Why Examine PD Residuals and Concentration Errors Together?
A staged PK/PD fit uses PK predictions as inputs to a PD analysis. If the PK predictions miss the concentrations that drive response, the resulting PD residuals may show a pattern that looks like a PD-model problem. Looking at the two kinds of discrepancies together can help locate where the mismatch occurs: concentration errors that align with PD residuals suggest an exposure-related contribution, while a PD pattern remaining after accounting for plausible exposure error points toward additional causes. Neither pattern, by itself, identifies a unique cause.
These diagnostics are especially useful because a staged analysis often treats the estimated PK predictions as fixed in the PD stage. Its PD residuals can therefore look acceptable even when uncertainty in those predictions has not been carried into the PD analysis. Residual plots are evidence about fit, not a substitute for representing uncertainty or for comparing estimation strategies.
The distinction between \(\delta C_{ij}\) and \(e^C_{ij}\) is essential. If \(C_{ij}^{obs}=C_{ij}^{true}+u_{ij}\), where \(u_{ij}\) is concentration measurement error, then \(e^C_{ij}=\delta C_{ij}+u_{ij}\). Thus an observed-minus-predicted concentration error is not automatically the true concentration discrepancy. They coincide only if concentration measurement error is absent or negligible.
What a Paired Residual Pattern Can Show
Suppose the response-generating model is \(Y_{ij}=f(C_{ij}^{true};\theta_{PD})+\epsilon_{ij}\), where \(\epsilon_{ij}\) is PD observation error with mean zero under the assumed model. If the fitted PD parameters equal the underlying parameters and \(\delta C_{ij}\) is small, a first-order expansion gives:
This is a local approximation, not an identity for every nonlinear model. It says that the response residual attributable to concentration misspecification depends both on the true concentration discrepancy and on the local slope of the PD relationship. A positive concentration discrepancy tends to produce a positive PD residual where the response increases with concentration; for a decreasing response, the expected sign reverses. Near an Emax plateau, the slope may be small, so even a noticeable concentration discrepancy can have little effect on the response prediction.
When the PD parameters are estimated rather than known, their estimation error also contributes. A first-order expansion around the fitted concentration and the underlying PD parameters yields:
Consequently, residual alignment with concentration errors is suggestive, not conclusive: PD parameter estimation, time trends, model misspecification, and random variation can also create structure. If \(u_{ij}\) is independent of \(\delta C_{ij}\), \(\epsilon_{ij}\), and the parameter-estimation contribution, it adds noise to \(e^C_{ij}\) and can weaken the visible association. Dependence between these quantities can complicate the pattern further.
Worked Example: Separating Observed Concentration Error From the PD-Relevant Discrepancy
Consider a hypothetical linear PD relationship \(f(C)=10+3C\), with response measured in response units and concentration in mg/L. For one observation, let \(\hat C=2.0\) mg/L, \(C^{true}=2.5\) mg/L, and \(C^{obs}=2.3\) mg/L. Suppose \(Y=17.4\) response units. The fitted PD prediction and residual are:
The true discrepancy is \(\delta C=2.5-2.0=0.5\) mg/L. The observed concentration error is \(e^C=2.3-2.0=0.3\) mg/L. In this example, the measurement error is \(u=C^{obs}-C^{true}=2.3-2.5=-0.2\) mg/L, so \(e^C=\delta C+u=0.5-0.2=0.3\) mg/L.
At the true concentration, the expected PD response under the specified model is \(10+3(2.5)=17.5\) response units. The observation is \(17.4\), so \(\epsilon=17.4-17.5=-0.1\) response units. The local expansion is exact for this linear model:
Using \(e^C=0.3\) mg/L in place of \(\delta C=0.5\) mg/L would give \(3(0.3)-0.1=0.8\), which does not reproduce the residual. The difference comes from concentration measurement error, not from an arithmetic inconsistency.
