Why Bootstrap the Derived Quantity?
A pharmacokinetic quantity such as elimination half-life or area under the concentration-time curve (AUC) is often calculated from several model parameters. Those parameters are estimated together, and their estimates can be correlated. Uncertainty in the derived quantity therefore depends not only on the uncertainty of each parameter but also on how the estimates vary together.
The key operation is to calculate the target quantity within each bootstrap refit, using that refit’s jointly estimated parameters. The resulting bootstrap distribution is a distribution of the derived quantity itself. Constructing an interval from that distribution retains the dependence represented by the joint refits and also accommodates nonlinear transformations more directly than transforming separate parameter intervals.
The earlier material on bootstrap confidence intervals establishes the general resampling idea and common interval types. Here the focus is the calculation of a new quantity from each refitted parameter vector. The intended quantity must be defined first: for example, a population-typical half-life under a specified model, or the linear-model AUC for a specified dose and route. These are not interchangeable targets.
Define the Quantity Before Resampling
For a one-compartment model with first-order elimination after an intravenous dose, the elimination rate constant is \(k_{el}=CL/V\), where \(CL\) is clearance and \(V\) is volume of distribution. The corresponding half-life is
This expression applies to the specified one-compartment, first-order elimination quantity. It should not be applied without justification to a terminal half-life from a multiexponential model, or to a model in which the relevant elimination process is not represented by this relationship. If clearance and volume are estimated on a log scale, first obtain their natural-scale estimates for this calculation.
For a linear pharmacokinetic model after an intravenous dose \(D\), total AUC is
This equation assumes the specified dose and route, linear pharmacokinetics, and clearance defined consistently with the AUC being calculated. For oral dosing, bioavailability generally also matters; \(D/CL\) alone is not the appropriate expression for systemic AUC unless the relevant bioavailability factor is included or otherwise fixed at one.
In each bootstrap replicate, calculate the target from the replicate’s parameter vector. For half-life, use that replicate’s \(V_b^*\) and \(CL_b^*\) together. For AUC, use that replicate’s clearance estimate and the prespecified dose. A bootstrap interval then describes sampling uncertainty in the specified derived quantity under the model and resampling scheme; it does not include uncertainty in a dose that was treated as fixed.
Worked Example: Calculating Half-Life From Paired Refit Estimates
Suppose a hypothetical one-compartment IV analysis estimates population clearance as \(5.0\) L/h and volume as \(50\) L. The fitted-model half-life is
Now consider three hypothetical bootstrap refits. Each row is one jointly refitted pair; values from different rows must not be mixed.
| Replicate | \(\widehat{CL}_b^*\) (L/h) | \(\hat V_b^*\) (L) | \(t_{1/2,b}^*\) (h) |
|---|---|---|---|
| 1 | 4.8 | 48 | 6.931 |
| 2 | 5.2 | 55 | 7.331 |
| 3 | 5.6 | 62 | 7.674 |
For replicate 1, \(0.693147(48/4.8)=0.693147(10)=6.931\) h. For replicate 2, \(55/5.2=10.5769\) h and \(0.693147(10.5769)=7.331\) h. For replicate 3, \(62/5.6=11.0714\) h and \(0.693147(11.0714)=7.674\) h. The last result rounds to \(7.674\) h, not \(7.673\) h.
These three values illustrate the transformation; they are far too few to form a useful interval. In an actual analysis, repeat the resampling and refitting many times, calculate half-life from every successful joint fit, and use the resulting distribution.
Why the Joint Estimates Matter
A useful local justification comes from the delta method. For \(g(CL,V)=\ln(2)V/CL\), the partial derivatives at the estimated values are
A first-order variance approximation is the variance of the linear change in \(g\). If the estimated clearance and volume have variances \(\operatorname{Var}(\widehat{CL})\), \(\operatorname{Var}(\hat V)\) and covariance \(\operatorname{Cov}(\widehat{CL},\hat V)\), that approximation is
The covariance term follows because the derivative with respect to clearance is negative whereas the derivative with respect to volume is positive. Thus, positive covariance between the estimates contributes negatively to the approximate variance of half-life; negative covariance contributes positively. This does not mean positive covariance always makes every derived quantity more precise. Its effect depends on the signs and sizes of the derivatives of the particular function.
