Why a Residual Is Not an Explanation
A concentration residual is the difference between an observed concentration and a model prediction. It summarizes disagreement, but by itself it does not say why the disagreement occurred. Some variation can arise because the laboratory measurement is imperfect. Other disagreement can arise because the structural model does not adequately describe the underlying concentration-time trajectory. These sources have different meanings and may call for different responses: improving or characterizing the assay is not the same as revising the pharmacokinetic model.
The distinction is especially important when a residual error model is being chosen. A residual variance can absorb both measurement noise and unresolved structure. If it is interpreted as assay precision without supporting information, the model may understate structural inadequacy. Conversely, if every residual pattern is attributed to the structural model, ordinary measurement noise may be mistaken for a pharmacokinetic feature.
These definitions locate the two sources at different stages. Structural discrepancy concerns the representation of the concentration-time process; assay error concerns observing a sampled concentration. Neither term automatically means a particular cause. For example, structural discrepancy might reflect an inadequate absorption model, an unrepresented process, or a systematic feature of the concentration-time curve. It does not, on its own, establish a biological mechanism.
A Model That Keeps the Sources Distinct
Let \(C_{ij}^{\mathrm{obs}}\) be the observed concentration for subject \(i\) at observation \(j\), and let \(f(t_{ij};\theta_i)\) be the structural model prediction at time \(t_{ij}\), given subject-specific parameters \(\theta_i\). Write the observation as
Here, \(\delta_{ij}\) represents structural model discrepancy on the concentration scale, and \(\epsilon_{ij}\) is assay error on that same scale. The residual is therefore \(r_{ij}=\delta_{ij}+\epsilon_{ij}\). This decomposition is a conceptual model: the components are not separately observed just because they have been given different names.
A basic variance decomposition requires explicit conditions. Suppose, conditional on the structural prediction and relevant design information, both components have mean zero, are uncorrelated with one another, and have finite variances. Then
If the components are correlated, a covariance term must be included:
The zero-mean condition is also substantive. A systematic difference between predictions and underlying concentrations is a mean discrepancy, not merely extra variance. For example, if the structural model consistently underpredicts concentrations near a particular time, the discrepancy may have a nonzero time-dependent mean. Increasing a constant residual variance can make the likelihood less surprised by those observations, but it does not remove the systematic pattern.
The scale must be consistent throughout. Under an additive model on the original concentration scale, the terms above have concentration units and the variances have squared concentration units. Under a log-concentration model, the decomposition is instead on the log scale, and the component variances are on that scale. A numerical assay variance measured on one scale cannot be subtracted from a total variance measured on another.
What Patterns Can Inform the Distinction?
Assay error is commonly represented as conditionally mean-zero measurement variation. Its variance may depend on concentration or other assay conditions, and this dependence should be characterized on the scale used by the model. Structural discrepancy, in contrast, may have recognizable organization: for example, residuals may tend to be positive over one part of the time course and negative over another. Such a repeated pattern across subjects is evidence that the structural predictions may be missing a feature.
A useful diagnostic question is not simply whether residuals are large, but whether they are organized in ways the assumed assay-error model does not predict. Plot residuals against time, predicted concentration, and relevant design features. Examine whether patterns recur across subjects and whether they persist after accounting for the specified random effects and residual model. These plots are clues, not definitive source labels: an omitted design feature, data-processing issue, or misspecified error scale can also produce structure.
Dependence across observations can also be informative. If assay errors are independent across observations conditional on the true concentrations, while a discrepancy process is correlated across nearby times, then a temporal covariance pattern may indicate that an independent-noise model is inadequate. But independence is an assumption, not a universal property of assays; laboratory batches, shared processing, or other procedures can create correlated assay errors. Dependence alone does not identify which source produced it.
The strongest separation often comes from information that does not rely on the same concentration-time residuals alone. For instance, independent assay validation may provide an estimate of assay precision under relevant conditions. That information can constrain the assay component, leaving the PK data to assess remaining disagreement. The validation must be applicable: assay performance at a different concentration range, sample matrix, or procedure may not describe the measurements in the analysis.
Worked Example: Using External Assay Information
Suppose a hypothetical analysis uses an additive error model on the original concentration scale. Its estimated total residual variance is \(25\;(\mathrm{mg/L})^2\). Independent validation under comparable conditions suggests assay-error variance of \(4\;(\mathrm{mg/L})^2\). Assume both components are mean-zero, uncorrelated, and expressed on the same scale, and treat the two variance estimates as applicable.
Under the variance decomposition,
The implied discrepancy standard deviation is \(\sqrt{21}=4.583\;\mathrm{mg/L}\), while the assay standard deviation is \(\sqrt{4}=2.000\;\mathrm{mg/L}\). The calculation suggests that assay noise alone would not account for the estimated total residual variance under the stated assumptions.
This is not proof that structural discrepancy is exactly \(21\;(\mathrm{mg/L})^2\). The subtraction is meaningful only if the external assay variance applies to these observations, the scales match, and the components are uncorrelated. Uncertainty in both estimates also matters; treating the point estimates as exact would overstate the precision of the conclusion.
Worked Example: A Repeating Time Pattern
In a hypothetical data set, model-based summaries across subjects show the following average residuals. Assume the predictions use the intended dosing and sampling times, and that the residuals have already been examined on the original concentration scale.
| Time after dose (h) | Mean observed concentration (mg/L) | Mean predicted concentration (mg/L) | Mean residual (mg/L) |
|---|---|---|---|
| 1 | 10.8 | 10.0 | 0.8 |
| 4 | 7.2 | 8.0 | -0.8 |
| 8 | 4.1 | 5.0 | -0.9 |
At 1 h, the mean residual is \(10.8-10.0=0.8\;\mathrm{mg/L}\). At 4 h, it is \(7.2-8.0=-0.8\;\mathrm{mg/L}\), and at 8 h, it is \(4.1-5.0=-0.9\;\mathrm{mg/L}\). The signs change with time: the predictions are low early and high later in this summary.
