Detecting Parameter Interactions with Second-Order Indices

Why Isolate Pairwise Interactions?

A global sensitivity analysis can show that two QSP parameters are influential without revealing whether their effects combine independently or interact. For example, a receptor abundance parameter and a binding-affinity parameter may each affect a predicted response, while their joint effect may also include a distinct contribution that cannot be assigned to either parameter alone. A second-order Sobol index quantifies that pure pairwise contribution for a specified output and input distribution.

The distinction matters when deciding which mechanisms to investigate, which parameter combinations to measure more carefully, or which terms might be simplified during model reduction. A pairwise index is a variance attribution under a defined input distribution; it is not evidence that a biological interaction is causal, and it is not a universal property of the model independent of that distribution.

Definition and Assumptions

Let \(Y=f(X_1,\ldots,X_d)\) be a scalar model output, where \(X_1,\ldots,X_d\) are uncertain parameters. The standard Sobol decomposition assumes that the inputs are mutually independent and that \(Y\) has finite, nonzero variance \(V=\operatorname{Var}(Y)\). The variance decomposition then assigns orthogonal components to individual inputs, pairs, triples, and larger subsets.

Equivalently, in the functional ANOVA decomposition, write \(f=f_0+\sum_i f_i(X_i)+\sum_{i<j} f_{ij}(X_i,X_j)+\cdots\), with components orthogonal under the independent input distributions. Then \(V_{ij}=\operatorname{Var}(f_{ij})\). The zero-mean component \(f_{ij}\) is what remains after removing the mean and both one-input effects from the conditional response given the pair.

The independence assumption is essential to this variance attribution. If parameters are correlated, the standard orthogonal decomposition does not apply as written: effects can be shared or assigned differently depending on the attribution method. For correlated inputs, the interpretation of \(S_{ij}\) given here should not be used without an appropriate alternative decomposition.

Deriving a Pick-and-Freeze Estimator

A second-order component can be isolated using two independent parameter samples. Let \(A\) and \(B\) be independent input vectors, each drawn from the joint input distribution. Define \(A_B^i\) as vector \(A\) with coordinate \(i\) replaced by \(B_i\), and \(A_B^{ij}\) as \(A\) with both coordinates \(i\) and \(j\) replaced by their values in \(B\). For a model output \(f(\cdot)\), form the contrast \(D_{ij}=f(A_B^{ij})-f(A_B^i)-f(A_B^j)+f(A)\).

Here is the justification. A functional ANOVA term that does not depend on \(X_i\) or does not depend on \(X_j\) cancels in the four-term contrast. The surviving terms must depend on both. In the expectation of \(f(B)D_{ij}\), orthogonality and independence make contributions involving additional coordinates vanish: those coordinates in the contrast are taken from \(A\), independently of the corresponding coordinates in \(B\), and the relevant ANOVA components have mean zero. The only surviving covariance is the pure \(f_{ij}\) component with itself, whose expectation is \(\operatorname{Var}(f_{ij})=V_{ij}\). Dividing by \(V\) gives the stated index.

With \(N\) independent base pairs \(A^{(r)},B^{(r)}\), estimate the numerator by the average of the products \(f(B^{(r)})D_{ij}^{(r)}\), and divide by an estimate \(\hat V\) of the output variance. This ratio is a consistent Monte Carlo estimator under the stated assumptions. It is not guaranteed to lie between zero and one in a finite run: sampling noise can make its numerator negative or larger than the estimated total variance. The true \(S_{ij}\), by contrast, is nonnegative.

Separating Pair Effects from Higher-Order Effects

A second-order index does not capture every interaction in which both selected parameters participate. For instance, a three-input component \(f_{ijk}(X_i,X_j,X_k)\) belongs to the third-order variance contribution, not to \(V_{ij}\). The total-order index for \(X_i\) includes all variance components involving \(X_i\), including pairwise and higher-order terms, but \(S_{ij}\) isolates only the pure two-input component.

Choosing Pairs, Diagnosing Estimates, and Comparing Approaches

Computing every pair requires \(d(d-1)/2\) pair-specific contrasts, each formed from model evaluations on hybrid inputs. For a large QSP parameter set, this can be expensive, so selected pairs are often chosen using mechanistic hypotheses or earlier screening results. Pair indices are most informative when the scientific question concerns a specific combination, such as two parameters controlling a shared pathway or feedback loop.

Check the estimator before interpreting a ranking. Confirm that the inputs were sampled independently from the intended distributions, that every hybrid vector changes only the specified coordinates, and that all runs use the same deterministic model settings or a consistent treatment of simulation noise. Compare estimates across independent batches or increasing sample sizes. A negative estimate is a warning about finite-sample noise, not evidence of a negative true variance component. A value above one can also arise from noisy numerator or denominator estimates and should prompt more computation and a check of the implementation.

First- and total-order Sobol indices answer broader questions: the main effect of one parameter, and the total variance contribution involving that parameter, respectively. Second-order indices add detail by isolating selected pure pair terms. The Fourier Amplitude Sensitivity Test offers a different variance-decomposition strategy, while pick-and-freeze estimates are convenient for targeted pairs when the model can be evaluated at hybrid inputs. Neither approach removes the need to define the output and input distributions carefully.

A practical limitation is that a pairwise table is incomplete when higher-order interactions matter. Also, indices depend on the parameter distributions and on the chosen output; changing either can change the variance allocation. Treat a large pair index as evidence of a substantial interaction contribution to variance under the specified analysis, not as a causal effect or a direct measure of biological mechanism strength.