Why Represent Structural Discrepancy?
A QSP model can reproduce many observed patterns yet still show systematic mismatch for a particular endpoint, dose, visit, or patient subgroup. If clinical predictions use only the model output and its usual sources of variability, they may understate that mismatch. Representing it explicitly makes the assumptions visible and allows us to assess how plausible mismatch patterns could change patient-level and trial-level predictions.
Here, structural discrepancy means residual systematic mismatch between a model’s prediction and the clinical quantity it is intended to predict. It is not the same as random observation noise, differences among patients, or uncertainty in estimated model parameters. A discrepancy term does not automatically correct a model: its form and magnitude require justification, and predictions should be examined under defensible alternatives.
The choice of scale matters. Additive discrepancy is expressed in the endpoint’s units; a value of \(5\) means five endpoint units. A multiplicative discrepancy is naturally expressed on a log scale; \(\delta(x)=\ln(1.1)\) corresponds to a factor of \(1.1\), or a 10% increase. The discrepancy may be treated as a fixed scenario, a function of patient or treatment characteristics, or an uncertain quantity over a set of plausible patterns. These interpretations are not interchangeable.
Assumptions and Propagation
Before propagating discrepancy, specify what is being predicted and what uncertainty is already represented. For a patient-level endpoint, let \(x_i\) denote that patient’s clinical setting, including relevant dose, visit, and covariates. A useful conditional model is
This expression assumes that the discrepancy is defined on the same endpoint scale as \(Y_i\), that the remaining error term has a stated distribution, and that the model prediction and discrepancy refer to the same clinical setting. If \(\epsilon_i\) has conditional mean zero, then the conditional mean of \(Y_i\), given \(x_i,\theta,\delta\), is \(m(x_i,\theta)+\delta(x_i)\). If the error is not mean-zero on this scale, the mean must include its actual expectation.
For an additive fixed discrepancy and an existing predictive distribution for \(Y_i\), propagation is often a shift: add \(\delta(x_i)\) to each simulated endpoint, or shift the distribution’s location by that amount. If \(\delta\) is itself uncertain, simulate or otherwise integrate over its specified uncertainty as well. The resulting distribution describes uncertainty under the chosen discrepancy assumptions; it does not establish that those assumptions are true.
A discrepancy shared across patients also creates dependence. If one uncertain discrepancy scenario shifts all patients in a simulated trial together, it should be drawn once per simulated trial, not independently for every patient. Drawing it independently would instead represent patient-specific mismatch and could substantially understate uncertainty in trial averages or event rates.
Worked Example: A Fixed Additive Discrepancy
Suppose the model predicts a hypothetical continuous endpoint for a patient as \(N(100,15^2)\), in endpoint units. For this exercise, assume that this distribution already includes parameter and patient-level predictive uncertainty, and that a plausible structural mismatch is a fixed additive \(+8\) units for every patient in the setting. Assume the discrepancy does not change the spread.
Adding a constant shifts the mean but leaves the variance unchanged. Thus the adjusted prediction is \(N(108,15^2)\). For the probability that the endpoint exceeds 120 units, standardize using the adjusted mean and standard deviation:
Under this hypothetical discrepancy scenario, the probability of exceeding 120 units is about 21.19%. Without discrepancy, the corresponding calculation is \(1-\Phi((120-100)/15)=1-\Phi(1.333)\approx0.0912\), or about 9.12%. The increase follows from the assumed upward shift; it is not evidence that the true clinical probability is 21.19%.
This calculation assumes the additive discrepancy is appropriate, fixed at \(+8\) in this setting, and does not alter predictive spread. If the discrepancy were uncertain or varied by dose, those possibilities would need to be included rather than hidden inside this single scenario.
Discrepancy That Varies Across Patient Groups
A single shift may be inadequate when mismatch depends on a covariate or clinical subgroup. In that case, define the discrepancy by the relevant setting and propagate it within each group. The overall endpoint distribution is then a mixture: each group’s adjusted distribution is weighted by its prevalence in the target population. The weights must describe the population being predicted, not merely the proportions in a convenient sample.
