Why Variability Must Be Counted by Source
A probability of target attainment (PTA) calculation depends on the distribution of exposure across the individuals or occasions represented in the simulation. A common mistake is to simulate individuals with variable pharmacokinetic parameters, calculate their exposures, and then add another broad variability term to those exposures without checking what that term represents. If it describes variability already present in the simulated parameters, it counts the same source twice and changes the exposure distribution for the wrong reason.
The key question is not simply whether exposure is variable. It is which source of variability the simulated quantity already contains, and which sources the target event requires. Individual differences in clearance can create variability in AUC. Residual error can make measured concentrations differ from model-predicted concentrations. Variability between occasions may matter if the target concerns a particular occasion. These sources are not interchangeable, and they should not be combined automatically.
Identify the Quantity and Its Sources
A population model commonly represents an individual parameter, such as clearance, as a typical value determined by covariates together with an individual-specific deviation. For example, a log-normal parameter model can be written as:
Here \(CL_i\) is individual \(i\)'s clearance, \(CL_{\mathrm{typ},i}\) is the typical clearance for that individual's covariates, and \(\eta_{CL,i}\) is the individual's deviation on the log scale. When the simulation draws \(\eta_{CL,i}\) and calculates an exposure using \(CL_i\), the resulting exposure already reflects the between-subject variability represented by that random effect. Adding an extra random effect intended to represent the same between-subject clearance variability duplicates it.
Residual error has a different role. It describes discrepancies between model-predicted and observed concentrations, often because of measurement error and other unexplained observation-level variation. A residual-error term is relevant if the event being evaluated concerns a measured concentration. It does not automatically belong in a PTA calculation for a model-predicted AUC or another latent exposure summary.
Other sources can be genuinely distinct. For example, between-occasion variability may matter when the question concerns exposure on a randomly selected treatment occasion and the model represents that variability separately from between-subject differences. Parameter uncertainty is different again: it describes uncertainty about the model parameters, not biological differences between two individuals. It should be tracked according to the estimand and simulation design, rather than inserted as though it were another independent person's random effect.
| Source | What it represents | Include in a PTA exposure calculation when... |
|---|---|---|
| Between-subject variability | Differences between individuals, such as different clearances | It is represented in the simulated individual parameters and the target concerns the population of individuals |
| Residual error | Difference between a predicted concentration and an observed concentration | The target event is explicitly about an observation, not just model-predicted exposure |
| Between-occasion variability | Differences across occasions for the same individual | The target concerns an occasion and the model includes a distinct occasion-level source |
| Parameter uncertainty | Uncertainty in estimated model parameters | The analysis is designed to propagate uncertainty in the estimated model, with that uncertainty kept distinct from individual variability |
Why Duplicate Terms Inflate the Distribution
Suppose a log exposure margin \(Z\), such as log exposure divided by a target, contains two genuinely distinct random contributions, \(A\) and \(B\). The variance of their sum is:
If \(A\) and \(B\) are independent, the covariance is zero and their variances add. But this rule does not justify adding a second term that is merely a duplicate of \(A\). If the same random contribution is included twice, then \(A+A=2A\), so:
Treating a duplicate as if it were an independent draw would instead give \(2\mathrm{Var}(A)\), which is still too large for a model that contains only one source. In practical simulations, an incorrectly enlarged spread can change the fraction of simulated individuals that cross a target. The direction and size of the change depend on the target and distribution; broader variability does not guarantee lower PTA in every setting.
Worked Example: A Second Random Effect Repeats the First
Hypothetical example. For a fixed regimen, suppose individual AUC divided by its target has a median of 1.25. A simulated individual-level random effect is already included, giving a standard deviation of 0.30 on the log scale. Assume the log AUC-to-target ratio is normally distributed around its log median. Calculate the PTA when the target is AUC at least the threshold, then show the effect of mistakenly adding an independent term with the same standard deviation.
With the individual-level variability included, the log ratio is normal with mean \(\log(1.25)\) and standard deviation 0.30. Therefore:
Since \(\log(1.25)=0.22314\), the standardized value is \(0.22314/0.30=0.7438\). Thus \(\mathrm{PTA}_{\mathrm{correct}}\approx 0.7715\), or approximately 77.2%.
Now suppose an additional independent log-scale term with standard deviation 0.30 is added, even though it is intended to represent the same between-subject variability already simulated. The resulting standard deviation is incorrectly calculated as:
The resulting PTA is \(\Phi(0.22314/0.4243)=\Phi(0.5260)\approx 0.7005\), or about 70.1%. The difference is not evidence that the regimen changed. It is an artifact of broadening the log-exposure distribution by adding a duplicate source.
Choose Error Terms to Match the Target
The simulation must match the quantity in the target event. If the target is based on model-predicted AUC, calculate AUC from the simulated individual parameters and evaluate whether that AUC meets the target. Do not add residual concentration error merely because the fitted model has a residual-error component. AUC is a summary of the model-predicted concentration-time profile; residual error describes observed concentrations around that profile.
If the target instead concerns whether a single measured concentration exceeds a limit, residual error may be relevant. In that case, the event is about an observation and should be formulated as such. This does not mean that residual error should be added to every exposure endpoint. It means the observation model is used when the target definition calls for an observation.
Worked Example: Residual Error for a Measured Concentration
Hypothetical example. For one individual, suppose the model-predicted concentration at a specified time is 10 mg/L. Assume the logarithm of the observed-to-predicted concentration ratio has mean 0 and standard deviation 0.20. Find the probability that a measured concentration is at least 8 mg/L. Then distinguish that probability from attainment based on the predicted concentration alone.