A mean-zero, independent assay-error model would not, by itself, explain why the same time-dependent pattern recurs across subjects. The pattern therefore motivates checking the structural model, time alignment, and other systematic features of the data. It does not establish that the structural model is the cause: a shared measurement or processing bias could also produce time-linked residuals. The diagnostic result is a reason to investigate, not a causal conclusion.
Why Residual Variance Alone Cannot Usually Separate the Sources
If the only information used is a single residual variance, two unknown component variances cannot generally be determined from it. Under the independent, mean-zero assumptions, the observed quantity is their sum. Knowing the sum does not reveal its unique parts.
For example, a total variance of \(25\;(\mathrm{mg/L})^2\) is compatible with discrepancy and assay variances of \(21\) and \(4\), of \(10\) and \(15\), or of \(0\) and \(25\), in squared concentration units. All three decompositions give the same total under this simple model. A numerical optimizer may return one pair if constraints or parameterization force it to, but a returned estimate is not evidence that the data identified the components.
Features beyond the overall variance can help only when they impose different predictions for the sources. Examples include independent assay validation, a defensible assumption that assay errors are independent across measurements while discrepancy is correlated over time, or repeated patterns across profiles that a shared structural discrepancy can explain. Each route depends on assumptions and study design. Without such information, reporting a separate estimate for each source can create false precision.
Worked Example: The Same Total Variance, Different Explanations
Suppose a hypothetical residual model estimates a total variance of \(16\;(\mathrm{mg/L})^2\). Assume, for this illustration, independent mean-zero assay error and discrepancy on the original concentration scale. Consider two possible decompositions.
Decomposition A: assay variance \(=4\;(\mathrm{mg/L})^2\), discrepancy variance \(=12\;(\mathrm{mg/L})^2\). The total is \(4+12=16\;(\mathrm{mg/L})^2\).
Decomposition B: assay variance \(=10\;(\mathrm{mg/L})^2\), discrepancy variance \(=6\;(\mathrm{mg/L})^2\). The total is \(10+6=16\;(\mathrm{mg/L})^2\).
If the residual data contain no other information that distinguishes the components, both explanations imply the same variance and cannot be preferred on that basis. To distinguish them, one would need relevant external assay information or additional structural assumptions that yield different observable predictions. The total residual variance alone is insufficient.
Diagnostics, Alternatives, and Limitations
A practical assessment should keep the decomposition and its assumptions visible. First, verify the measurement scale and the form of the residual model. Next, examine residuals against time, predicted concentration, and design features for systematic patterns. Then compare the observed variation with any assay-performance information that is relevant to the study. Finally, state which parts of the interpretation are data-supported and which depend on assumptions.
The main alternatives are not interchangeable ways to “fix” the same problem. One approach is to use a residual model that represents the total unexplained variation without claiming a source-specific decomposition. Another is to incorporate externally supported assay precision and assess the remaining discrepancy. A third is to revise the structural model when reproducible patterns point to inadequacy in its concentration-time predictions. A more elaborate correlated discrepancy model may be considered only when there is a defensible basis for its structure and enough information to estimate it.
Common failure modes include:
- Calling all residual variation assay error. A large residual variance may include structural inadequacy. The correction is to describe it as total residual variation unless assay precision is independently supported.
- Calling every trend model error. A pattern may instead reflect a measurement, processing, or design issue. Check those possibilities before assigning a cause.
- Subtracting variances from incompatible scales. Variances on the original and log concentration scales cannot be directly compared or subtracted.
- Assuming independence without justification. If the components are correlated, the covariance term changes the decomposition.
- Treating a fitted split as identified. A software estimate does not overcome a likelihood that depends only on the sum of the component variances.
The distinction is useful because it clarifies what evidence a model needs, but it has limits. Structural discrepancy is often not directly measured, and a flexible discrepancy term can compete with the structural parameters rather than clarify them. Results can therefore depend strongly on assumptions about its mean, variance, and dependence over time. The defensible conclusion may be that residual variation is present but its sources are not separately identified.
Check Your Understanding
Answer each question using the assumptions stated in the question. Solutions follow the questions.
- If residual variance is \(36\;(\mathrm{mg/L})^2\) and applicable independent assay-error variance is \(9\;(\mathrm{mg/L})^2\), what discrepancy variance is implied under the additive decomposition?
- Why can a residual variance of \(16\;(\mathrm{mg/L})^2\) not, by itself, identify assay and discrepancy variances separately?
- Mean residuals are repeatedly positive early after a dose and negative late after a dose. What does this pattern suggest, and what does it not prove?
- Can an assay variance on the log-concentration scale be subtracted from a total residual variance on the original concentration scale? Explain.
Solutions. (1) Under the mean-zero, uncorrelated, same-scale assumptions, the implied discrepancy variance is \(36-9=27\;(\mathrm{mg/L})^2\). (2) Any nonnegative pair summing to 16 gives the same total variance under the simple model, so the total alone cannot distinguish the pair. (3) The repeated time pattern is evidence to investigate structural inadequacy or another time-linked systematic feature; it does not prove the structural model is the cause, because measurement or processing effects may also be patterned. (4) No. The two variances use different scales and units of interpretation, so their subtraction has no valid variance-decomposition meaning.