Reading Exposure-Dependent Patterns
For each observation, examine residuals against predicted concentration, time, and relevant design features; when concentrations are observed, examine \(e^C\) against predicted concentration and time as well. Also inspect paired \(r^{PD}\) and \(e^C\), preferably with points identified by individual. A trend in the average residual across exposure levels can signal exposure-dependent bias, but the expected direction and size depend on the PD function and its local slope.
Residuals measured in response units can have different variability across observations. If the model specifies a response standard deviation \(\sigma_{ij}\), a standardized residual is \(r^{PD}_{ij}/\sigma_{ij}\); interpret it only under that variance specification. A change in residual spread with concentration can indicate an inadequate variance model, even if the residual average is near zero. Likewise, concentration errors should be judged against the expected concentration measurement and prediction variability, rather than by their raw magnitudes alone.
Worked Example: An Exposure-Related Residual Pattern
In a hypothetical analysis, the fitted PD relationship is an Emax model \(f(C)=10+40C/(4+C)\), with response in arbitrary response units and \(C\) in mg/L. In four low-exposure observations, the PD residuals are \(0.2,-0.3,0.1,0.0\), and the concentration errors are \(0.1,-0.1,0.2,0.0\) mg/L. The means are:
In four high-exposure observations, suppose the PD residuals are \(2.0,2.4,1.8,2.2\) response units and the concentration errors are \(0.8,1.0,0.7,0.9\) mg/L. Then:
The simultaneous positive averages at high exposure are a diagnostic signal: the PK predictions may be low in this range, while the PD predictions are also low. At a representative predicted concentration of 9 mg/L, the Emax slope is \(40(4)/(4+9)^2=160/169=0.947\) response units per mg/L. If the observed error were a good proxy for \(\delta C\), a discrepancy of 0.85 mg/L would correspond locally to about \(0.947(0.85)=0.805\) response units. That is smaller than the mean PD residual of 2.10 response units. The calculation is only a local comparison; it does not prove that the remaining difference is a PD-model defect. Measurement error, the approximation, PD parameter uncertainty, or other systematic structure could contribute. The pattern warrants investigation, not a causal conclusion.
Individual-Level Mismatch and Practical Diagnostics
A pooled residual average can conceal opposing individual patterns. For example, some individuals may have mostly positive PD residuals and others mostly negative residuals, leaving the overall mean close to zero. Plotting residuals and concentration errors by individual can reveal this cancellation and show whether the same individuals have systematic concentration and response discrepancies.
Worked Example: Detecting Opposing Individual Patterns
Suppose a hypothetical staged fit gives the following three PD residuals, in response units, and concentration errors, in mg/L, for each of three individuals:
| Individual | PD residuals | Mean PD residual | Concentration errors | Mean concentration error |
|---|---|---|---|---|
| A | 1.2, 1.0, 0.8 | 1.00 | 0.4, 0.5, 0.3 mg/L | 0.40 mg/L |
| B | -1.1, -0.9, -1.0 | -1.00 | -0.4, -0.3, -0.5 mg/L | -0.40 mg/L |
| C | -0.1, 0.2, -0.1 | 0.00 | 0.0, 0.1, -0.1 mg/L | 0.00 mg/L |
For A, the mean residual is \((1.2+1.0+0.8)/3=1.00\) response unit and the mean concentration error is \((0.4+0.5+0.3)/3=0.40\) mg/L. For B, the corresponding means are \((-1.1-0.9-1.0)/3=-1.00\) response unit and \((-0.4-0.3-0.5)/3=-0.40\) mg/L. For C, both means are zero. The pooled PD residual mean is \((3-3+0)/9=0\) response units, and the pooled concentration-error mean is also zero. Thus the pooled means alone conceal the opposing patterns. If the PD response increases with concentration, A and B show signs consistent with exposure discrepancies contributing to their response mismatch. Individual PD effects, other model features, and noise remain possible explanations.
