Modeling Discrepancy Between Linked Components

Why Model Linked-Component Discrepancy?

A linked model can be internally coherent and still systematically miss observations at the point where one component hands information to another. For example, a PBPK model may predict a clearance that is consistently lower than the clinical estimate, or a population PK model may provide exposures that lead an exposure-response model to underpredict biomarker response. Treating every mismatch as random noise can make predictions look less precise than warranted or obscure a directional error. But adding a flexible correction without care can hide which component is failing.

A discrepancy term represents systematic mismatch between a linked model’s prediction and the quantity it is intended to reproduce. It is not a new observation, and it is not automatically evidence that a particular component is wrong. Its interpretation depends on where it enters the model, what data inform it, and which assumptions constrain it.

Definition: Let component A supply an output \(q_A(x;\theta_A)\) under conditions \(x\), and let component B transform that output into a prediction \(m_{AB}(x)=h(q_A(x;\theta_A),x;\theta_B)\). A discrepancy term \(d_{AB}(x)\) represents systematic mismatch at this link. For an outcome \(y\), one possible observation model is \(y=m_{AB}(x)+d_{AB}(x)+\epsilon\), where \(\epsilon\) represents remaining observation or process error. The discrepancy belongs to the specified interface; it should not be silently interpreted as a correction to either component’s parameters.

The distinction between \(d_{AB}(x)\) and \(\epsilon\) is substantive. The discrepancy describes a structured offset or pattern that may persist across observations in similar conditions. The error term describes variation remaining after that structure is represented. If the discrepancy is set to zero, a systematic error may be forced into the residual variance. If the discrepancy is too flexible, it may instead absorb component misspecification or unsupported extrapolation.

Choosing the Scale and Structure

Use the scale of the linked quantity to decide how to define discrepancy. If additive differences are meaningful, an additive term is natural: a response prediction in concentration units can be corrected by an offset in concentration units. For positive quantities such as clearance or exposure, a log-scale term often gives a more interpretable multiplicative correction. With \(d_{\log}(x)\) on the log scale, the implied multiplicative factor is \(\exp(d_{\log}(x))\).

A constant discrepancy is appropriate only when a common offset or multiplier across the stated conditions is scientifically plausible. When mismatch is expected to vary with a covariate, a structured function can represent that pattern, for example \(d_{AB}(x)=\gamma_0+\gamma_1x\). The number and form of terms should be limited by the available information. A discrepancy function that can reproduce nearly any observed outcome may fit the calibration data while saying little about predictions in new conditions.

Modeling principle: Place the discrepancy at the narrowest interface supported by evidence, define its scale and units, and state what it is allowed to vary with. Keep it distinct from uncertainty in \(\theta_A\) or \(\theta_B\), and from residual error. Where possible, evaluate each component against data that directly inform that component before attributing a linked mismatch to the interface.

This principle matters because different explanations imply different future predictions. A PBPK clearance bias, a population PK structural error, and an exposure-response offset may produce similar discrepancies in one linked outcome, yet behave differently when dose, population, or endpoint changes. A correction at the final outcome can fit the observed data without resolving which explanation is credible.

Worked Example: Estimating a Constant Additive Offset

Suppose a linked model predicts concentrations of 8.0, 10.0, and 12.0 mg/L in three matched conditions. Hypothetical observations are 9.2, 11.0, and 13.4 mg/L. Assume the interface discrepancy is a common additive offset \(\delta\), and, for this calculation, the three errors are independent with equal variance.

The residual differences before correction are \(r_i=y_i-m_i\): \(1.2\), \(1.0\), and \(1.4\) mg/L. Under the assumed model, \(r_i=\delta+\epsilon_i\), so the least-squares estimate is their mean:

\hat{\delta}=\frac{1.2+1.0+1.4}{3} =\frac{3.6}{3}=1.2\ \mathrm{mg/L}.

