Subject-Level Nonparametric Bootstrap

Why Resample Subjects?

A population pharmacokinetic (PK) dataset is usually grouped by subject. Each subject may contribute several concentrations, recorded at different times and under a particular dosing history, along with covariates and other study information. Those observations are not independent rows: they belong to the same person and are linked through that subject’s latent characteristics, dosing, sampling schedule, and within-subject variability. If the inferential target is uncertainty arising from sampling subjects from a population, the subject—not the individual concentration record—is the natural resampling unit.

The subject-level nonparametric bootstrap approximates repeated sampling from the population by repeatedly drawing subjects, with replacement, from the observed subjects. Each draw carries along that subject’s full observed data. The model is then refitted to each resampled dataset. The resulting estimates show how the fitted estimates vary across bootstrap versions of the study under this resampling scheme.

The method is most directly justified when the observed subjects can reasonably be treated as independent, representative sampling units from the population relevant to the analysis. The fitted model, estimation method, data-processing rules, and target parameter should remain fixed across replicates. Resampling is not a remedy for an unrepresentative sample, a misspecified model, or a poorly identified parameter.

Constructing a Subject-Level Resample

Write the complete record for subject \(i\) as \(D_i\). This record may include the subject’s concentration measurements, sample times, doses, covariates, occasion information, and relevant missing-data pattern. One bootstrap replicate draws \(n\) subject indices independently from the set \(1,\ldots,n\), each with probability \(1/n\). The selected records, including repeated selections, form the replicate dataset.

The star denotes a bootstrap quantity or dataset. A subject selected twice contributes two copies of that subject’s full record. In the fitting dataset, those copies must have distinct bootstrap subject identifiers. They are separate sampled units in the resample, even though their recorded observations are identical. In a nonlinear mixed-effects analysis, the model should treat the copies as separate subjects, with a separate subject-level random effect for each copy under the fitted model.

Resampling whole records preserves the observed within-subject structure: the number and timing of that subject’s measurements, the ordering of their records, the observed dose history, and the association among the recorded values. It also preserves each selected subject’s covariate values and observed pattern of missing measurements. It does not create new concentration profiles or new residual errors. The same profile may appear more than once in a replicate, while another observed profile may not appear at all.

Why Refitting Produces an Uncertainty Distribution

Let \(\hat{\theta}\) denote the estimate of a population model parameter or parameter vector from the original dataset. For bootstrap replicate \(b\), fit the same model using the same estimation procedure and denote the result by \(\hat{\theta}^{*(b)}\). Repeating the sample-and-refit process \(B\) times gives \(B\) estimates. Their variation approximates how the estimator would vary if comparable studies were repeatedly sampled from a population represented by the observed subjects.

A brief derivation makes clear why the whole-subject scheme corresponds to resampling the empirical subject distribution. Define \(M_i\) as the number of times subject \(i\) is selected in one replicate. The vector of counts has a multinomial distribution with \(n\) draws and equal selection probabilities \(1/n\). For a likelihood that factors into independent subject contributions, the original log likelihood is a sum of subject-level contributions. Resampling subjects repeats each selected subject contribution according to its count:

Here, \(\ell_i(\theta)\) is the contribution of subject \(i\), and \(\ell^*(\theta)\) is the resampled log likelihood. In a population PK likelihood, a subject contribution accounts for that subject’s observed records under the model, including the model’s treatment of within-subject dependence and random effects. Repeating the contribution is equivalent to including a distinct copy of that subject in the resample. The refitted estimate is the value of \(\theta\) that optimizes the resampled fitting criterion, not an estimate calculated by merely duplicating the original parameter values.

This argument relies on subjects being the independent sampling units for the target of inference. If the data were sampled through dependent groups or a design with distinct sampling strata, simple independent subject resampling may not reproduce the sampling design. Those situations call for a resampling scheme that reflects the design. The subject-level method also assumes that the original estimation procedure can be applied meaningfully to the resampled datasets.

Refitting a Population PK Model

For a population PK analysis, the replicate workflow is conceptually straightforward. Start with the analysis dataset and a clearly defined subject-level record. Draw \(n\) subject indices with replacement. Copy all records for each selected index into the replicate, assigning a new identifier to every sampled copy. Apply the same prespecified data handling and model-fitting procedure as in the original analysis. Save the estimate of interest and relevant information about the fit. Repeat for the desired number of replicates.

In population PK applications, parameters of interest may include typical clearance, a between-subject variability parameter, or a covariate-effect coefficient. The bootstrap distribution is formed from the corresponding refitted estimates. This tutorial’s focus is the construction and rationale of that distribution; particular interval methods and detailed rules for handling failed replicates require separate treatment.

Diagnostics, Alternatives, and Limitations

A plausible bootstrap distribution should be interpreted alongside the quality of its resamples and refits. Check that each replicate contains the intended number of sampled subjects, that duplicate copies have distinct identifiers, and that all records belonging to each selected subject were retained. Confirm that the same model, estimation settings, and prespecified data processing were applied in every replicate. Examine the replicate estimates for unusual concentration, boundary accumulation, extreme values, or distinct clusters, and track convergence and other fit diagnostics. A narrow distribution does not by itself prove that uncertainty is small: if the original subjects do not represent the target population, or the model is weakly identified, resampling cannot repair that problem.

The main alternative resampling units answer different questions. Resampling individual concentration records breaks the observed subject profiles and can treat measurements from one subject as if they were independent sampled units. That is generally unsuitable when the target uncertainty concerns sampling subjects. An analytic covariance estimate instead relies on the fitted model and its approximation to estimator variability; it does not generate resampled datasets. A parametric bootstrap generates data from a fitted model rather than resampling observed subjects, so it relies on a different representation of the data-generating process. The choice among approaches depends on the sampling design and the source of uncertainty being assessed.

Subject-level resampling has important limits. It uses only the observed subject profiles, so the empirical distribution cannot contain subject types or profile patterns absent from the study. With few subjects, bootstrap samples can have limited diversity and replicate estimates may be unstable. Resampling observed records also does not simulate fresh measurement errors or new within-subject trajectories conditional on covariates. Finally, this scheme presumes that subjects are independently sampled units; dependence induced by design or recruitment structure can make simple independent resampling inappropriate.