Why Propagate Uncertainty by Resampling Subjects?
A model-derived exposure ratio depends on more than the final contrast reported by the fitted model. It also depends on which subjects entered the study, the repeated concentration measurements within each subject, the fitted pharmacokinetic parameters, and the calculation of exposure from those parameters. A standard error for one fitted coefficient may not capture the uncertainty in this entire chain, especially when the exposure metric is a nonlinear function of fitted parameters.
The subject-level bootstrap approximates sampling variation by repeatedly drawing subjects from the observed study and repeating the analysis. The essential design rule is that a subject contributes a complete cluster of observations: their periods, formulations, sampling times, concentration records, and replicate measurements remain associated. This preserves the within-subject structure that supports a crossover or replicate comparison.
The method targets sampling uncertainty under the study design and the analysis procedure. It does not, by itself, account for uncertainty about which structural pharmacokinetic model is correct. Nor can resampling repair non-identifiability, model misspecification, biased sampling, or a poorly supported design.
What Must Be Resampled and Recomputed?
Suppose subjects have been assigned to sequences in a crossover study. If sequence counts are part of the planned design, resample subjects within each sequence, keeping the original number of subjects in that sequence. This retains the sequence allocation in each bootstrap sample. Within a selected subject, retain the records for every period and formulation as observed. In a replicate design, keep all replicate periods and formulation records for that subject together as well.
For each bootstrap sample, refit the same prespecified model and recalculate the same exposure estimand. For example, if the model gives a log-scale formulation contrast \(\hat{\beta}^{*}_b\) in bootstrap sample \(b\), the corresponding geometric mean ratio is \(GMR^{*}_b=\exp(\hat{\beta}^{*}_b)\). The star marks a bootstrap estimate; it does not mean that the parameter is known or fixed.
The sequence of operations matters. Resampling only the original fitted parameter estimates would not reflect the variation caused by refitting the model to a different set of subjects. Similarly, calculating an exposure ratio once and then resampling concentration-time observations separately would not preserve the participant-level dependence.
Why the Bootstrap Distribution Can Represent Sampling Variation
Let the observed subjects be \(n\) independent draws from the target population under the study design. Write \(Z_i\) for the complete record of subject \(i\), including all relevant periods and measurements. The analysis produces an estimate \(\hat{\theta}=T(Z_1,\ldots,Z_n)\), where \(T\) includes model fitting and calculation of the exposure contrast.
The empirical distribution assigns probability \(1/n\) to each observed subject record. A bootstrap sample draws \(n\) records from this empirical distribution with replacement and applies the same analysis function \(T\). The resulting variation in \(\hat{\theta}^{*}\) approximates how \(\hat{\theta}\) would vary across repeated studies, provided the observed subjects reasonably represent the target population and the estimator is sufficiently stable.
Here \(B\) is the number of bootstrap samples. This variance summarizes the spread of the bootstrap estimates; it is not a measure of model fit. The argument is an empirical approximation, not a guarantee: if the study has few independent subjects, important subgroups are absent, or model fits frequently fail, the bootstrap distribution may be a poor guide to repeated-sampling uncertainty.
Worked Example: Resampling Paired Subject Contrasts
Consider a hypothetical four-subject crossover study. Suppose the fitted model produces the following log exposure contrasts, defined as test formulation minus reference formulation. A log contrast is positive when the test exposure is larger.
| Subject | Log exposure contrast |
|---|---|
| 1 | -0.04 |
| 2 | 0.02 |
| 3 | 0.07 |
| 4 | 0.01 |
The mean log contrast is \((-0.04+0.02+0.07+0.01)/4=0.015\), so the estimated geometric mean ratio is \(\exp(0.015)=1.015\). Now take one hypothetical bootstrap draw of subject indices \(1,1,3,4\). The complete records for subjects 1, 3, and 4 are selected, with subject 1 appearing twice; subject 2 is absent from this draw.
This is one bootstrap estimate, not an alternative estimate to average into the original result. Repeating the subject draw and refitting the model produces a distribution of estimates. In an actual analysis, the full observation records—not only a precomputed contrast—are resampled and the model is refitted in each draw.
From Fitted Curves to a Bootstrap Exposure Ratio
A practical implementation must repeat every data-dependent step that defines the estimand. For each bootstrap sample, fit the specified concentration-time model, obtain its fitted parameters, derive exposure for each formulation as prescribed by the analysis, and compute the formulation contrast. If the estimand is the ratio of population geometric means, calculate the contrast on the log scale and exponentiate it. Use the same model specification, covariate handling, constraints, and exposure calculation as in the primary analysis.
The following hypothetical calculation illustrates how resampling subjects affects an exposure contrast. Assume model-derived AUC values, in \(\mathrm{mg\cdot h/L}\), for three subjects under test and reference formulations. For illustration, the model has already been fitted to the records in each period.
Worked Example: Recomputing a Ratio After a Bootstrap Draw
The original model-derived AUC values are: subject 1, test 100 and reference 96; subject 2, test 110 and reference 110; and subject 3, test 90 and reference 100 \(\mathrm{mg\cdot h/L}\). The individual log ratios are \(\ln(100/96)=0.04082\), \(\ln(110/110)=0\), and \(\ln(90/100)=-0.10536\).
The original mean log ratio is \((0.04082+0-0.10536)/3=-0.02151\). Its geometric mean ratio is \(\exp(-0.02151)=0.9787\). Now suppose a bootstrap sample selects subjects \(1,1,3\). Both formulation records and all other records for subject 1 are retained each time that subject is selected. The resampled mean log ratio is:
In the full model-based workflow, the selected subject records are used to refit the pharmacokinetic model before deriving the AUC values and ratio. The arithmetic above isolates the ratio calculation so its effect is transparent. The AUC units cancel in the ratio, so the geometric mean ratio is dimensionless.
