Separating Parameter Uncertainty from Output Variability

Why Separate the Two Sources of Spread?

A PBPK model can produce different outputs for two fundamentally different reasons. First, the population-level inputs may not be known precisely: for example, the typical tissue-to-plasma partition coefficient or the population distribution of an enzyme activity may be uncertain. Second, individuals can differ even if those population-level inputs were known: people have different body sizes, organ flows, and enzyme activities. A single simulation that varies both sources produces a spread of outputs, but that spread alone does not say which source contributed what.

The distinction matters when interpreting predictions. Uncertainty in a population-level input expresses limited knowledge about the model or target population. Interindividual variability describes a distribution of outcomes across people under a specified population model. The two may both affect a prediction interval, but they answer different questions and may call for different actions: collecting better evidence for an uncertain input is not the same as describing variation across patients.

The introductory material on epistemic uncertainty and interindividual variability establishes the conceptual distinction. Here, the focus is how to quantify it when both sources enter a model prediction. A nested design makes the conditioning explicit: an outer simulation draws a population-level input set; an inner simulation draws individuals conditional on that set.

The Variance Decomposition

Suppose \(\theta\) has a distribution representing current uncertainty in population-level inputs. For each fixed value of \(\theta\), the model and population assumptions induce a conditional distribution for \(Y\) through variation in \(U\). Define the conditional mean and variance as \(E[Y\mid\theta]\) and \(\operatorname{Var}(Y\mid\theta)\). The marginal output distribution combines the uncertainty over \(\theta\) with the individual variation conditional on each \(\theta\).

The first term is the between-input component: it measures how much the expected output for the population shifts as the uncertain population-level inputs change. The second is the within-population component: it is the average individual-level output variance across the possible input sets. Both terms are in squared output units. Their square roots are component standard deviations, but those standard deviations do not generally add to the total standard deviation.

The decomposition follows by adding and subtracting the conditional mean. Write \(Y-E[Y]=\{Y-E[Y\mid\theta]\}+\{E[Y\mid\theta]-E[Y]\}\), square, and take expectations. The cross-product has expectation zero because, conditional on \(\theta\), \(E[Y-E[Y\mid\theta]\mid\theta]=0\). The remaining squared terms give the two components:

This result is an identity for the joint predictive distribution implied by the specified uncertainty distribution for \(\theta\) and the specified individual distribution for \(U\). It does not establish that either distribution is scientifically appropriate. If the outer distribution is a subjective or evidence-based representation of uncertainty, its variance component describes that representation; it is not automatically a frequentist confidence interval or a measure of biological differences between people.

Designing the Nested Simulation

For each outer draw \(m=1,\ldots,M\), sample a set of population-level inputs \(\theta_m\). Holding that set fixed, draw \(K\) individuals \(U_{mk}\), \(k=1,\ldots,K\), from the intended target-population distribution and calculate \(Y_{mk}=f(\theta_m,U_{mk})\). The individual draws should reflect the conditional population model, including any specified dependencies among individual characteristics. The outer draws should reflect uncertainty in population-level quantities, not randomly redraw an individual trait in a way that changes the meaning of the population model.

With equal inner sample size \(K\), let \(\bar{Y}_m\) be the mean of the \(K\) outputs under outer draw \(m\), and let \(s_m^2\) be their sample variance. Define \(\bar{\bar{Y}}\) as the mean of the \(M\) conditional means, \(B\) as the sample variance of those means, and \(W\) as the average within-set sample variance. Since each \(\bar{Y}_m\) has conditional sampling variance approximately \(\operatorname{Var}(Y\mid\theta_m)/K\), \(B\) includes both between-input variation and finite-inner-sample noise. A method-of-moments estimate subtracts that noise:

Thus the estimated total variance is approximately \(B-W/K+W\). These estimates rely on independent outer draws, conditionally independent inner individuals, and an equal inner sample size. They are Monte Carlo estimates, not exact values. In particular, the between-input estimate can be negative when the true component is small relative to simulation noise. A negative result is a warning that the design does not resolve that component; it is not evidence that a variance is physically negative.

Heterogeneous Variability and Interpretation

The conditional variance need not be constant across \(\theta\). For example, uncertainty in a population-level clearance parameter may change not only the mean exposure but also how sensitive exposure is to individual differences in clearance. The decomposition remains valid: the within-population term is the average of the conditional variances, not a single variance assumed to hold for every outer draw. Reporting only one conditional simulation or calculating a within variance at one “typical” parameter set can therefore miss important heterogeneity.

A predictive interval from all nested outputs combines both sources, but its interpretation must be stated carefully. It describes the distribution generated by the specified uncertainty distribution for \(\theta\) and the specified population model for individuals. It is not automatically a confidence interval for an unknown parameter, and it does not mean that parameter uncertainty is biological variation among people. If the scientific question is the range of outcomes across individuals assuming a known population model, condition on a fixed \(\theta\). If the question is how uncertain the predicted population mean is, examine \(E[Y\mid\theta]\) across outer draws instead.

Diagnostics, Alternatives, and Limitations

Nested simulation separates components only if the simulation design respects the intended conditioning. Check that the population-level inputs stay fixed within each inner set, that individual traits are sampled from the intended population, and that dependencies among individual traits are maintained. Check conditional means and variances across outer draws, not just the pooled histogram. If the conditional means have not stabilized as \(K\) increases, inner Monte Carlo noise can obscure the between-input component. If the component estimates change materially as \(M\) increases, the outer simulation may not adequately explore the uncertainty distribution, especially when outputs respond nonlinearly to uncertain inputs.

• Pooling too early. The variance of all \(Y_{mk}\) values estimates total predictive spread, but does not identify its sources. Preserve the outer-draw labels and calculate conditional summaries first.
• Calling every random input “variability.” A random draw in code is not automatically biological variation. Classify an input by what its distribution represents, not by how it is sampled.
• Confusing numerical noise with a component. Small \(K\) makes conditional means noisy; small \(M\) makes their variance estimate unstable. Increase simulation effort or report that a component is unresolved.
• Ignoring changing conditional variance. Use the average conditional variance across outer draws. A variance computed at one nominal input set may not represent it.

A one-level Monte Carlo simulation can efficiently estimate the overall predictive distribution but generally cannot allocate its variance to the two sources unless it retains labels or uses another decomposition design. Fixing \(\theta\) and simulating many individuals estimates conditional variability, not uncertainty across population-level inputs. Conversely, varying \(\theta\) while using only a conditional mean per outer draw can assess variation in the predicted population mean, but does not estimate individual variability. Nested simulation is useful because it represents both conditioning levels directly, though it costs more model evaluations.

The decomposition also depends on the model and distributions supplied. It does not decide whether the outer uncertainty distribution is well supported, whether the individual distribution represents the target population, or whether an output is clinically relevant. Separate measurement error from biological variation when the question requires it; otherwise, including observation noise in the inner simulations will inflate the component labeled as within-population output variability. The result is a structured accounting of simulated spread, not a substitute for evaluating the inputs and assumptions.