Worked Example: A Subgroup-Dependent Discrepancy
Consider a hypothetical population with 60% in subgroup A and 40% in subgroup B. Before discrepancy, endpoints in subgroup A are normally distributed with mean 50 units and standard deviation 10 units; subgroup B has mean 70 units and standard deviation 12 units. Assume fixed additive discrepancies of \(+5\) units in A and \(-8\) units in B, with no change in either subgroup’s standard deviation. Find the population probability of an endpoint above 60 units.
After adjustment, the subgroup means are 55 and 62 units. In subgroup A, the standardized threshold is \((60-55)/10=0.5\), so the exceedance probability is \(1-\Phi(0.5)\approx0.3085\). In subgroup B, it is \((60-62)/12=-0.1667\), so the exceedance probability is \(1-\Phi(-0.1667)=\Phi(0.1667)\approx0.5662\).
The adjusted mixture therefore has an exceedance probability of about 41.16%. For comparison, without discrepancy the subgroup probabilities are \(1-\Phi(1)\approx0.1587\) and \(1-\Phi(-0.8333)=\Phi(0.8333)\approx0.7977\); their weighted average is approximately \(0.60(0.1587)+0.40(0.7977)=0.4143\). The overall probabilities are similar, but the subgroup-specific probabilities change substantially in opposite directions. Reporting only an overall average could conceal this pattern.
The calculation assumes the subgroup proportions apply to the intended population, the subgroup predictive distributions are appropriate, and the specified discrepancies apply within those groups. If the discrepancy also varies continuously with covariates, the same principle applies, but predictions must be averaged over the target covariate distribution.
Shared Discrepancy and Trial-Level Predictions
Uncertain discrepancy can affect trial-level predictions differently from patient-level predictions. Consider a simplified setting in which every patient in a simulated trial shares the same uncertain additive discrepancy \(\delta\), while individual errors \(\epsilon_i\) are independent. Suppose \(\delta\sim N(0,\tau^2)\), \(\epsilon_i\sim N(0,\sigma^2)\), and both are independent. Then \(Y_i=m+\delta+\epsilon_i\) has marginal variance \(\tau^2+\sigma^2\). But the patients’ outcomes are correlated because they share \(\delta\).
The dependence and its effect on the trial mean follow directly. For distinct patients \(i\) and \(j\), conditional independence of their individual errors gives:
For a trial of \(n\) patients, \(\bar{Y}=m+\delta+\frac{1}{n}\sum_{i=1}^n\epsilon_i\). Independence of \(\delta\) and the individual errors, and independence among the \(\epsilon_i\), therefore imply:
This is different from \((\tau^2+\sigma^2)/n\), which would apply if both sources were independently redrawn for each patient. The shared discrepancy does not average away as the trial gets larger.
Worked Example: Shared Discrepancy in a Trial Mean
Suppose a hypothetical endpoint has model mean \(m=40\) units, shared discrepancy \(\delta\sim N(0,6^2)\) units across the trial, and independent patient-level errors \(\epsilon_i\sim N(0,8^2)\) units. Consider a trial of \(n=25\) patients. Find the probability that its mean endpoint exceeds 46 units.
The trial mean has mean 40 units. Its variance accounts for the shared discrepancy once and averages the independent patient errors:
Consequently, the standardized threshold is \((46-40)/6.210\approx0.9662\), and
Under the stated shared-discrepancy model, the probability is about 16.70%. If the discrepancy were incorrectly treated as independent across patients, the variance of the mean would instead be \((36+64)/25=4\), giving a standard deviation of 2 units and an exceedance probability \(1-\Phi(3)\approx0.00135\). The large difference is due to the dependence assumption, not a change in the patient-level marginal variance.
Here, the normal distribution for \(\delta\) is a hypothetical representation of uncertainty across discrepancy scenarios. In a decision analysis, its probabilities need a defensible basis; it should not be presented as an empirically established frequency distribution merely because it is convenient to simulate.