A practical diagnostic sequence is to check the sign conventions and data alignment first, then inspect concentration and PD residuals separately, followed by paired patterns and individual-level plots. Check whether patterns change with exposure, time, or sampling design; verify the assumed variance behavior; and consider whether a few influential observations dominate the appearance. If PK concentrations are available only at some times, be explicit about which observations contribute to each diagnostic. A paired plot using observed concentrations is informative only when those measurements correspond appropriately to the PD observation and their measurement error is understood.
Confirm that positive \(e^C\) means observed concentration exceeds prediction and positive \(r^{PD}\) means observed response exceeds prediction. Match times and individuals correctly.
Look for shifts in mean, changes in spread, curvature, time trends, and individual clustering. Judge standardized residuals only against a stated variance model.
Compare the direction of PD residuals with concentration errors in light of the local PD slope, and account for concentration measurement error and parameter estimation.
Check data, model structure, and variance assumptions. Where appropriate, compare the staged result with a simultaneous fit, while recognizing that residual patterns alone do not quantify all uncertainty omitted by staging.
Assumptions, Alternatives, and Limits
Interpreting these patterns requires correctly aligned observations, a stated definition of the PK prediction, and a PD model whose response direction and local slope are understood. The Taylor argument assumes a small enough \(\delta C\) for a first-order approximation to be useful and differentiability of \(f\) near \(\hat C\). Interpreting a residual as an unexpected response also depends on the assumed PD error distribution and variance structure. If residuals are correlated within an individual, they should not be treated as independent evidence simply because they are plotted as separate points.
The alternative of inspecting only PD residuals can reveal a poor response fit but cannot show whether exposure prediction errors align with it. Inspecting only concentration errors can reveal PK prediction problems but cannot show whether they matter for the response at the local PD slope. Joint inspection is more informative about alignment, but it remains a diagnostic rather than a formal decomposition of error. Simultaneous estimation represents PK and PD information within one estimation framework; comparing its residual displays with staged-fit displays can help assess robustness, but no visual comparison alone guarantees correct uncertainty estimates.
A common mistake is to substitute \(e^C=C^{obs}-\hat C\) directly for \(\delta C=C^{true}-\hat C\) in the response expansion. That is valid only when concentration measurement error is negligible: in general, \(e^C=\delta C+u\). Another mistake is to treat residual correlation as proof that PK misspecification caused PD bias. A shared time trend, an omitted individual effect, correlated errors, or fitting to the same observations may produce similar patterns. Residual diagnostics help identify where a model deserves scrutiny; they do not establish causation, and a clean residual plot does not show that staged uncertainty has been fully propagated.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If \(C^{true}=5.0\), \(\hat C=4.5\), and \(C^{obs}=4.8\) mg/L, calculate \(\delta C\), \(e^C\), and the concentration measurement error \(u\).
- For an increasing linear PD model with slope 2 response units per mg/L, a true concentration discrepancy of \(-0.4\) mg/L, and PD error of \(0.1\) response units, what first-order PD residual is expected if PD parameters are known?
- Why can a pooled mean PD residual of zero fail to show individual-level mismatch?
- What does a positive PD residual paired with a positive concentration error establish, and what does it not establish?
Solutions. (1) \(\delta C=5.0-4.5=0.5\) mg/L, \(e^C=4.8-4.5=0.3\) mg/L, and \(u=C^{obs}-C^{true}=4.8-5.0=-0.2\) mg/L; indeed, \(e^C=\delta C+u=0.5-0.2=0.3\) mg/L. (2) The slope contribution is \(2(-0.4)=-0.8\) response units, so the expected first-order residual is \(-0.8+0.1=-0.7\) response units. (3) Positive and negative individual residual means can cancel in the pooled mean; inspect residuals by individual as well as overall. (4) The signs are consistent with an exposure discrepancy contributing to the response residual if the PD relationship is increasing. They do not prove that explanation: measurement error, parameter estimation, other model misspecification, or correlated errors can also produce the pattern.