Why parameter uncertainty changes PTA
A simulated probability of target attainment (PTA) depends on the population pharmacokinetic model used to generate exposures. If that model’s parameters are estimated from data, the estimates are uncertain. A PTA calculated only at the fitted point estimates describes the model conditional on those particular values; it does not show how much PTA might change if plausible parameter values were different. This distinction matters when an estimated PTA lies near a dose-selection criterion.
There are two different sources of variation to keep separate. Between-subject variability describes differences among individuals under a specified population model. Parameter-estimation uncertainty describes uncertainty about the population-model parameters themselves because they were estimated from finite data. Monte Carlo error is a third source: the numerical noise from using a finite number of simulation replicates. Increasing the number of simulated subjects reduces Monte Carlo error, but does not remove parameter-estimation uncertainty.
For example, \(T\) might be all AUC values at or above a threshold. The probability in the definition is over the individual-level variability represented in the model, conditional on the parameter vector. Parameter uncertainty adds a separate outer layer: the estimated vector \(\hat{\boldsymbol{\theta}}\) is not known to equal the true vector. The uncertainty distribution used for that outer layer must reflect the inferential framework and the fitted model.
How to propagate uncertainty
A common approach uses the estimated joint sampling covariance matrix of the model parameters. In a large-sample approximation, the fitted vector may be treated as approximately multivariate normal around the true vector, with covariance estimated from the model fit. A parametric bootstrap is another option: simulate data under the fitted model, refit the model to each simulated data set, and retain the resulting parameter estimates. If the analysis is Bayesian, posterior draws of the parameter vector provide the outer draws instead. These approaches are related but have different interpretations; they should not be described as interchangeable probability statements about a fixed parameter.
For each outer draw, simulate individuals and calculate PTA conditional on that draw. The outer draws must preserve parameter dependence. Drawing each parameter independently when the estimates are correlated can create combinations that are much more or less plausible than the fitted joint uncertainty supports. Positive parameters and proportions may also need to be drawn on suitable transformed scales and converted back before simulation.
Define dose \(D\), the exposure measure, target set \(T\), and the population model that generates individual exposures.
Obtain joint draws \(\boldsymbol{\theta}^{(b)}\) from an appropriate covariance-based approximation, bootstrap, or posterior distribution. Respect parameter constraints and dependence.
For each draw \(b\), simulate \(M\) individuals using \(\boldsymbol{\theta}^{(b)}\) and the model’s individual-level variability, then calculate the fraction attaining \(T\).
Report the plug-in PTA and an uncertainty summary appropriate to the inferential framework. Check that both inner simulation error and outer-draw error are small enough for the intended decision.
Writing \(I_{bm}=1\) when individual \(m\) in outer draw \(b\) attains the target, and \(I_{bm}=0\) otherwise, the conditional Monte Carlo PTA is
The collection \(\hat q_1,\ldots,\hat q_B\) shows how the simulated PTA varies over parameter draws, with some additional inner Monte Carlo error unless \(M\) is sufficiently large. Under a Bayesian analysis, summaries of this collection describe posterior uncertainty in PTA, conditional on the model and data. Under frequentist approaches, parameter bootstrap results can be used to construct an interval for the sampling uncertainty of the estimated PTA. A fraction of bootstrap draws above a decision threshold is not automatically the probability that the true PTA exceeds that threshold.
A short justification using the delta method
When PTA is a smooth function of the estimated parameters, a first-order approximation gives a quick estimate of parameter-driven uncertainty. Let \(\widehat{\Sigma}_{\theta}\) be the estimated covariance matrix of \(\hat{\boldsymbol{\theta}}\), and let \(\nabla q(\hat{\boldsymbol{\theta}})\) be the gradient of PTA with respect to the parameters, evaluated at their estimates. A first-order Taylor expansion gives
If the estimator is approximately unbiased with covariance \(\widehat{\Sigma}_{\theta}\), the variance of this linear approximation is
This follows because the variance of a linear combination \(\mathbf{a}^{\mathsf T}\hat{\boldsymbol{\theta}}\) is \(\mathbf{a}^{\mathsf T}\widehat{\Sigma}_{\theta}\mathbf{a}\). The off-diagonal terms in the covariance matrix therefore contribute whenever parameter estimates are correlated. The approximation is most useful when uncertainty is moderate and \(q\) changes smoothly near the fitted values. It can be poor near boundaries or when PTA is highly nonlinear in the parameters; simulation or bootstrap propagation is then more informative.