The corrected predictions are \(8.0+1.2=9.2\), \(10.0+1.2=11.2\), and \(12.0+1.2=13.2\) mg/L. The corrected residuals are \(0.0\), \(-0.2\), and \(0.2\) mg/L. The correction removes the average offset in these data; it does not prove that the linked model is unbiased in other conditions. In particular, the remaining pattern and the component-level evidence still matter.

Estimating and Interpreting a Constant Offset

The constant-offset calculation extends directly when observations have different known or estimated error variances. Suppose \(r_i=y_i-m_i=\delta+\epsilon_i\), with independent errors \(\epsilon_i\) having mean zero and variance \(s_i^2\). The weighted least-squares estimate minimizes the sum of squared residuals weighted by inverse variance. Differentiating that objective gives the estimate below.

Q(\delta)=\sum_{i=1}^{n}\frac{(r_i-\delta)^2}{s_i^2}, \qquad \frac{dQ}{d\delta}=-2\sum_{i=1}^{n}\frac{r_i-\delta}{s_i^2}=0, \qquad \hat{\delta}= \frac{\sum_{i=1}^{n}r_i/s_i^2}{\sum_{i=1}^{n}1/s_i^2}.

Thus observations with smaller error variance contribute more to estimating the common offset. If the variances are treated as known and the errors are independent normal variables, the variance of this estimate is \(1/\sum_i(1/s_i^2)\). This uncertainty describes estimation of the offset under the stated model; it does not include uncertainty about whether a constant offset is the right discrepancy structure.

Worked Example: A Multiplicative Clearance Discrepancy

A PBPK component predicts clearance values of 4.0, 5.0, and 6.0 L/h for three matched hypothetical settings. Corresponding clinical estimates are 4.8, 6.0, and 7.2 L/h. Suppose a common multiplicative discrepancy is plausible and define it on the natural-log scale: \( \log(CL_{\mathrm{obs}})=\log(CL_{\mathrm{PBPK}})+\delta_{\log}+\epsilon \).

Each observed-to-predicted ratio is \(4.8/4.0=1.20\), \(6.0/5.0=1.20\), and \(7.2/6.0=1.20\). Therefore the common log discrepancy is

\hat{\delta}_{\log}=\log(1.20)=0.1823, \qquad \exp(\hat{\delta}_{\log})=1.20.

The linked predictions would be multiplied by 1.20: for example, \(5.0\ \mathrm{L/h}\times1.20=6.0\ \mathrm{L/h}\). In this calibration set, the clinical estimates are 20% higher than the PBPK predictions. This is a description of the matched settings, not a universal 20% correction for other populations or conditions. The model also assumes that the discrepancy is multiplicative and common across these settings.

Diagnosing Mismatch Without Hiding Its Source

A discrepancy term is most useful when its placement and form can be defended. Check the linked quantity, units, population, and clinical conditions before fitting a correction: a mismatch caused by incompatible exposure definitions or nonaligned conditions is not evidence for a biological bias term. Earlier material on harmonizing quantities and aligning populations is directly relevant to making that comparison meaningful.

Then ask whether the discrepancy is supported by data at the interface. For example, if component A predicts a PK quantity, compare that quantity with data that directly inform PK where possible, rather than relying only on a downstream clinical endpoint. If a discrepancy is estimated only from the final endpoint, it may compensate for errors in several upstream and downstream components. A fitted correction alone cannot identify which one is responsible.

Inspect whether the assumed discrepancy form is plausible across the conditions represented in the data. A constant offset should not leave a clear trend with dose or population covariates; a multiplicative correction should be judged on the ratio scale. Such checks assess the chosen discrepancy representation, not every possible source of model error. When the data cover only a narrow range, apparent constancy within that range does not establish constancy beyond it.

