Why a Workflow Matters in Sparse PK Studies
In a sparse-sampling population pharmacokinetic study, each participant may contribute only a few concentrations, often at different sampling times. The data can still support population-level estimation because information is shared across subjects through the model, but the limited records per person make the analysis sensitive to which subjects are represented. A bootstrap analysis must therefore preserve the structure of the observations and apply the same estimation procedure consistently in every replicate.
The focus here is a subject-level nonparametric bootstrap: resample subjects, retain each selected subject’s records as a unit, refit the specified population PK model, check the resulting fits, and use the valid estimates to summarize uncertainty. The interval reflects uncertainty under that resampling scheme and fitting procedure. It does not repair a weakly identified parameter, guarantee that the model is appropriate, or account automatically for choices that were made once and then held fixed.
Specify the Analysis Before Resampling
Write down the analysis whose uncertainty is to be assessed. For example, let \(y_{ij}\) denote the concentration recorded for subject \(i\) at time \(t_{ij}\), and let \(x_i\) denote that subject’s covariates. A population PK model may describe concentrations using population parameters \(\theta\), subject-specific random effects \(\eta_i\), and residual error. The target might be a component of \(\theta\), such as typical clearance \(CL\), or a derived quantity calculated from the fitted model.
The bootstrap workflow requires a fixed fitting specification: the structural and statistical model, covariate relationships, estimation method, parameter constraints, initialisation rules, and criteria for declaring convergence. If these choices change from replicate to replicate, differences among estimates can reflect changing procedures rather than resampling uncertainty. If model selection is part of the stated analysis, it must be repeated within each replicate to represent that selection process; otherwise, the resulting uncertainty is conditional on the selected model.
The basic subject bootstrap assumes that subjects are independent sampling units and are representative of the target subject population. Within-subject observations need not be independent for this resampling step: their grouping is retained, and their dependence is handled by the PK model and its residual structure. If the study sampled subjects from distinct strata or clusters, the resampling scheme must respect that design rather than treating all subjects as interchangeable. The target population and any stratification must be specified before interpreting the result.
Worked Example: Resampling Complete Sparse Records
Suppose a hypothetical study has six subjects. Each contributes two concentrations, with each subject’s own sampling times and covariates stored in the same record. A bootstrap draw selects subject labels \(A,A,C,D,F,F\).
There are still six bootstrap subject slots, but only four distinct original subjects. The numbers of copies are \(A:2\), \(C:1\), \(D:1\), and \(F:2\). Because each selected copy contributes both observations, the replicate has \(6\times2=12\) concentration records. The two copies of \(A\) must be given distinct bootstrap identifiers, such as \(A_1\) and \(A_2\), so a fitting program treats them as separate subjects with separate random effects.
This replicate does not contain 12 independently resampled concentrations. Resampling the 12 records separately could pair a concentration with the wrong sampling time or covariate, break the within-subject grouping, and create a dataset that does not represent the original sampling process. Resampling whole subject records preserves the observed sparse design within each selected subject.
Construct and Fit Each Replicate
A practical workflow separates the resampling step from the fitting and checking steps. The fitted estimate for replicate \(b\) can be denoted \(\hat{\theta}^{*(b)}\), where the asterisk marks a bootstrap quantity. The same procedure is applied to every replicate; it is not enough to resample once and then calculate an interval from the original fit.
Record the target parameter or derived quantity, the complete model and estimation settings, and the convergence and validity criteria.
For each replicate, draw \(n\) subject records from the observed \(n\) records. Apply any design-based stratification consistently with the sampling scheme.
Include all observations and covariates for every selected copy. Assign a distinct identifier to each copy, including duplicate selections.
Run the prespecified estimation procedure on the replicate. Save estimates, uncertainty summaries, objective and convergence information, and any warning or boundary flags.
Apply the prespecified validity criteria. Do not silently remove failed or problematic fits; retain their status and investigate how often each problem occurs.
Use the valid replicate estimates for the chosen standard error or interval, and report the number attempted, the number valid, and the treatment of problematic replicates.
Refitting is essential when the target is uncertainty in the estimator produced by the population model. Holding the original fitted parameters fixed and merely predicting or resampling residuals answers a different question. The earlier material on refit-versus-fixed-parameter bootstrap distinguishes these targets; here, the workflow uses refitting because each resampled subject dataset must contribute to a new population parameter estimate.
Why Subjects, Not Concentrations?
The resampling unit follows the independent sampling unit. Under the basic setup, the observed subjects \(D_1,\ldots,D_n\) are treated as draws from a population distribution of subject records. A bootstrap sample approximates repeated sampling from that empirical distribution. Resampling individual concentrations instead would treat measurements from one subject as if they were independent subjects, which generally understates the role of between-subject variability and ignores the original observation pattern.
A small calculation shows why a bootstrap replicate may contain substantially fewer distinct subjects than its nominal sample size. For any one of \(n\) observed subjects, the probability of not selecting that subject in one draw is \(1-1/n\). With \(n\) draws with replacement, its probability of being absent is \((1-1/n)^n\), so its probability of appearing at least once is \(1-(1-1/n)^n\). By adding these inclusion probabilities over all \(n\) subjects, the expected number of distinct subjects is
where \(U\) is the number of distinct original subjects in a replicate. For \(n=6\), \(E(U)=6[1-(5/6)^6]\approx3.991\). Thus, a replicate with six subject slots typically contains only about four distinct original subjects. This is a property of ordinary sampling with replacement, not an error in the software. In sparse studies, however, it can make particular replicates especially weak for estimating some model components.