Worked Example: A Delta-Method PTA Uncertainty Approximation
Consider a hypothetical lognormal exposure model in which \(\log(\mathrm{AUC})\sim N(\mu,\sigma^2)\), where \(\sigma\) is the between-subject standard deviation on the log scale. The target is \(\mathrm{AUC}\geq 50\) mg·h/L. Suppose the fitted values are \(\hat{\mu}=\log(60)=4.0943\) and \(\hat{\sigma}=0.35\), and the standard error of \(\hat{\mu}\) is 0.08. For this illustration, treat \(\sigma\) as fixed and assume the estimate of \(\mu\) is approximately normal with the stated standard error.
The conditional PTA is the probability that a normal log exposure exceeds \(\log(50)\):
For uncertainty in \(\mu\) alone, the derivative is \(\partial q/\partial\mu=\phi(z)/\sigma\), where \(z=(\mu-\log(50))/\sigma\), and \(\phi\) is the standard normal density. Here \(\phi(0.5209)\approx0.348\), so the estimated standard error of PTA is approximately
A normal-approximation 95% interval is \(0.699\pm1.96(0.0795)\), or approximately 0.543 to 0.855. This interval is an approximation to uncertainty from estimating \(\mu\) under the stated assumptions; it does not include uncertainty in \(\sigma\), other model parameters, or model structure. In context, it indicates substantial parameter-estimation uncertainty around the estimated PTA of 69.9%, not a 95% probability that a fixed unknown PTA lies in this particular interval.
Nested simulation with more than one uncertain parameter
A simulation can propagate uncertainty in multiple parameters, including variance parameters, without requiring an analytic derivative. The following small ensemble illustrates the arithmetic. It is deliberately too small to provide a reliable uncertainty interval; an actual analysis would use enough outer and inner replicates to control simulation error.
Worked Example: Propagating Joint Draws Into PTA
Use the same hypothetical lognormal model and AUC target of 50 mg·h/L. Suppose three illustrative joint parameter draws are \((\mu,\sigma)=(4.02,0.38)\), \((4.10,0.34)\), and \((4.17,0.32)\). For each draw, the conditional PTA is \(\Phi((\mu-\log(50))/\sigma)\).
For the first draw, \(z=(4.02-3.9120)/0.38=0.2842\), giving PTA \(\Phi(0.2842)\approx0.612\). For the second, \(z=(4.10-3.9120)/0.34=0.5529\), giving PTA approximately 0.710. For the third, \(z=(4.17-3.9120)/0.32=0.8062\), giving PTA approximately 0.790.
The average across these three illustrative conditional PTAs is \((0.612+0.710+0.790)/3=0.704\), compared with the plug-in result of approximately 0.699 in the preceding example. The draw-specific values range from 0.612 to 0.790. That range is a demonstration of sensitivity to the parameter draws, not a confidence interval: three draws are insufficient for a dependable interval, and the draws have not been established as a calibrated uncertainty distribution.
In a full nested simulation, each outer draw would use simulated individuals rather than evaluating this particular normal probability analytically. The within-draw fraction estimates PTA conditional on that parameter vector; variation across outer draws reflects parameter uncertainty as well as any remaining inner Monte Carlo error.
How parameter uncertainty can affect a dose decision
A dose may satisfy a prespecified PTA criterion at the fitted parameters but fail to do so for a meaningful portion of plausible parameter draws. Conversely, a dose that misses the criterion at the plug-in estimates may meet it under some draws. The important question is not only how the point estimate compares with the criterion, but also how sensitive that comparison is to parameter uncertainty. This is especially relevant when the PTA curve is steep near the selected dose.
