Why the Design Must Travel With the Subject
In an NLME analysis, observations are interpreted in relation to a subject’s dose history, sampling schedule, and covariates. These features help determine what the model predicts for that subject. A bootstrap replicate that pairs an observation with the wrong dose or time does not merely rearrange the data: it can describe a different experiment and therefore a different source of uncertainty.
The central task is to decide which parts of the study are held fixed and which are resampled or simulated. In a subject-level nonparametric bootstrap, a subject’s records travel together as a cluster. In a parametric bootstrap, the design is usually held at its observed values while new outcomes are generated under the fitted model. In either case, the replicate must preserve the design elements relevant to the chosen target.
An observation time is meaningful relative to the dosing history. For example, a concentration measured 4 hours after a dose is not equivalent to one measured 4 hours before a dose. Similarly, a covariate used to define clearance or volume must remain attached to the appropriate subject, and time-varying covariates must stay aligned with the occasions at which they were recorded.
State the Bootstrap Target First
Let \(D\) denote the design information and \(Y\) the observed outcomes. The fitted parameter estimate can be written as \(\hat{\theta}=T(Y;D)\), emphasizing that the estimate depends on both the outcomes and the design. Many NLME bootstrap analyses seek uncertainty conditional on the study’s realized design: the dose regimens, sampling schedules, covariates, and group structure are treated as fixed, while outcomes vary.
For a parametric bootstrap targeting that conditional uncertainty, a replicate generates \(Y^*\) under the fitted model at the retained design \(D\), then refits the model to obtain \(\hat{\theta}^*=T(Y^*;D)\). Depending on the specified conditional or unconditional bootstrap, the simulation may condition on subject-level random effects or generate new random effects. That choice concerns the outcome-generating distribution; it does not license accidental changes to the design.
A subject-level nonparametric bootstrap instead samples subjects with replacement. The entire selected subject record, including its outcomes and design, is copied as a unit. The sampled subjects’ designs may therefore differ from replicate to replicate, but each copied outcome remains paired with its original design. This is distinct from independently resampling outcome rows, dose records, or observation times.
There are settings where the design itself is part of the intended source of variation—for example, when inference is meant to average over a population of study schedules. That requires an explicit design-resampling scheme. It is not achieved by casually shuffling records. The law of total variance makes the distinction clear:
Holding \(D\) fixed targets variation conditional on the realized design. Resampling or simulating \(D\) can add variation in the second term, and may also change the average conditional variation. The appropriate choice depends on the scientific target, not on convenience.
What to Retain in a Replicate
A robust implementation represents the design explicitly rather than reconstructing it from outcome rows. In an event-based dataset, dose administrations and observations may be separate records. A dose record needs its amount, time, route or other relevant attributes; an observation record needs its time, outcome type, and subject or occasion identifiers. The replicate must preserve their chronological relationship.
- Dose history: amount, administration time, route, infusion duration, and any dose modifications relevant to the model.
- Observation schedule: actual sample times, nominal times if they are analytically relevant, and the ordering or occasion of measurements.
- Covariates: baseline covariates attached to the correct subject and time-varying covariates attached to the correct occasions.
- Grouping and structure: treatment or dose group, site or stratum where relevant, occasion identifiers, and any repeated-dose or crossover structure.
The observed outcome values are handled differently across bootstrap types. A nonparametric subject bootstrap copies them with the complete subject cluster. A parametric bootstrap generates new outcomes while retaining the chosen design. A design field should not be regenerated merely because it appears in the same data table as the outcome; its role in the bootstrap target must be decided explicitly.
Specify whether uncertainty is conditional on the observed design or includes variation in design features.
Keep each subject’s dose events, observation schedule, covariates, outcomes, and relevant group labels linked.
Resample whole subject clusters for a nonparametric bootstrap, or simulate outcomes at the retained design for a parametric bootstrap.
Check subject identifiers, event order, dose-to-sample relationships, covariate alignment, group counts, and the expected number of records.
Worked Examples
Worked Example: Simulating Outcomes at the Observed Sampling Times
Suppose a hypothetical subject receives a single 100 mg intravenous bolus at time zero. The fitted subject-specific parameters are \(V=20\) L and \(CL=2\) L/h, and samples are scheduled at 2, 6, and 10 hours. Under a one-compartment model after an IV bolus,
Here, \(D/V=100/20=5\) mg/L and \(CL/V=2/20=0.1\) h\(^{-1}\). The model predictions at the retained times are:
In a parametric replicate, new outcomes are simulated for this subject at 2, 6, and 10 hours, with the dose still recorded at time zero. If a residual-error model is applied, it generates deviations around these predictions; it does not move the sample times. Simulating a value at 5 hours instead of the scheduled 6-hour sample would change the design unless 5 hours were deliberately part of the target design.
This setup assumes the dose and sampling schedule are regarded as fixed for the conditional target, and that the stated one-compartment prediction is the applicable structural model. The arithmetic confirms the predictions from the displayed parameter values; no new empirical study is implied.
Worked Example: Resampling Whole Subjects With Unequal Records
Consider a hypothetical nonparametric bootstrap from three subjects. Subject A has two observations at 1 and 8 hours after a 100 mg dose. Subject B has three observations at 2, 6, and 12 hours after a 50 mg dose. Subject C has one observation at 4 hours after a 100 mg dose. Suppose one replicate samples subjects in the order B, C, B.