Under the stated observation model, the log of the observed concentration has mean \(\log(10)\) and standard deviation 0.20. The standardized threshold is:
Therefore the probability of an observed concentration at least 8 mg/L is \(1-\Phi(-1.1157)=\Phi(1.1157)\approx 0.868\), or 86.8%. By contrast, the model-predicted concentration is 10 mg/L, which is above 8 mg/L; for this fixed individual's predicted value, attainment is 1. The two answers differ because they concern different quantities: an observation that includes residual error versus a deterministic model prediction. Residual error belongs in the first calculation, not automatically in a PTA based on predicted exposure.
Combine Distinct Sources, Not Labels
A variability term should be included because it represents a distinct mechanism required by the target, not simply because the model or software has a place to enter it. If two sources are distinct and independent on the scale of the calculation, their variances can be added on that scale. If they are correlated, the covariance term must also be considered. If they are the same source under different labels, they should not be added as if independent.
Worked Example: Between-Subject and Between-Occasion Variation
Hypothetical example. Suppose a regimen has a median AUC 10% above a target, so the median AUC-to-target ratio is 1.10. On the log scale, between-subject variation has standard deviation 0.20 and a distinct, independent between-occasion source has standard deviation 0.10. The target concerns one randomly selected occasion. Calculate the PTA with both sources and compare it with the result from mistakenly counting between-subject variability twice.
Because the target concerns a randomly selected occasion, and the two stated sources are distinct and independent, the total log-scale variance is \(0.20^2+0.10^2=0.0500\). The total standard deviation is \(\sqrt{0.0500}=0.2236\). The PTA is:
Thus approximately 66.5% of the simulated individual-occasion outcomes meet the target under these assumptions. If between-subject variation is mistakenly counted a second time as an additional independent source, the standard deviation becomes \(\sqrt{0.20^2+0.10^2+0.20^2}=0.3000\). The resulting calculation is \(\Phi(0.09531/0.3000)=\Phi(0.3177)\approx 0.625\). This lower value reflects the duplicate term, not an additional biological process. The correct calculation includes the 0.10 between-occasion term because it is separately defined and relevant to the target.
A Practical Audit Before Reporting PTA
Before running or interpreting a PTA simulation, write down the event being counted and trace each random quantity entering it. This makes it easier to see whether variability is already propagated through simulated parameters, added at the observation level, or introduced as a separate process. The following sequence is useful:
Specify whether it concerns model-predicted AUC, a concentration at a time, an observed measurement, or an event on a particular occasion.
Record which individual parameters, covariates, occasions, or other terms vary across simulated outcomes.
For every additional term, state what it represents and why that source is needed for the target event.
Combine variances only for distinct terms on the same scale, and account for covariance when terms are correlated.
Temporarily remove questionable terms and check whether the distribution changes for an explainable reason.
Useful diagnostics include comparing the simulated parameter distributions with their intended distributions, checking whether the exposure spread is plausible, and plotting or summarizing predicted exposure separately from simulated observations. If exposure variability is unexpectedly large, investigate duplicated random effects, repeated resampling, and residual error applied to an endpoint that is already a model-predicted summary. The comparison of PTA with median exposure is also informative: a median alone does not determine PTA, so an unexplained change in spread can materially alter the reported attainment.
Common Pitfalls and Alternatives
- Adding residual error to every endpoint. This is wrong when the target is a model-predicted exposure and residual error only describes observed concentrations. Use the observation model only when the target is defined in terms of an observation.
- Calling parameter uncertainty another individual effect. Uncertainty about estimated parameters is not the same as variability among individuals. If it is propagated, keep parameter draws distinct from individual-level draws and describe what probability the resulting summary represents.
- Assuming every source is independent. Independent variance addition is valid only when the terms are independent on the relevant scale. Correlated random effects require their covariance to be represented.
- Assuming more variability must reduce PTA. A broader distribution can move more probability into either tail. Whether PTA rises or falls depends on the target, the location of the distribution, and the event being evaluated.
The main alternative to a single pooled exposure distribution is a nested simulation: draw model parameters to represent parameter uncertainty, then draw individuals conditional on those parameters, and include observation-level error only for observation-based targets. This can separate uncertainty about the PTA estimate from variability among individuals. It is more computationally demanding and requires a clear specification of the parameter uncertainty distribution. Neither approach repairs a duplicated random effect; the sources must first be identified correctly.
These calculations also have limits. A log-normal approximation may not describe a complex exposure distribution well, and a simple variance sum is inadequate when sources are dependent or the model is nonlinear on the scale used. In those settings, simulate the specified joint model rather than replacing it with an unsupported single standard deviation. Report the target definition and the sources included so that the PTA can be interpreted in context.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- AUC is calculated from simulated individual clearances that include between-subject variability. Should residual concentration error automatically be added to determine PTA for a predicted AUC threshold?
- Two independent, distinct log-scale sources have standard deviations 0.12 and 0.16. What is their combined standard deviation?
- A model-predicted concentration is below a threshold, but residual error could produce an observed concentration above it. Are predicted-concentration attainment and observed-concentration attainment the same event?
- Why is adding a second independent draw with the same standard deviation not a valid way to represent an individual effect that is already simulated?
Solutions. (1) No. The simulated individual clearances already affect predicted AUC; residual error is relevant only if the target is defined for an observation. (2) The combined standard deviation is \(\sqrt{0.12^2+0.16^2}=\sqrt{0.0400}=0.200\), under the stated independence assumption. (3) No. One event concerns a model prediction and the other concerns an observation that includes residual error. (4) The second draw increases the distribution's variance as though there were another source. It does not represent the single individual effect already included, so it artificially broadens the exposure distribution.