Percentile and Basic Bootstrap Intervals in Population PK

From Bootstrap Replicates to Interval Endpoints

A bootstrap analysis produces a distribution of estimates by repeatedly resampling the study data under a chosen resampling scheme and refitting the population PK model. The earlier material on bootstrap standard errors and confidence intervals distinguishes the spread of these estimates from the interval one ultimately reports. Here the focus is on two ways to turn the bootstrap distribution into endpoints: the percentile interval and the basic bootstrap interval.

Let \(\hat{\theta}\) be the estimate of a population PK parameter from the original data. For \(B\) successful bootstrap refits, write the estimates as \(\hat{\theta}^{*(1)},\ldots,\hat{\theta}^{*(B)}\). For a nominal \(100(1-\alpha)\%\) interval, let \(q_{\alpha/2}^{*}\) and \(q_{1-\alpha/2}^{*}\) be the corresponding lower and upper quantiles of the bootstrap estimates. These are estimated from the bootstrap distribution, using a stated quantile convention such as the software's empirical quantile rule.

The order of the quantiles matters. The basic interval subtracts the upper bootstrap quantile to obtain its lower endpoint, and subtracts the lower bootstrap quantile to obtain its upper endpoint. This reverses the bootstrap deviations around \(\hat{\theta}\). For a 95% interval, \(\alpha=0.05\), so the relevant quantiles are the 2.5th and 97.5th percentiles.

Assumptions and a Justification for the Basic Interval

Neither formula guarantees a valid confidence interval simply because it can be calculated. The bootstrap procedure must represent the sampling process that produced the data. For a subject-level nonparametric bootstrap, this typically means that subjects are independent sampling units and that resampling whole subjects preserves the within-subject concentration profiles. If the design has strata or clusters, resampling must respect that structure. A parametric bootstrap instead relies on the fitted model being a suitable data-generating approximation. In either case, each replicate must apply the same estimation procedure and model specification as the original analysis.

The interval also relies on the bootstrap distribution providing a useful approximation to the sampling behavior of \(\hat{\theta}\). This can be doubtful with too few independent subjects, frequent failed fits, weakly identified parameters, or a parameter close to a constraint. A large number of replicates reduces Monte Carlo error in the estimated quantiles, but cannot repair a resampling design that fails to represent the study or an estimator whose sampling behavior is poorly approximated by the bootstrap.

The basic formula follows from approximating the distribution of estimation error. If \(\theta\) is the target parameter, suppose the conditional bootstrap distribution of \(\hat{\theta}^{*}-\hat{\theta}\) approximates the sampling distribution of \(\hat{\theta}-\theta\). Then the central bootstrap error interval is approximately \([q_{\alpha/2}^{*}-\hat{\theta},\,q_{1-\alpha/2}^{*}-\hat{\theta}]\). Reversing the sign to express an interval for \(\theta\) gives:

This is a justification based on an approximation, not an exact finite-sample result. Its accuracy depends on how well the bootstrap error distribution represents the estimator's sampling error. The percentile construction makes a different approximation: it uses quantiles of the bootstrap estimates directly as quantiles for the parameter. Neither construction automatically corrects every form of bias or skewness.

Calculating and Interpreting the Intervals

Bias, Skewness, and Diagnostic Checks

The bootstrap distribution can be biased relative to the original estimate: for example, its mean may differ from \(\hat{\theta}\), or its quantiles may be displaced relative to \(\hat{\theta}\). In the basic interval, the quantile deviations are reflected around \(\hat{\theta}\); a bootstrap distribution shifted above \(\hat{\theta}\) can therefore yield endpoints shifted below it. In the percentile interval, the quantiles are used directly. These behaviors describe the constructions; they do not establish which interval has better coverage for a particular PK estimator.

Skewness also affects the intervals differently. A percentile interval preserves the quantile spacing and asymmetry of the bootstrap estimates. A basic interval reverses the direction of that asymmetry around \(\hat{\theta}\). If the estimator has substantial bias, strong skewness, or irregular behavior, both intervals can have poor coverage. Asymmetric endpoints alone do not prove that an interval is wrong, but a striking difference between percentile and basic intervals is a useful signal to investigate.

Before interpreting either interval, check the bootstrap output and the analysis specification:

• Resampling unit and design: confirm that resampling reflects the sampling design, including subject-level dependence and any strata or clusters that must be preserved.
• Successful refits: review the number and pattern of failed or excluded replicates. Quantiles from the remaining fits may be misleading if failures are common or related to parameter values.
• Distribution shape: examine a plot or summary of the bootstrap estimates for skewness, outliers, multiple modes, or a pile-up near a constraint.
• Quantile stability: assess whether the reported tail quantiles are stable enough for the intended precision. Increasing \(B\) can reduce simulation noise, but does not resolve limited information in the original study.
• Scale and plausibility: check units and whether the endpoints make scientific and mathematical sense for the parameter. Do not silently truncate an interval to make it plausible; report the issue and assess the method and model.

For comparison, a normal-approximation interval uses the estimate and its standard error, commonly as \(\hat{\theta}\pm z_{1-\alpha/2}SE\). It is symmetric on the parameter scale and depends on a suitable normal approximation for the estimator. Percentile and basic intervals instead use bootstrap quantiles; they can reflect non-normal shape, but their performance still depends on the bootstrap approximation. The standard error alone does not determine either interval's endpoints.

Common Mistakes and Limitations

• Using the basic quantiles in the wrong order. The basic lower endpoint is \(2\hat{\theta}-q_{1-\alpha/2}^{*}\), not \(2\hat{\theta}-q_{\alpha/2}^{*}\). Reflection reverses the order.
• Treating percentile and basic intervals as interchangeable. They use the same bootstrap quantiles but make different constructions, so skewness and bias can make their endpoints differ substantially.
• Assuming a large \(B\) guarantees coverage. More replicates improve the numerical estimation of bootstrap quantiles; they do not fix an unsuitable resampling scheme, model misspecification, or weakly identified parameters.
• Ignoring parameter scale. Percentile intervals transform consistently under strictly increasing transformations when the quantiles are transformed accordingly. Basic intervals generally do not.
• Calling the observed interval a probability statement about a fixed parameter. A 95% confidence procedure describes long-run coverage under its assumptions, not the probability that this particular interval contains the parameter.

These intervals are useful summaries of bootstrap uncertainty, but they do not diagnose a model's predictive adequacy or validate the model against an external dataset. When the interval methods disagree markedly or show signs of irregularity, report the evidence and investigate the fitting and resampling behavior rather than choosing an interval solely because it looks more reassuring.