Worked Example: Checking a Batch of Fits
Suppose a hypothetical analysis attempts \(B=1{,}000\) subject-bootstrap refits. The prespecified checks classify 932 as valid, 21 as converged but at a parameter boundary relevant to the target, and 47 as failed to converge. These categories sum to \(932+21+47=1{,}000\).
The valid-fit proportion is \(932/1{,}000=0.932\), or 93.2%. Boundary estimates account for \(21/1{,}000=0.021\), or 2.1%, and convergence failures account for \(47/1{,}000=0.047\), or 4.7%. The combined proportion not classified as an ordinary valid fit is \(68/1{,}000=6.8\%\).
These counts do not by themselves establish that the interval is trustworthy or untrustworthy. The analyst should inspect whether failures share a cause, such as missing covariate support or an unstable variance component, and whether boundary fits are plausible under the model constraints. If only the 932 valid estimates are used for an interval, that interval describes the distribution conditional on obtaining a fit meeting those validity rules. Reporting only the interval while omitting the other 68 outcomes would conceal an important feature of the workflow.
From Replicate Estimates to an Interval
For a scalar target \(\theta\), let \(\hat{\theta}^{*(1)},\ldots,\hat{\theta}^{*(B)}\) be the valid bootstrap estimates. A percentile interval uses the empirical quantiles of this distribution. For a nominal \(100(1-\alpha)\%\) interval, its endpoints are the \(\alpha/2\) and \(1-\alpha/2\) quantiles. The quantile calculation convention should be stated, especially when the number of valid replicates is small.
The following small numerical example uses linear interpolation between ordered values to show the arithmetic. Eight replicates are far too few for a dependable 95% interval in an actual analysis; the calculation is illustrative only.
Worked Example: A Percentile Interval for Typical Clearance
Suppose eight valid hypothetical bootstrap estimates of typical clearance, in L/h, are already sorted: \(3.8,\ 4.0,\ 4.1,\ 4.2,\ 4.3,\ 4.4,\ 4.6,\ 4.9\). For this illustration, define the interpolated quantile position as \(h=1+(B-1)p\), where \(p\) is the desired quantile probability.
For the lower endpoint of a 95% percentile interval, \(p=0.025\), so \(h=1+7(0.025)=1.175\). This lies 0.175 of the way from the first estimate, 3.8 L/h, to the second, 4.0 L/h: \(3.8+0.175(4.0-3.8)=3.835\) L/h.
For the upper endpoint, \(p=0.975\), so \(h=1+7(0.975)=7.825\). This lies 0.825 of the way from the seventh estimate, 4.6 L/h, to the eighth, 4.9 L/h: \(4.6+0.825(4.9-4.6)=4.8475\) L/h. Under this stated interpolation rule, the illustrative interval is approximately 3.835 to 4.848 L/h.
The interval is a procedure-based uncertainty summary, not a statement that there is a 95% probability that the fixed population clearance lies within these realized endpoints. Its quality depends on the study design, the model, the resampling scheme, and the stability and number of valid refits. With only eight replicates, the tail quantiles are particularly crude.
Diagnostics, Alternatives, and Limitations
A workflow should make failures visible before interval construction. Useful checks include the valid-fit proportion, boundary estimates, warnings, parameter correlations or extreme estimates, and whether particular covariate levels or sampling patterns disappear from problematic replicates. Compare the bootstrap estimates with the original estimate and inspect the distribution for skewness or a long tail. These checks can reveal instability, but a visual inspection cannot prove model adequacy or identify every source of bias.
The subject-level nonparametric bootstrap is not the only approach. An asymptotic standard error uses the fitted model’s local uncertainty approximation and can be efficient when that approximation is reliable, but sparse data or weakly informed parameters may make the approximation questionable. A parametric bootstrap generates new subject records from the fitted model and therefore depends more directly on the assumed data-generating model. A subject-level nonparametric bootstrap retains observed subject records and covariates, but it cannot create covariate patterns or sampling designs absent from the study. These alternatives target uncertainty under different assumptions; none automatically resolves weak identification.
In particular, sparse observations can leave a parameter poorly distinguished from other parameters or from random-effect and residual components. A bootstrap may then produce frequent boundary estimates, failed fits, or highly variable estimates. Increasing the number of attempted replicates reduces Monte Carlo noise in the bootstrap summaries, but it does not add information to the original study or make an unidentified parameter identifiable. If failures are numerous or concentrated in a scientifically important part of the parameter space, report that limitation and reconsider whether the requested interval is interpretable.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- A study has 40 subjects with three concentrations per subject. How many units are drawn in one ordinary subject-level bootstrap replicate, and how many records does each selected subject copy contribute?
- Why must duplicate selections of the same original subject receive distinct bootstrap identifiers during a population-model refit?
- A replicate converges but estimates a variance component at its allowed lower boundary. Should it automatically be counted as an ordinary valid estimate? Explain.
- What does a percentile interval summarize, and why does a larger number of replicates not cure weak parameter identification?
Solutions. (1) Draw 40 subject records with replacement; each selected copy contributes that subject’s three concentrations, so the replicate has 120 records if every subject has exactly three. (2) Each copy represents a separate bootstrap subject and must receive its own subject-specific random effect; reusing one identifier would combine the copies as though they were one person. (3) No. Apply the prespecified validity rules, retain and report the boundary status, and assess its frequency and relevance to the target. (4) It summarizes quantiles of the refitted bootstrap estimates under the chosen resampling and fitting procedure. More replicates improve Monte Carlo precision of that summary, but they do not add information to the observed study or resolve confounding among weakly identified parameters.