Worked Example: The Covariance Contribution to Half-Life Uncertainty
Consider hypothetical joint estimates near \(CL=5.0\) L/h and \(V=50\) L, with \(SE(\widehat{CL})=0.50\) L/h, \(SE(\hat V)=5.0\) L, and correlation \(0.80\). The covariance is \(0.80(0.50)(5.0)=2.0\) L2/h. At these values, the derivatives are \(-1.3863\) h per (L/h) for clearance and \(0.13863\) h/L for volume.
The first-order variance including covariance is
The approximate standard error is \(\sqrt{0.1922}=0.4384\) h. If the covariance were incorrectly set to zero, the approximate variance would instead be \(0.4805+0.4805=0.9610\) h2, with standard error \(0.9803\) h. This calculation isolates the covariance contribution; it is not a bootstrap interval and relies on a local linear approximation. The bootstrap approach retains the joint variation without requiring this approximation to define the interval.
Forming a Percentile Interval
A percentile interval uses empirical quantiles of the bootstrap-derived values. One simple convention is the nearest-rank empirical quantile: sort \(q_1^*,\ldots,q_B^*\) as \(q_{(1)}^*\le\cdots\le q_{(B)}^*\), and define the \(p\) quantile as \(q_{(\lceil Bp\rceil)}^*\). With this convention, a 95% percentile interval has endpoints \(q_{(\lceil0.025B\rceil)}^*\) and \(q_{(\lceil0.975B\rceil)}^*\), provided \(B\) is large enough that the indices are at least one. Software may use an interpolation convention instead, so report the convention used.
This interval is formed on the scale of the derived quantity. For example, a half-life interval has units of time and an AUC interval has units of concentration multiplied by time. It is not generally equivalent to transforming the endpoints of separate marginal intervals for the input parameters. A rectangular combination of marginal endpoints may pair values that do not occur together in the joint refits, and its extremes need not match the quantiles of the derived quantity.
Worked Example: A Percentile Interval for IV AUC
Suppose the prespecified IV dose is \(500\) mg and the original fitted clearance is \(5.0\) L/h. The fitted-model AUC is \(500/5.0=100\) mg·h/L. In a hypothetical bootstrap run with \(B=1000\) successful refits, calculate \(AUC_b^*=500/\widehat{CL}_b^*\) mg·h/L for every replicate. Some replicate calculations are shown below.
| Replicate | \(\widehat{CL}_b^*\) (L/h) | \(AUC_b^*\) (mg·h/L) |
|---|---|---|
| 1 | 4.4 | 113.636 |
| 2 | 5.2 | 96.154 |
| 3 | 4.8 | 104.167 |
| 4 | 5.6 | 89.286 |
| 5 | 4.6 | 108.696 |
For example, replicate 4 gives \(500/5.6=89.2857\), or \(89.286\) mg·h/L. Suppose sorting all 1000 derived values gives \(AUC_{(25)}^*=82.4\) mg·h/L and \(AUC_{(975)}^*=124.7\) mg·h/L. Under the stated nearest-rank convention, the 95% percentile interval is therefore \([82.4,\ 124.7]\) mg·h/L. In repeated use of the stated bootstrap procedure, this interval method is designed to cover the target AUC at approximately the nominal rate under its assumptions; it does not mean there is a 95% probability that this fixed target lies in this particular calculated interval.