Choosing and Checking a Discrepancy Representation
Discrepancy patterns should be tied to evidence and intended use. Predictive checks can reveal where model predictions systematically miss observed clinical outcome distributions, but a visible mismatch alone does not identify its cause or uniquely determine a discrepancy function. Consider whether the pattern is plausibly additive or proportional, whether it changes with dose or patient characteristics, and whether it is shared across a trial or specific to individuals. Then propagate a small set of plausible alternatives and report how consequential the choice is.
Specify the clinical endpoint, time, treatment setting, and target patient population. The discrepancy must refer to this same prediction target.
Record which patient, residual, and parameter uncertainties are already in the predictive distribution. Do not add them a second time when discrepancy is propagated.
State discrepancy scale, magnitude, dependence on setting, and whether it is fixed, shared, or patient-specific. Explain the evidence or sensitivity rationale for each pattern.
Recalculate patient and trial summaries under each pattern, preserving any dependence structure, and identify conclusions that change materially.
Useful diagnostics include plotting or tabulating observed-versus-predicted patterns by dose, visit, and subgroup; checking whether a proposed discrepancy reverses direction across settings; and comparing the adjusted distributions with clinically plausible ranges. A discrepancy that improves one subgroup while producing implausible predictions elsewhere is not a general correction. Sensitivity to its size and shape should be made explicit, especially for tail probabilities and threshold-based decisions.
| Approach | What it represents | Key limitation or distinction |
|---|---|---|
| Output discrepancy | Systematic mismatch in the prediction for a specified setting | Its form and uncertainty require justification; it is not automatically identified by a model fit. |
| Parameter uncertainty | Uncertainty about parameter values within the chosen model structure | It does not, by itself, represent a missing mechanism or systematic structural mismatch. |
| Residual or patient variability | Remaining observation-level or between-patient variation | Random variation around predictions cannot substitute for a systematic pattern of mismatch. |
| Alternative model structures | Different mechanistic or mathematical representations | Comparing models can expose structural sensitivity, but does not eliminate the need to state assumptions about each model. |
A common mistake is to add a broad random error term and call the resulting wider prediction interval “structural uncertainty.” That reasoning confounds variability with systematic mismatch. The correction is to specify a discrepancy pattern on the endpoint scale, state whether it is shared or patient-specific, and propagate it in a way that preserves those assumptions. Another pitfall is to treat a sensitivity distribution as if it were data-calibrated; label scenario-based probabilities accordingly and distinguish them from probabilities supported by an estimated predictive model.
Discrepancy propagation is limited by the evidence available to define its magnitude and dependence on setting. It cannot guarantee accurate predictions outside the evidence used to motivate the patterns, nor can it identify which biological mechanism is missing. Its value is disciplined sensitivity analysis: it shows whether a clinical conclusion is robust to plausible residual mismatch rather than implying that one arbitrary adjustment repairs the model.
Check Your Understanding
Use the stated assumptions in each question. Brief solutions follow the questions.
- A model predicts a normal endpoint with mean 30 units and standard deviation 4 units. A fixed additive discrepancy of \(-2\) units is assumed. What is the adjusted mean and standard deviation?
- Why is a discrepancy shared across all patients in a simulated trial not divided by the sample size when calculating the variance of the trial mean?
- In a two-subgroup prediction, why must subgroup-specific exceedance probabilities be weighted by target-population proportions?
- What does an additive discrepancy of \(+3\) units mean, and how does that differ from a multiplicative factor of \(1.03\)?
- A broadening of the predictive distribution is proposed to address a consistent underprediction in one subgroup. What potential conceptual error should be checked?
Solutions. (1) The adjusted mean is \(30-2=28\) units; the standard deviation remains 4 units under the fixed-shift assumption. (2) The same uncertain shift affects every patient, so it remains in the trial mean; only independent patient-level errors average down. (3) The population probability is a mixture over the population being predicted, so each subgroup’s contribution is its probability multiplied by its target-population weight. (4) The additive term shifts the endpoint by 3 endpoint units; the multiplicative factor increases it by 3% relative to its value. (5) Broadening alone represents more variation, not necessarily a systematic subgroup-specific mismatch; specify and propagate the discrepancy pattern separately.