Representing Dependence Among Uncertain Inputs

Why Dependence Among Inputs Matters

A PBPK model may require uncertain values for several physiological or drug-related inputs at once. Their marginal distributions describe uncertainty in each input separately, but they do not say which combinations of values are plausible. That second part is the dependence structure. It matters when inputs share a biological determinant, come from the same experiment, or are linked by a physiological constraint.

For example, organ size and organ blood flow may vary together across individuals, while two measurements from one experiment may share calibration error. Conversely, inputs that are treated as uncertain for different reasons might reasonably be modelled as independent, if there is evidence for that assumption. Dependence should be justified from biology, study design, or external data; it should not be added merely because a joint model can accommodate it.

The marginal distributions selected in earlier material specify each input’s scale and range. Here the focus is how to join those marginals into a coherent joint distribution. The issue is separate from how the resulting values are generated or propagated through the PBPK model.

Covariance and Correlation Matrices

For inputs with finite means and variances, pairwise dependence can be summarized by covariance. The covariance of \(X_i\) and \(X_j\) is the expected product of their deviations from their means. Positive covariance indicates that values above their respective means tend to occur together; negative covariance indicates that a high value of one tends to accompany a low value of the other. Covariance has units equal to the product of the inputs’ units, so its magnitude is hard to compare across differently scaled inputs.

The correlation is covariance divided by the product of the two standard deviations. It is unitless and lies between \(-1\) and \(1\). A correlation near \(1\) represents strong positive linear association; a correlation near \(-1\) represents strong negative linear association. Correlation describes linear association, not every possible form of dependence.

The covariance matrix \(\Sigma\) collects the variances on its diagonal and covariances off the diagonal. The correlation matrix \(R\) has ones on the diagonal and pairwise correlations off the diagonal. A valid covariance matrix must be symmetric and positive semidefinite: for every vector \(a\), \(a^\mathsf{T}\Sigma a\geq 0\). This condition ensures that no linear combination of inputs has a negative variance. A proposed matrix can have individually plausible pairwise correlations and still fail this joint validity condition.

A multivariate normal model is one specific joint model described by a mean vector and a covariance matrix. Under that model, the covariance matrix determines the joint distribution, and zero covariance between two components implies their independence. Outside a multivariate normal model, covariance does not fully determine the joint distribution, and zero covariance does not in general imply independence. This distinction is important for bounded, skewed, or heavy-tailed PBPK inputs.

Preserving Marginals With a Copula

A covariance-based multivariate normal model is convenient when its marginal shapes and support are appropriate. But a normal distribution can assign probability to impossible values, such as negative organ volume or negative clearance. Transforming inputs can address support constraints, while a copula offers a general way to specify dependence separately from the marginal distributions.

A copula joins marginal cumulative distribution functions into a joint distribution. For continuous inputs, the probability integral transform converts each input \(X_i\) to a uniform variable \(U_i=F_i(X_i)\), where \(F_i\) is its marginal cumulative distribution function. A copula is the joint distribution of these uniform variables. This construction lets an analyst retain, for example, a positive skewed marginal for one input and a bounded marginal for another, while specifying a dependence pattern between them.

A Gaussian copula is a common choice. It starts with a multivariate normal vector \(Z\) with a specified correlation matrix \(R\), transforms each component to \(U_i=\Phi(Z_i)\), and then maps \(U_i\) through the inverse marginal distribution \(F_i^{-1}\). The resulting \(X_i=F_i^{-1}(U_i)\) has the chosen marginal distribution. The latent normal correlation is not generally equal to the Pearson correlation of the transformed inputs, especially when the marginals are skewed or bounded. It must not be reported as though it were the transformed inputs’ correlation without checking.

How Dependence Changes Predicted Exposure

A local approximation makes the effect of dependence explicit. Let \(Y\) be a PBPK prediction, such as log exposure, and approximate its change near a reference input vector by a linear combination of input deviations. If \(g_i\) is the local sensitivity of \(Y\) to \(X_i\), then the approximate variance of \(Y\) is the sum of the individual variance contributions and all pairwise covariance contributions. For two inputs, the covariance contribution is \(2g_1g_2\operatorname{Cov}(X_1,X_2)\). Its sign depends on both the input association and the directions of the model sensitivities.

This is the variance of a linearized prediction. It is exact if \(Y\) is an affine function of the inputs; for a nonlinear PBPK model it is a local approximation, whose accuracy can diminish over broad uncertainty ranges or near nonlinear transitions. It nevertheless shows why independent sampling is not automatically conservative: it removes covariance terms, which can either increase or decrease predictive variance.

Choosing and Checking a Dependence Structure

A defensible dependence model begins by identifying why inputs might be related. Shared physiology, common body-size scaling, a conservation constraint, paired measurements, or a common assay error can create different kinds of association. The analyst should distinguish these explanations: an empirical association in one population may not represent a mechanistic constraint, and an association caused by shared measurement error may not describe the true biological inputs.

A covariance matrix is a compact choice when the relevant joint distribution is adequately described by its means, variances, and covariances, as in a multivariate normal model. A copula is useful when the selected marginals have different shapes or supports and dependence needs separate specification. Neither device repairs unsupported assumptions. For example, a Gaussian copula can represent rank association, but it may inadequately represent strong joint tail dependence. A single correlation can also miss nonlinear relationships, thresholds, or constraints that make some combinations impossible.

Practical checks should include whether the matrix is symmetric and positive semidefinite, whether each marginal has the intended support, and whether transformed values retain the intended marginal distributions. For a copula, distinguish the latent dependence parameter from dependence measured on the original input scale. When data are available, compare the proposed relationships with observed joint patterns and their uncertainty; when they are not, describe the evidence and assumptions transparently. Also inspect plausible joint draws or input combinations for biological impossibilities, such as organ size and flow combinations that violate the intended physiology.

The main alternative is to assume independence, which is a simple joint model but still an assumption. It may be appropriate when independence is scientifically defensible or when the available evidence cannot support a more structured model and the limitation is made explicit. It should not be selected merely because marginal distributions are available. A further alternative is a mechanistic or conditional joint model that directly encodes shared drivers or constraints. That can be more interpretable but requires more structure and evidence than a correlation matrix. Whichever approach is used, sensitivity of the exposure prediction to plausible dependence assumptions is informative when dependence is poorly known.

Common Mistakes and Limitations

• Assuming separate marginals imply independence. They specify each input alone, not how inputs co-vary. State the joint assumption separately.
• Equating zero correlation with independence. This is valid for jointly normal inputs, but not generally. Nonlinear dependence can remain when covariance is zero.
• Using an invalid correlation matrix. Pairwise correlations in the range \([-1,1]\) are not enough. Check symmetry and positive semidefiniteness of the entire matrix.
• Confusing latent and observed-scale correlation. A copula’s parameter may describe dependence on a latent scale. Calculate or otherwise assess the dependence induced among the actual PBPK inputs.
• Calling a mathematical association biological evidence. A valid joint distribution is not proof that its relationships apply to the intended population, dose, or experimental setting.
• Treating a local variance approximation as a complete uncertainty analysis. For nonlinear responses, broad input uncertainty can make the linear approximation inaccurate. Its result is a diagnostic or approximation, not a substitute for evaluating model predictions over the relevant uncertainty range.