Assumptions, Diagnostics, and Alternatives
The interval’s validity depends on more than applying the formula correctly. The bootstrap resampling scheme must reflect the independent sampling units and study design; in clustered or repeated-measures data, resampling individual observations as though they were independent can distort uncertainty. Each replicate must be refitted using the same model specification and the same definition of the estimand. The target must be meaningful for all parameter values represented in the bootstrap distribution, such as positive clearance and volume when calculating the stated half-life.
Inspect both the refits and the derived-value distribution. Record the number and reasons for failed fits, check for implausible parameter estimates, and look for boundary estimates or optimizer warnings. Plot or summarize the derived values for skewness, long tails, multiple modes, or extreme values. A small number of unusual parameter combinations can create a long tail in \(V/CL\), even when the marginal distributions of \(V\) and \(CL\) appear less unusual. Increasing the number of bootstrap replicates can reduce Monte Carlo noise in estimated endpoints, but cannot repair a misspecified model, an inappropriate resampling scheme, or systematic fitting failures.
Do not silently discard failed replicates if failure is associated with particular parts of the sampling distribution. Investigate the causes, use stable fitting settings where justified, and report the failure rate and handling. If the target is undefined or scientifically implausible for a substantial part of the fitted distribution, this may signal weak identification or a mismatch between the estimand and model. In that case, an interval alone does not resolve the problem.
A Wald interval for a derived quantity can be built from a delta-method standard error, but it relies on a local linear approximation and a chosen reference distribution. A percentile bootstrap interval instead uses the empirical distribution of the refitted derived values and can reflect nonlinear transformation and asymmetry. BCa intervals adjust percentile endpoints for estimated bias and acceleration, but require their own assumptions and calculations. Profile-likelihood intervals address uncertainty through the likelihood for a target value, with nuisance parameters reoptimized as needed. These approaches answer related uncertainty questions but can differ in their behavior; the earlier material on comparing Wald, bootstrap, and profile intervals discusses their general trade-offs. None makes an unsuitable model or poorly supported estimand reliable by itself.
Common Mistakes
- Combining separate marginal endpoints. The maximum of \(V/CL\) is not necessarily obtained from the upper endpoint for \(V\) and lower endpoint for \(CL\) when those estimates are dependent. Calculate the function for each paired refit instead.
- Breaking the pairing. Randomly matching clearance from one replicate with volume from another destroys the dependence captured by the joint refits. Keep estimates from the same refit together.
- Using a formula outside its assumptions. \(AUC=D/CL\) is not a general oral-dose formula when bioavailability is unknown, and \(\ln(2)V/CL\) is not a universal half-life formula for every structural model.
- Interpreting the interval as certainty about a parameter. A calculated 95% interval does not assign a 95% probability to the fixed target being inside it.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A one-compartment IV model has a bootstrap replicate with \(CL=4.0\) L/h and \(V=40\) L. What is the replicate half-life?
- Why should clearance and volume from the same bootstrap refit be used together when calculating half-life?
- For a fixed IV dose of \(600\) mg, a replicate has \(CL=6.0\) L/h. Calculate its AUC and give the units.
- What is one reason an interval formed by transforming the endpoints of marginal clearance and volume intervals may differ from the bootstrap interval for half-life?
- A notable fraction of refits fail, and the successful replicate half-lives are strongly right-skewed. Name two diagnostic or reporting actions that should follow.
Solutions. (1) \(t_{1/2}=0.693147(40/4.0)=6.931\) h. (2) The jointly refitted estimates retain the dependence between clearance and volume; pairing values from different refits would alter that dependence and thus the distribution of their ratio. (3) \(AUC=600/6.0=100\) mg·h/L. (4) Marginal endpoints can combine parameter values that do not occur together in the joint bootstrap refits, whereas the bootstrap-derived interval uses the empirical distribution of the paired transformations. (5) Investigate why fits fail and whether failure depends on parameter values; report the failure rate and handling. Also inspect and summarize the skewed derived-value distribution, since the long tail can materially affect the interval endpoints.