Representing Correlated Parameter Uncertainty

Why Dependence Between Inputs Matters

A PBPK parameter set is a collection of inputs, not a list of isolated uncertain quantities. Organ volumes and blood flows may be constrained by shared physiology; several parameters may be measured in the same assay; and physicochemical quantities may be linked through a common underlying property. If those inputs are uncertain, their joint uncertainty can therefore include dependence. Treating each marginal distribution as correct does not ensure that independently combining the inputs produces plausible parameter sets.

This tutorial focuses on representing that dependence. Covariance and correlation summarize particular aspects of joint variation, while rank correlation summarizes the tendency of paired values to move together in rank. None of these summaries, by itself, specifies every feature of a joint distribution. A useful representation must be compatible with the marginal distributions and with the biological or measurement relationships that motivate it.

The finite-second-moment condition ensures that the centered product in the covariance has a finite expectation: by the Cauchy–Schwarz inequality, \(E[|(X-E[X])(Y-E[Y])|]\) is finite when both variances are finite. Finite means alone are not sufficient. If either standard deviation is zero, the corresponding variable is constant with probability one and Pearson correlation is undefined. A positive covariance means that deviations from the respective means tend to have the same sign; a negative covariance means they tend to have opposite signs. These are summaries of joint variation, not proof that one parameter causes the other to change.

Covariance Structures and Basic Checks

For a vector of uncertain inputs \(X_1,\ldots,X_k\), the covariance matrix has the variance of each input on its diagonal and the covariance of each pair in the corresponding off-diagonal positions. It is symmetric. It must also be positive semidefinite: for every vector of constants \(a\), the variance of the linear combination \(a^\mathsf{T}X\) cannot be negative. This mathematical requirement rules out some collections of pairwise correlations even when every individual correlation lies between \(-1\) and \(1\).

Standardizing each input by subtracting its mean and dividing by its standard deviation gives a correlation matrix. Its diagonal entries are 1, and its off-diagonal entries are Pearson correlations. A proposed matrix should be checked as a whole for positive semidefiniteness, not just entry by entry. For three variables, for example, every two-variable correlation can be valid while the full three-variable matrix is not.

The arithmetic above also illustrates why dependence should be chosen for a reason rather than added as a cosmetic adjustment. If a total volume were exactly fixed, then \(Y=C-X\) for a constant \(C\), and \(\operatorname{Cov}(X,Y)=-\operatorname{Var}(X)\). The example's correlation of \(-0.60\) is not that exact constraint: it represents a less-than-perfect inverse association. The proposed correlation should be supported by the physiological constraint or data that motivate it.

Physiological and Measurement Sources of Dependence

Several common features of PBPK inputs can make independent uncertainty questionable. Organ volumes and blood flows may be linked by whole-body totals or by shared size-related physiology. Tissue fractions that must add to a fixed total cannot all vary independently. Parameters derived from a common assay or from the same individuals may share measurement error or biological variation. Binding inputs can also be associated because they reflect shared underlying properties or were obtained under related conditions. Earlier material on joint physiological constraints and on measurement error, biological variability, and knowledge uncertainty provides relevant distinctions: the source of dependence matters when deciding what the joint uncertainty is intended to represent.

A physiological constraint does not automatically imply a particular Pearson correlation. The relationship might be nonlinear, bounded, or different across regions of the parameter range. Likewise, shared measurement error may produce dependence even when two underlying physiological quantities are not causally linked. A defensible dependence structure should therefore state its basis, such as a fixed-sum constraint, paired observations, a joint assay, or a clearly described modeling assumption.

Rank Correlation When the Relationship Is Not Linear

Pearson correlation describes linear association on the scale where it is calculated. It can be small even when two variables have a strong nonlinear relationship. Rank correlation instead describes how consistently larger values of one input are paired with larger or smaller values of another. Spearman's rank correlation is the Pearson correlation of the variables' ranks; for \(n\) paired observations with no ties, it can be calculated from rank differences \(d_i\) as \(1-6\sum_i d_i^2/[n(n^2-1)]\). With ties, use the Pearson correlation of the assigned ranks rather than this no-ties shortcut.

Rank-based dependence can be useful when the intended relationship is monotone but not well represented as linear on the original scale. It still does not fully specify the joint distribution. In particular, a rank correlation does not determine how strongly extreme values co-occur, and it does not by itself define how to combine bounded, lognormal, or other marginal distributions into a joint model. A dependence model must preserve the chosen marginals and respect the support and constraints of each input.

Derivation: Why the Covariance Term Appears

The variance formula used above follows directly from the definition. Assume \(X\) and \(Y\) have finite second moments, and write \(\mu_X=E[X]\) and \(\mu_Y=E[Y]\). Centering the sum and expanding its square gives $$ \begin{aligned} \operatorname{Var}(X+Y) &=E[((X-\mu_X)+(Y-\mu_Y))^2]\\ &=E[(X-\mu_X)^2]+E[(Y-\mu_Y)^2]\\ &\quad+2E[(X-\mu_X)(Y-\mu_Y)]\\ &=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y). \end{aligned} $$

The cross-product expectation is precisely the covariance, so it disappears only when the covariance is zero. Independence, when the relevant expectations exist, implies zero covariance; zero covariance alone does not generally imply independence. This distinction matters because an uncertainty description based only on covariance can miss nonlinear dependence.

Diagnostics, Alternatives, and Limitations

Before using a dependence structure, check that it is mathematically possible and interpretable. Confirm that the covariance matrix is symmetric and positive semidefinite, that each variance is nonnegative, and that any reported Pearson correlations are between \(-1\) and \(1\). Check the units and scales: a covariance between a volume and a flow has units of volume-times-flow, while a correlation does not. For rank-based descriptions, state how ties and transformations are handled. Then ask whether the implied pairings are physiologically plausible, particularly near bounds or under fixed totals.

Several alternatives answer different questions. Independent marginals are appropriate when independence is a defensible assumption, not merely when dependence information is unavailable. A covariance matrix is compact and informative for linear co-variation, especially when the variables are on well-chosen scales. Rank correlation can express monotone association when a linear pattern is unsuitable. A mechanistic constraint may be preferable when a known identity, such as a fixed total, determines how inputs must move together. These approaches are not interchangeable: matching pairwise correlations does not guarantee matching joint behavior.

A common failure is to specify plausible-looking correlations pair by pair without checking the matrix as a whole. Another is to impose a multivariate normal model on the original scale when some inputs must be positive or bounded. Transformations can help represent positive inputs, but correlations depend on scale, and transformations do not automatically solve all constraints. Finally, correlation is a summary, not an explanation: it cannot distinguish shared physiology from shared measurement error, and it does not imply causation.