Worked Example: Detecting a Dose-Dependent Offset

An exposure-response component predicts a hypothetical biomarker change of 2.0, 4.0, and 6.0 units at doses of 10, 20, and 30 mg. Observed changes are 2.3, 4.8, and 7.3 units. Consider the structured additive discrepancy \(d(D)=\gamma_0+\gamma_1(D-10)/10\), where \(D\) is dose in mg.

The observed-minus-predicted differences are \(0.3\), \(0.8\), and \(1.3\) units. At 10 mg, \(d(10)=\gamma_0=0.3\), so \(\hat{\gamma}_0=0.3\) units. At 20 mg, the scaled dose term is one, giving \(0.8=0.3+\gamma_1\); hence \(\hat{\gamma}_1=0.5\) units per 10-mg increment. The 30-mg difference checks the same form: \(0.3+2(0.5)=1.3\) units.

The correction reproduces these three differences exactly, but three observations do not establish that the linear discrepancy generalizes. With more data, one would examine whether the form is supported over the relevant dose range and whether the estimated trend is distinguishable from measurement and model uncertainty. A flexible dose correction should not be used to conceal a misspecified exposure-response relationship without evidence that the mismatch belongs at this interface.

Alternatives, Limitations, and Common Pitfalls

A discrepancy term is one option, not a default fix. The main alternatives answer different questions:

Approach	What it represents	Main caution
No discrepancy term	The linked components are assumed to reproduce the target quantity apart from specified error.	Systematic mismatch may be forced into residual variation.
Recalibrate component parameters	Parameters of a component are updated using relevant data.	Do not recalibrate parameters using downstream outcomes if those data cannot identify the component’s contribution.
Revise component structure	The component’s functional form or mechanistic assumptions are changed.	A revised model requires evidence and evaluation; a correction term is not itself a structural explanation.
Add a constrained discrepancy	A defined interface offset or trend accounts for systematic mismatch not represented by the linked components.	Weak constraints or sparse data can make it absorb errors from multiple components.
Increase residual variance	Unexplained observation-level variation is allowed to be larger.	Variance inflation alone does not represent a persistent directional bias.

A common mistake is to see an improved fit after adding discrepancy and conclude that the upstream component has been validated. The reasoning is wrong because the correction can compensate for an upstream error, a downstream error, or both. The correction improves agreement under its fitted conditions; it does not identify the source of the mismatch. A better interpretation names the interface, scale, and conditions covered, and keeps component-specific evidence visible.

Other limitations are practical. Discrepancy estimates may be weakly identified when model parameters and correction terms can produce similar changes in predictions. They may also be unstable when data are sparse or concentrated in a narrow range. If the correction is carried into a new scenario, its applicability must be justified; an estimated offset under one population or endpoint is not automatically transportable to another. Finally, discrepancy terms do not repair errors in the mapping of quantities, dependence, or uncertainty between components.

Key takeaway: A discrepancy term makes a defined mismatch explicit; it does not explain that mismatch by itself. Specify its location, scale, structure, and evidential basis, and avoid allowing it to erase evidence about which component may need revision.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions.

A linked prediction is 15 mg/L and the matched observation is 18 mg/L. What is the additive discrepancy for this observation, and what is its multiplicative ratio?
Why can a flexible correction fitted only to a clinical endpoint fail to identify whether the PBPK or exposure-response component is biased?
In a constant-offset model with independent errors of variance \(s_i^2\), should an observation with smaller or larger \(s_i^2\) receive more weight? State the weight.
When might a log-scale discrepancy be more interpretable than an additive discrepancy?

Solutions. (1) The additive difference is \(18-15=3\) mg/L, and the ratio is \(18/15=1.20\). (2) Errors in either component can produce the same endpoint mismatch, and an endpoint-only correction can compensate for their combined effect without separating their contributions. (3) The smaller-variance observation receives more weight; its inverse-variance weight is \(1/s_i^2\). (4) For a positive quantity such as clearance, a log-scale term represents a multiplicative factor, making a common proportional difference meaningful across different absolute values.