Why Residual Patterns Matter
A PD model can produce plausible average predictions and still miss an important feature of the response course. It may predict a response too early, fail to follow a baseline trend, or impose the wrong shape on the exposure-response relationship. A single overall fit statistic can conceal these problems. Residual patterns help reveal where and under what conditions predictions depart systematically from observations.
Consider a longitudinal response \(Y_{ij}\) measured for individual \(i\) at time \(t_{ij}\). Let the model prediction be \(\hat{Y}_{ij}\), based on the model and the information used for prediction. In a population analysis, this might be a population prediction; a conditional prediction may also incorporate an individual’s estimated random effects. The prediction type matters: conditional predictions can describe observed individuals well while concealing shortcomings in the population model.
Residuals are not automatically independent or normally distributed. Repeated observations from one individual may be correlated, and the residual variance may depend on the predicted response. Therefore, plots should be read with the observation model in mind. A systematic pattern is evidence to investigate, not by itself proof of a particular mechanism.
What a Pattern Can Tell You
Under a correctly specified conditional mean model, the expected residual at a given time and exposure should generally be near zero: \(E(Y_{ij}-\hat{Y}_{ij}\mid t_{ij},C_{ij},\text{model information})\approx 0\). If residuals instead vary systematically with time or concentration, the model may be missing a time course or an exposure-response feature. This justification assumes the predictions are appropriate for the diagnostic and that the relevant covariates and observation model have been handled adequately.
Here \(m\) is the model-predicted mean response and \(\epsilon_{ij}\) is observation error. If the mean function is correct and the error has conditional mean zero, then \(E(e_{ij}\mid t_{ij},C_{ij})=0\). A smooth trend in residuals against time or concentration challenges that condition. Random scatter does not establish that the model is correct, but structured scatter is a reason to investigate.
- Possible response lag: residuals may change sign between rising and falling exposure phases. An immediate-effect model can predict too much response while exposure is rising and too little while exposure is falling, or the reverse depending on response direction.
- Possible baseline drift: residuals may move consistently with elapsed time, including at low exposure or in a placebo group. A time-varying baseline or disease course may be missing.
- Possible exposure-response misspecification: residuals may vary with concentration or predicted effect, such as systematic overprediction at high exposure when the model assumes a linear relationship that actually saturates.
These clues overlap. For example, a concentration profile that changes over time can make exposure-response misspecification look like time misspecification. Individual residual sequences, concentration histories, baseline observations, and control data can help separate explanations. If exposure measurements or predictions are uncertain, apparent PD patterns may also reflect upstream PK error.
Worked Example: A Sign Reversal Suggests a Delayed Response
Suppose a hypothetical biomarker is defined so that larger values represent greater drug-related suppression. An immediate-effect model predicts response from the current concentration. For one individual, the residual standard deviation is treated as 4 response units for this diagnostic. The following values are hypothetical.
| Time (h) | Concentration (mg/L) | Predicted response | Observed response | Residual | Standardized residual |
|---|---|---|---|---|---|
| 1 | 2 | 10 | 4 | -6 | -1.50 |
| 2 | 8 | 30 | 23 | -7 | -1.75 |
| 4 | 8 | 30 | 38 | 8 | 2.00 |
| 8 | 2 | 10 | 19 | 9 | 2.25 |
At 1 and 2 h, the raw residuals are \(4-10=-6\) and \(23-30=-7\). At 4 and 8 h, they are \(38-30=8\) and \(19-10=9\). Dividing each residual by 4 gives standardized residuals \(-1.50,-1.75,2.00,2.25\). The model overpredicts early and underpredicts later. Because response remains high after concentration has fallen, a delayed response is plausible.
This sequence is a diagnostic clue, not a formal test for lag. There are few observations, and time trends, baseline change, or an incorrect exposure-response shape could also contribute. The next checks would include individual profiles and residuals against both time and concentration, ideally alongside an alternative model that represents delayed response.
Prediction-Corrected Visual Checks
A visual predictive check compares observed response distributions with distributions generated by simulations from the fitted model. When predictions differ substantially among individuals within a time bin, raw response quantiles can be difficult to interpret: variability in typical predicted response can obscure whether the model reproduces variability around those predictions. A prediction-corrected visual predictive check (pcVPC) rescales observations and simulations to a common reference prediction within each bin before comparing quantiles.
For a proportional response scale, one common correction is \(Y_{\mathrm{pc}}=Y\times \tilde{P}/P_i\), where \(P_i\) is the individual prediction and \(\tilde{P}\) is the median prediction in that time bin. Apply the same correction to simulated responses using their corresponding predictions. For an additive scale, a corresponding correction may be \(Y_{\mathrm{pc}}=Y-P_i+\tilde{P}\). The scale must be chosen to suit the model and response: proportional correction is unsuitable when values can be near zero or when variability is additive.
