Bootstrap Standard Errors and Confidence Intervals

What Bootstrap Uncertainty Measures

In population pharmacokinetics, an estimated parameter such as clearance, a typical volume, or an interindividual variability parameter is uncertain because a different study sample could produce a different estimate. A subject-level or parametric bootstrap represents that sampling process by generating replicate datasets and refitting the model. The resulting refitted estimates show how the estimation procedure varies under the chosen bootstrap scheme.

The central distinction is between the estimate from the original dataset and the estimates from the bootstrap datasets. Write the original-data estimate of a scalar parameter as \(\hat{\theta}\), and the estimate from bootstrap replicate \(b\) as \(\hat{\theta}^{*(b)}\), for \(b=1,\ldots,B\). The bootstrap replicates are used to estimate the sampling variability of the estimator; they do not replace the original-data estimate.

This standard error is not the standard deviation of individual concentrations, nor is it the standard error of the bootstrap mean \(\bar{\theta}^{*}\). The latter would describe Monte Carlo precision in estimating the bootstrap mean, approximately \(\widehat{SE}_{\mathrm{boot}}/\sqrt{B}\), whereas the bootstrap standard error describes variation in the fitted parameter across hypothetical repeated datasets.

Assumptions and a Normal-Approximation Interval

A bootstrap standard error and an interval based on it are meaningful only relative to the bootstrap design and the refitting procedure. The resampling scheme must reflect the sampling structure relevant to the inferential target. For example, when subjects are the independent sampling units, resampling individual records rather than whole subjects would generally fail to preserve within-subject concentration profiles. Stratification or clustering, when required by the design, must also be respected.

Each replicate must fit the same model and return an estimate with the same definition, parameterization, and units as the original estimate. A collection made only of converged fits can be misleading if convergence failures are more common for particular datasets or parameter values. A large number of replicates does not repair a resampling scheme that fails to represent the study, a systematically misspecified model, or selectively missing refits.

A simple confidence interval can be formed by assuming that the estimator’s sampling distribution is approximately normal and centered sufficiently close to \(\hat{\theta}\). For a two-sided interval with nominal confidence level \(1-\alpha\), the normal-approximation bootstrap interval is:

where \(z_{1-\alpha/2}\) is the corresponding standard normal quantile. For a 95% interval, \(z_{0.975}\) is approximately 1.96. This construction uses the bootstrap to estimate the standard error but relies on a normal, symmetric approximation for the interval. A bootstrap standard error by itself does not uniquely determine a confidence interval: other interval constructions use the replicate distribution differently and may be preferable when it is skewed. Those constructions should not be confused with the normal-approximation interval shown here.

Calculating the Bootstrap Standard Error

The calculation is a direct application of the sample standard deviation. Its justification is that the bootstrap replicates are treated as draws from the estimated sampling distribution induced by the chosen resampling procedure. If that distribution has standard deviation \(\tau\), its empirical variance uses deviations from the empirical mean; dividing the sum of squared deviations by \(B-1\) is the usual sample-variance estimate of \(\tau^2\). Taking the square root returns the estimated spread in the parameter’s units.

Worked Intervals and Comparison With Covariance-Based Results

A covariance-based standard error is typically obtained from an estimated covariance matrix for the fitted parameters; for a scalar parameter, its standard error is the square root of the corresponding diagonal variance. Under regularity conditions and a sufficiently informative sample, likelihood-based estimates are often approximately normal, and the inverse information matrix supplies an asymptotic covariance estimate. The bootstrap instead estimates variability by repeating the fitting procedure on bootstrap datasets. These approaches can agree when the model is well behaved and the asymptotic approximation is adequate, but they rely on different approximations.

Diagnostics, Alternatives, and Limitations

Before reporting a bootstrap standard error or interval, inspect the refitted estimates rather than relying only on their summary. Useful checks include a plot of the replicate estimates, their range and quantiles, and the frequency and pattern of failed fits. Look for skewness, multiple clusters, extreme values, estimates at parameter bounds, and differences in convergence behavior across replicate datasets. Such features may make a standard deviation a limited summary of uncertainty or make a symmetric interval misleading.

The number of successful replicates also matters. Increasing \(B\) reduces Monte Carlo noise in the estimated bootstrap standard error and in features of the replicate distribution. It does not increase the information in the original study, remove bias, or correct model misspecification. In particular, the tails of an estimated distribution are harder to characterize precisely than its center, so a replicate count adequate for a standard error may still be inadequate for a tail-based interval.

The main comparison is with the asymptotic covariance-based standard error and its associated Wald interval. The covariance approach is computationally convenient and can be informative when the likelihood is regular, the estimate is well inside its parameter space, and the local quadratic approximation is credible. The bootstrap has the advantage of repeating the refitting procedure and can reveal departures from that local approximation. It can also be computationally expensive and sensitive to the resampling scheme, refit failures, and unstable or non-identifiable parameters. Neither approach automatically accounts for model misspecification or uncertainty about which model should have been selected.

These calculations matter because uncertainty communicates how much a fitted parameter may vary under repeated sampling, and that uncertainty can affect scientific interpretation and downstream decisions. Report the original estimate, the bootstrap standard error with units and scale, the interval construction, the number and handling of refits, and a comparison with covariance-based results when relevant. For parameters fitted on a transformed scale, state the scale on which the standard error and interval were calculated; if the interval is transformed back, report the transformation and original-scale units.