A1), paired within study, AUC(0-t). Bias terms use studies evaluable under the rule; BE-rate terms use all studies. Cells: estimate (MCSE). Selection share: (A2s - A0) / (A2 - A0), with its delta-method MCSE; left blank when the total is within 2 MCSE of 0.

Single-study ADaM demonstration

The same rule applied to one study from the repository's PK sandbox, generated with the new washout_h option of simulate_adam.R (the sandbox's own PK model, median half-life about 8 h, washout 48 h). ADPC period-2 pre-dose rows then carry the period-1 residual: quantified when it is at or above the LLOQ, and reported as <LLOQ (counted as 0, M13A) when it is below. ADPP gains PDCMXR (pre-dose concentration / Cmax of the same period), ANL02FL (bioequivalence analysis flag, "Y" or null) and PKEXRSN (the reason, as in the repository's flagship study).

With washout_h = 48, 11 of 40 participants have a period-2 pre-dose concentration above 5% of that period's Cmax (largest ratio 0.107). Their ADPP rows carry a null ANL02FL in both periods, and the period-2 rows give the reason in PKEXRSN; 29 participants remain for the bioequivalence analysis.

Participants excluded from the bioequivalence analysis in the ADaM demonstration (out/adam_demo/adpp.csv; PDCMXR = pre-dose concentration / Cmax, period 2).

Variable names: the ADaMIG for Non-compartmental Analysis Input Data (ADNCA) v1.0 defines NCAXFL, NCAwXRS and PKSUMXF to flag concentration records left out of the NCA or of PK summaries (as listed in the Pinnacle 21 and Certara webinar "What is ADNCA", September 2024; the CDISC document itself was not re-read for this page). This rule removes a participant from the bioequivalence model, not from the NCA, so the demonstration uses an ADPP analysis flag and the reason variable of the repository's flagship study instead. Record-level ADaM flags take "Y" or null. With washout_h unset the generator's output is byte-identical to the committed sandbox files (check T10).

What this shows and does not show

SAP language

Wording a statistical analysis plan could use for the primary analysis and the two sensitivity analyses studied here:

"In this single-dose 2x2 study, a period whose predose concentration exceeds 5% of that period's Cmax will be excluded for that participant (ICH M13A); because both periods are required, the participant will be excluded from the bioequivalence analysis, and exclusions will be reported by sequence. As sensitivity analyses, the primary analysis will be repeated including all participants and after subtracting the extrapolated residual (predose concentration times exp(-lambda_z t), lambda_z from that participant's period-1 terminal phase). The bioequivalence conclusion rests on the primary analysis."

CSR appendix shell

Shells for reporting the rule and the sensitivity analyses; xx marks a value filled from the study's ADPP.

Table 14.2.x.1. Participants excluded from the bioequivalence analysis under the pre-dose rule, by sequence (PK population)

Table 14.2.x.2. Bioequivalence of test versus reference: primary analysis and sensitivity analyses

Footnotes for the shell: model with sequence, participant within sequence, period and formulation; limits compared unrounded; the conclusion rests on the primary analysis. A by-participant listing (16.2.x) gives USUBJID, sequence, period, pre-dose concentration, Cmax, their ratio and PKEXRSN for every participant excluded.

Provenance and credit

Established prior art, credited first:

• Carryover is aliased with sequence in the 2x2 design: Grizzle 1965, Biometrics 21:467-480; Freeman 1989, Statistics in Medicine 8:1421-1432.
• Carryover in bioequivalence crossovers: Senn, D'Angelo and Potvin 2004, Pharmaceutical Statistics 3:133-142; Ocaña, Sánchez and Carrasco 2015, Pharmaceutical Statistics 14:400-408, the closest prior art for the consumer-risk algebra above.
• Two one-sided tests: Schuirmann 1987, Journal of Pharmacokinetics and Biopharmaceutics 15:657-680. The pre-dose rule, the 12-participant floor and the coverage condition: ICH M13A (Step 4 23 Jul 2024; FDA final 30 Oct 2024; EU effective 25 Jan 2025).
• Adjusting for incomplete washout with a model (predict, then subtract or integrate): Huang et al., PAGE 32 (2024), abstract 10872; a Simulations Plus abstract at PKUK; and the KEYNOTE-555 protocol (NCT03665597), which adjusts when pre-dose is at least 5% of Cmax. These three come from the study plan and were not re-read for this page.
• The failure mode itself: Zhang Z (FDA CDER, Office of Generic Drugs). Common bioequivalence study design deficiencies and how to avoid them. SBIA Generic Drug Forum, 22-23 April 2026, slide 8 (fda.gov/media/194074). It reports a crossover that failed because an inadequate washout led to many participants being excluded under the 5% of Cmax recommendation, and advises setting the washout from the variability of the half-life rather than its mean alone and pre-specifying alternatives such as subtraction. This page puts numbers on that failure mode.
• Reporting of simulation studies: Morris, White and Crowther 2019, Statistics in Medicine 38:2074-2102.

What is new here is narrow. The failure mode and the advice to pre-specify alternatives are in the FDA presentation above; what this page adds is the operating characteristics of the M13A exclusion rule itself (consumer and producer risk, exclusion and non-evaluable rates) at borderline washouts under clearance variability, and the split of the rule's leftover bias into selection and retained residual. Searches of the literature and the BEBAC forum on 2026-10-25 found discussion of the rule but no published simulation of these quantities; that is not proof that none exists. The algebra, the 5% rule and predict-then-subtract are all established.

Design. Comparing a naive, a censored and a model-adjusted analysis follows the design of my cluster randomized trial simulation study, where contamination plays the role residual drug plays here.

Iben P. Contamination-resilient cluster randomized trial design under spatial interference: a simulation study of contamination control. Statistics in Medicine, submitted. Preprint: doi:10.5281/zenodo.23486177 (latest version: doi:10.5281/zenodo.23486176). Study code and data: doi:10.17605/OSF.IO/XPA4S. That manuscript acknowledges Dr. Kathryn Colborn, who posed the original question, and the co-authors of an earlier unpublished trial-data manuscript: Banaei-Kashani, Kechris, Echodu, Eganyu, Bever and Yeka.

How it was built and checked. The study plan was prepared with AI assistance and scored by two AI-agent reviewers; one of them wrote the Python check of the selection effect that T9 compares against, and an earlier, unvalidated Python pilot supplied the expected patterns. This module was drafted with AI assistance. T9 is a second implementation of the same plan, not independent double programming by a second person. Human sign-off is not yet recorded.

Run it

Rscript sim_lifecycle/carryover_oc/R/run_oc.R            # grid and ADaM demonstration (about 10 s on 4 cores)
Rscript sim_lifecycle/carryover_oc/tests/test_carryover.R  # checks; writes the pass table
Rscript sim_lifecycle/carryover_oc/R/tables.R              # tables, figures and this page's numbers

Run from the repository root, in that order. Base R 4.6.0 only (the parallel package ships with R); OC_CORES=1 runs serially with the same results. The PKNCA cross-check in T2 runs when PKNCA is installed and is skipped otherwise.

• R/dgm.Rsettings, participants, profiles, residual, assay, closed form
• R/nca_m13a.RNCA for many profiles at once (M13A conventions)
• R/analyze.R2x2 estimator, arms, MCSEs, decomposition, streams
• R/run_oc.Rthe grid and the ADaM demonstration
• R/tables.RT1 to T5, F1 to F3, generated page blocks
• tests/test_carryover.RT1 to T12 known-answer checks and controls
• out/oc_results.csvevery cell, washout, arm and endpoint, with MCSEs
• out/oc_bias_decomposition.csvselection and retained residual, paired, with MCSEs
• out/known_answer_checks.csvthe pass table above
• out/adam_demo/adpp.csvADPP with PDCMXR, ANL02FL and PKEXRSN

Tables T1 to T5 are also in out/tables/ as CSV; run facts are in out/run_info.csv.