Worked Example: Correcting a Response for Varying Predictions
Suppose, in a hypothetical 4–6 h bin, an individual has observed response \(Y=18\) response units and prediction \(P_i=16\) response units. The median prediction in that bin is \(\tilde{P}=20\) response units. Under the proportional correction:
The correction maps the observation to the bin’s reference prediction scale. Suppose the corresponding simulated pcVPC has a median of 21 response units and a 90% simulation interval from 18 to 24 response units. The corrected observation, 22.5 response units, lies within that interval and is near its upper end. This single point does not establish adequate fit; the diagnostic compares observed quantiles across bins with simulation-based uncertainty intervals across the full data set.
A pcVPC is a distributional check, not a replacement for individual residual plots. Aggregation can hide a poor fit for a subgroup or a short-lived delay. Conversely, a few unusual residuals may have little effect on population quantiles. Use the checks together and inspect whether the simulation intervals reflect the design, sampling schedule, and variability model.
Time-Stratified Predictions
Time-stratified predictions make temporal lack of fit easier to see than a single overall comparison. Divide follow-up into meaningful intervals—such as baseline, early response, near expected peak, and recovery—and compare observed summaries with model predictions in each interval. The strata should be chosen to reflect the response process and sampling design, rather than selected after inspecting the largest discrepancies.
For each stratum, examine the number of observations, observed mean or median, predicted summary, and uncertainty. For a longitudinal case, it is also useful to show individual trajectories or residuals by time stratum. If a discrepancy occurs at a similar time in both treated and placebo individuals, baseline drift or a shared time effect becomes more plausible. If it tracks exposure across different times, an exposure-response feature deserves closer inspection. Neither pattern alone identifies cause.
Worked Example: Detecting a Saturating Exposure-Response Shape
Suppose hypothetical data are summarized at four concentrations. A linear model predicts \(5+2C\), where response is in units and \(C\) is in mg/L. The observations and calculations are:
| Concentration \(C\) (mg/L) | Observed response (units) | Linear prediction (units) | Residual (units) |
|---|---|---|---|
| 0 | 5 | 5 | 0 |
| 2 | 10 | 9 | 1 |
| 5 | 15 | 15 | 0 |
| 10 | 17 | 25 | -8 |
At \(C=10\) mg/L, for example, the prediction is \(5+2(10)=25\) units and the residual is \(17-25=-8\) units. Residuals are mostly near zero at lower concentrations but strongly negative at the highest concentration. This pattern is consistent with overprediction from an assumed linear relationship as the response approaches a plateau.
As a targeted revision, consider the saturating function \(5+15C/(3+C)\). Its predictions at \(C=0,2,5,10\) mg/L are \(5.00,11.00,14.38,16.54\) units, respectively. The corresponding residuals are \(0,-1.00,0.625,0.462\) units. This model reduces the high-concentration discrepancy in these summaries. It is not thereby established as the correct model: uncertainty, residual structure, individual trajectories, and predictions in other time strata must also be checked.
Revising and Reassessing Without Overinterpreting
A useful diagnostic sequence moves from pattern to hypothesis, then from hypothesis to a targeted revision. Avoid adding several mechanisms at once: if the fit improves, it will be unclear which change mattered. A revision should have a plausible PD interpretation and should be reassessed using both the evidence that suggested it and complementary checks.
Check timing, baseline observations, exposure rise and decline, sampling density, and whether the prediction is population-level or conditional.
Plot raw or standardized residuals against time, concentration, and prediction; inspect individual sequences and compare relevant time strata.
For example, propose a delayed response for a phase-related sign reversal, a baseline time trend for a shared temporal drift, or a saturating function for exposure-dependent overprediction.
Compare residual patterns, time-stratified predictions, and simulation-based checks; also examine parameter plausibility and uncertainty.
Diagnostics can fail in predictable ways. Conditional predictions that use estimated individual effects may absorb population-level misspecification. Sparse sampling may miss a delayed peak. A time trend in treatment exposure can confound time and concentration patterns. A pcVPC can conceal individual failures through aggregation, while a residual plot can look dramatic because of a few influential observations. Compare prediction types and investigate influential data, but do not discard observations solely because their residuals are large.
Residual plots, pcVPCs, and time-stratified predictions answer related but different questions. Residual plots locate systematic discrepancies at the observation level. A pcVPC checks whether simulated response distributions reproduce observed distributions after prediction correction. Time stratification reveals when predictions diverge. Formal likelihood-based comparisons can supplement these checks when models are suitable for comparison, but a better numerical fit does not guarantee a scientifically adequate model.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- An immediate-effect model overpredicts early response and underpredicts response after concentration has declined. What explanation is plausible, and why is it not conclusive?
- For an observation of 12 units, individual prediction 10 units, and bin median prediction 15 units, calculate the proportional prediction-corrected response.
- Residuals are negative at high concentrations but near zero at low concentrations. Name one possible model feature to investigate.
- Why should residual plots and a pcVPC be considered together?
Solutions. (1) A delayed response is plausible because the observed response appears to follow exposure in time, but baseline drift, exposure-response misspecification, and other errors can create similar patterns. (2) \(12(15/10)=18\) corrected response units. (3) A saturating exposure-response relationship is one possibility; the pattern should be checked across individuals and time strata. (4) Residual plots locate observation-level or temporal patterns, whereas a pcVPC compares simulated and observed distributions; aggregation in the latter can hide individual patterns, and isolated residuals need not imply a distributional failure.