Worked Example: Checking a PTA Criterion Across Parameter Draws
Continue the hypothetical model above and suppose the criterion is PTA of at least 0.90. Assume AUC scales proportionally with dose, so increasing dose from 600 mg to 900 mg adds \(\log(900/600)=\log(1.5)=0.4055\) to \(\mu\), while leaving \(\sigma\) unchanged. This dose-proportionality assumption is imposed for the example; it would need to be justified for an application.
At the fitted values, the 900 mg dose gives \(z=(4.0943+0.4055-3.9120)/0.35=1.679\), so the plug-in PTA is approximately \(\Phi(1.679)=0.953\). It exceeds the 0.90 criterion.
For the three illustrative draws, the corresponding \(z\)-values at 900 mg are \(1.351\), \(1.746\), and \(2.073\). Their PTAs are approximately 0.912, 0.960, and 0.981, respectively. All three exceed 0.90, but this tiny illustrative ensemble cannot establish that the dose is robust to parameter uncertainty. With many calibrated draws, one would summarize the uncertainty in PTA and assess whether the decision remains supported. In a frequentist bootstrap analysis, the proportion of bootstrap results that meet the criterion is a stability diagnostic, not by itself a probability statement about the true PTA.
Diagnostics, alternatives, and limitations
Before interpreting propagated uncertainty, check that the procedure is representing the intended uncertainty. Parameter draws should use the joint covariance structure or joint refit results, and transformed draws should map back to admissible parameter values. A large number of outer draws cannot repair a misspecified model or an invalid uncertainty approximation. Also check inner and outer simulation error separately: increasing \(M\) stabilizes each conditional PTA, whereas increasing \(B\) stabilizes summaries across parameter draws.
| Approach | What it provides | Main caution |
|---|---|---|
| Plug-in simulation | PTA conditional on fitted parameter values | Does not quantify parameter-estimation uncertainty |
| Delta method | Local approximation to the variance of PTA | Can be inaccurate for nonlinear or boundary-sensitive PTA |
| Parameter bootstrap | Sampling variation in the PTA estimator when the model is refit | Depends on the fitted model and bootstrap procedure |
| Posterior draws | Posterior distribution of PTA under the specified model and prior | Depends on model and prior; interpret as Bayesian uncertainty |
Common warning signs include implausible parameter draws, a highly skewed or discontinuous PTA distribution, and uncertainty intervals that change markedly when replicate counts increase. Implausible draws may indicate that a normal approximation on the original scale is unsuitable for a constrained parameter. A PTA distribution that changes with replicate counts may reflect inadequate Monte Carlo precision rather than parameter uncertainty. If the model is weakly informed by the data, parameter uncertainty may be wide; reporting only a precise-looking plug-in PTA conceals that limitation.
Parameter uncertainty is not a substitute for assessing whether the model adequately represents the target population or exposure process. Nor should it be confused with adding another layer of random variation to each simulated individual: the outer draw concerns uncertainty in estimated model parameters, while the inner simulation represents individuals under a given model. Keep the layers distinct to avoid counting variability twice.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- What does a plug-in PTA condition on, and what additional uncertainty is propagated by outer parameter draws?
- Why can independently drawing correlated parameter estimates distort the PTA uncertainty distribution?
- A bootstrap analysis produces PTA values of 0.72, 0.80, and 0.88 in three example replicates. What can and cannot be concluded from these values?
- If the number of simulated individuals per parameter draw is increased, which source of error is reduced?
- When may the delta method be a poor approximation for PTA uncertainty?
Solutions. (1) A plug-in PTA conditions on the fitted parameter vector; outer draws propagate uncertainty in that estimated vector. (2) Independent draws discard covariance, producing combinations that may not reflect the joint uncertainty supported by the fit. (3) The values illustrate variation across those replicates, but three results do not provide a reliable interval or calibrated probability; interpretation depends on the bootstrap design. (4) Increasing individuals per draw reduces inner Monte Carlo error, not parameter-estimation uncertainty. (5) It may be poor when PTA is strongly nonlinear, uncertainty is large, or parameter constraints and boundaries make a local linear approximation inadequate.