The replicate contains seven observations: three from the first copy of B, one from C, and three from the second copy of B. Thus the total is \(3+1+3=7\) observations. Each copy of B retains its 50 mg dose and its 2, 6, and 12 hour sampling times; C retains its 100 mg dose and 4 hour time. A and its two observations are absent from this replicate.
The two sampled copies of B must receive distinct replicate identifiers, such as B_1 and B_2, so the model treats them as two sampled clusters rather than joining their records into one subject. Their source design and outcomes are copied together. This is appropriate for a subject-level nonparametric bootstrap under the assumption that subjects are the independent sampling units; row-wise resampling would break the within-subject structure.
Worked Example: Why Dose-Group Allocation Must Not Be Accidentally Pooled
Suppose a hypothetical study has four subjects: three receive 50 mg and one receives 100 mg. All have a sample at 2 hours, and for illustration their \(V=20\) L and \(CL=2\) L/h. The predicted concentration at 2 hours is proportional to dose:
while
The mean of these four design-specific predictions is \((3(2.0468)+4.0937)/4=2.5585\) mg/L. If a resampling procedure instead produces two 50 mg and two 100 mg subjects, the corresponding mean is \((2(2.0468)+2(4.0937))/4=3.0703\) mg/L. The difference is \(3.0703-2.5585=0.5118\) mg/L, caused here solely by the changed dose-group composition.
If the target is conditional on the observed group allocation, resample within dose group so each replicate retains three low-dose and one high-dose subject. If the target includes variation in group allocation, an unstratified design may be intentional, but its implications must be stated. This example assumes the same structural parameters across subjects and uses model predictions only to isolate the consequence of changing design composition.
Diagnostics and Common Failure Modes
Design preservation is partly a data-engineering problem. A replicate can look plausible in a table while containing a dose linked to the wrong subject, a sample recorded before the administration that is supposed to precede it, or a baseline covariate duplicated incorrectly. Checks should be automated and run before each fit, not left to visual inspection of a few replicates.
- Check counts: verify expected subjects, observations, dose events, and group or stratum counts for the selected bootstrap scheme.
- Check linkage: confirm every outcome, dose, and time-varying covariate has the intended subject and occasion identifier.
- Check chronology: sort events by subject and time, then verify that dose and observation ordering is consistent with the study records and model convention.
- Check support: detect impossible times, missing doses, duplicated event rows, or covariate values outside the intended design.
- Check design summaries: compare dose-group allocation, sampling-time patterns, and covariate distributions with what the bootstrap scheme is meant to preserve.
A common mistake is to resample observation rows independently because the software expects one row per outcome. The wrong reasoning is that rows are interchangeable measurements. In repeated-measures NLME data, rows from one subject are dependent and have linked design records; independent row resampling destroys that dependence and can attach outcomes to inappropriate histories. The correction is to sample the subject as the cluster, or to construct a parametric replicate by simulating outcomes at the specified subject-level design.
Alternatives and Limitations
The main approaches answer related but distinct questions. A parametric bootstrap retains specified design records and generates outcomes under the fitted model; it relies on the adequacy of that model and its simulation assumptions. A subject-level nonparametric bootstrap retains each selected subject’s observed design and outcomes, allowing more empirical variation in subject composition, but it requires a suitable population of independent subject clusters. Stratification can preserve important group allocation, while changing the resampling scheme and the variation represented.
| Approach | What is resampled or generated? | Design implication |
|---|---|---|
| Parametric bootstrap | Outcomes under the fitted model | Usually retains the observed design for a conditional target |
| Subject-level nonparametric bootstrap | Whole subject clusters | Copies each selected subject’s outcomes and design together |
| Stratified subject bootstrap | Whole subjects within specified groups | Preserves chosen stratum counts while resampling subjects |
| Design-resampling scheme | Design features as well as, or before, outcomes | Can represent design variation, but requires a justified design-generating mechanism |
No bootstrap scheme repairs a design that is poorly represented in the original study. Sparse sampling, few subjects in a dose group, covariate imbalance, or a schedule that provides little information about a parameter can remain limiting. Preserving a design correctly is necessary for a meaningful bootstrap, but it does not establish that the fitted model is adequate or that the uncertainty estimate is precise.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In a parametric bootstrap conditional on the observed design, which parts are ordinarily retained and which are generated anew?
- A subject has a dose at time zero and samples at 1 and 6 hours. Why is sampling rows independently a problem for a subject-level nonparametric bootstrap?
- Two copies of one subject are selected in a bootstrap replicate. How should their identifiers be handled, and what records should each copy contain?
- When might stratified resampling by dose group be appropriate?
- Does preserving the observed design guarantee that the bootstrap uncertainty estimate is reliable? Give one reason.
Solutions. (1) Retain the specified dose history, sample times, covariates, and relevant group structure; generate outcomes under the fitted model, with random effects treated according to the chosen conditional or unconditional setup. (2) It can separate outcomes from their subject’s dose history and destroy within-subject dependence, so resample the full subject cluster instead. (3) Give the copies distinct replicate identifiers and include the complete selected subject record for each copy, keeping each outcome linked to its design. (4) It is appropriate when the target conditions on observed group counts or when preserving group representation is part of the planned bootstrap. (5) No. For example, a misspecified model or sparse sampling can still make the resulting uncertainty estimate inadequate.