Intervals, Diagnostics, and Failure Modes
A common summary is the percentile interval. Sort the \(B\) bootstrap estimates and use the empirical \(\alpha/2\) and \(1-\alpha/2\) quantiles as the endpoints of a \(100(1-\alpha)\%\) interval. For a ratio, it is often more natural to form the interval on the log scale and exponentiate both endpoints; this ensures positive limits. Percentile intervals are easy to interpret, but their coverage can be poor when the estimator is strongly biased, highly skewed, or close to a parameter boundary.
Worked Example: Interpreting a Bootstrap Interval
Suppose a hypothetical analysis uses \(B=2{,}000\) successful subject-level bootstrap samples. The 2.5th and 97.5th percentiles of the bootstrap log contrasts are \(-0.08338\) and \(0.07696\). Exponentiating gives:
This is a 95% bootstrap percentile interval for the model-derived test-to-reference geometric mean ratio. Under the usual repeated-sampling interpretation and the assumptions of the bootstrap procedure, the interval method is intended to cover the target ratio in about 95% of repeated studies. It is not a statement that there is a 95% probability the fixed ratio lies in this particular interval.
If the relevant equivalence limits are 0.80 and 1.25, this illustrative interval is wholly inside those limits. That comparison describes the interval produced by this analysis; it does not establish that the bootstrap procedure has adequate coverage or substitute for the applicable bioequivalence analysis requirements.
Inspect the bootstrap distribution, not only its endpoints. Useful diagnostics include the proportion of failed model fits, the frequency of estimates at parameter constraints, extreme or multimodal contrasts, and whether the bootstrap samples retain the intended sequence counts. Check that the same estimand and calculation are used in every successful replicate. A cluster of failed fits or a long tail driven by a few unstable fits can make a reported interval misleading.
A failed fit should not be silently replaced by the original estimate or discarded without reporting. First determine whether failures arise from an implementation error, insufficient information in some resamples, or genuine instability of the fitted model. Report the number and proportion of failures, investigate their causes, and assess how conclusions change under defensible handling choices. Increasing \(B\) reduces Monte Carlo noise in estimated quantiles, but it does not fix unstable estimation or a weak design.
How This Approach Differs From Alternatives
| Approach | How uncertainty is obtained | Key consideration |
|---|---|---|
| Model-based standard error | Uses the fitted model's estimated sampling variance, often with a standard-error approximation for a contrast. | Can be efficient when the model and variance approximation are adequate; a nonlinear derived metric may require careful propagation. |
| Subject-level nonparametric bootstrap | Resamples observed subject records and refits the analysis. | Preserves observed within-subject dependence but relies on subjects being representative and the fit being stable. |
| Parametric bootstrap | Simulates subject records from a fitted probability model, then refits the analysis. | Can use the full specified model, but its results depend on the adequacy of that model's distributional assumptions. |
A model-based standard error may be preferable when its assumptions are credible and its variance formula directly covers the derived estimand. A subject-level bootstrap is useful when the analysis pipeline is complicated but subjects are the independent sampling units. Neither method automatically includes uncertainty about alternative structural models; that is a distinct question requiring additional analysis.
Common Mistakes and Scope
- Resampling concentration records independently. This breaks the association of repeated measurements within subjects and can understate or distort uncertainty. Resample complete subject records.
- Resampling only within formulation. In a crossover study this can detach a subject's test and reference observations. Resample subjects, preserving their formulation and period records together.
- Holding fitted parameters fixed. Recalculating exposure without refitting omits variation due to model estimation. Refit the model in each bootstrap sample.
- Calling the result structural model uncertainty. The subject-level bootstrap measures sampling variation conditional on the chosen analysis procedure; it does not compare competing structural models or correct misspecification.
- Assuming more resamples solve every problem. More bootstrap draws improve numerical stability of the empirical distribution, not the information content of a small or poorly designed study.
The main assumptions are that subjects are independent sampling units, the observed subject records represent the target population sufficiently well, the resampling scheme reflects the study design, and the fitted analysis is defined and reasonably stable across resamples. If subjects were sampled in higher-level clusters, or the design requires additional allocation constraints, resampling must respect those independent units and constraints rather than treating all records as freely exchangeable.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A four-period replicate study records two formulations for each participant. What should be resampled, and what records should remain together?
- A bootstrap analysis yields a 90% interval for the test-to-reference GMR of 0.94 to 1.12. State its interpretation without assigning a probability to the fixed parameter.
- Why is it insufficient to resample observed fitted AUC ratios without refitting the pharmacokinetic model?
- A large number of bootstrap samples have nonconvergent model fits. Does increasing the number of attempted samples resolve the problem? What should be investigated?
Solutions. (1) Resample participants as complete clusters. Each selected participant contributes all four periods, both formulation records, and their associated concentration measurements. (2) Under the assumptions of the procedure, the 90% interval method is intended to cover the target GMR in about 90% of repeated studies; this interval is not a 90% probability statement about the fixed GMR. (3) Resampling ratios without refitting omits variation in fitted pharmacokinetic parameters and therefore does not propagate uncertainty through the complete analysis. (4) No. More attempts may reveal the same instability more precisely. Investigate model identifiability, convergence behavior, resample composition, constraints, and whether the data support the fitted model; report failures and assess their effect on conclusions.