Why Examine These Residual Relationships?
A PD model can reproduce an overall response distribution yet systematically miss for particular doses, baseline response levels, or patient groups. A residual plot can reveal such patterns: for example, predictions might be too low at one dose level, or residuals might become more variable as baseline response increases. These checks complement plots against concentration and predicted effect by asking whether prediction errors vary with treatment assignment or patient characteristics.
A pattern is a diagnostic clue, not a causal conclusion. Dose can be associated with patient characteristics because of the dose-assignment design; a residual trend against dose therefore does not by itself show that the dose-response relationship is misspecified. Likewise, residual differences between covariate groups may reflect other differences between those groups. Interpretation requires the model, sampling design, and residual definition to be considered together.
The prediction target must be held consistent: for example, do not mix conditional predictions that use individual-specific estimates with population predictions that do not. Use a residual definition appropriate to the response distribution and variance model, as established in the earlier material on PD residuals. Standardization can make residual magnitudes more comparable when the model specifies different residual variances across observations, but it does not remove a misspecified mean trend or make dependent observations independent.
Reading a Plot Against Dose
Plot residuals against the dose relevant to the model and design. If dose is a set of assigned levels, display the levels as categories or at their actual numeric values with the discreteness made clear. If dose varies continuously, retain its numeric scale. With repeated observations per patient, points from one patient are not independent; use patient identifiers or suitable summaries to see whether an apparent pattern is driven by a few patients.
Under a correctly specified conditional mean model, residuals should not show a systematic mean trend over dose after accounting for the model structure. A rising or falling residual center can indicate dose-related underprediction or overprediction. A changing vertical spread instead suggests that residual variability may depend on dose. These are different patterns: a trend in the center concerns the mean structure; a fan-shaped spread concerns the variance structure.
Worked Example: Detecting a Dose-Related Mean Pattern
In a hypothetical continuous-PD analysis, the model provides predicted responses and standard errors at three assigned dose levels. The following observations are independent patients, and the displayed standardized residuals use the same model-based residual definition throughout.
| Dose (mg) | Observed response | Predicted response | SE (response units) | Standardized residual |
|---|---|---|---|---|
| 10 | 14 | 12 | 2 | 1.00 |
| 10 | 11 | 12 | 2 | -0.50 |
| 20 | 17 | 16 | 2 | 0.50 |
| 20 | 14 | 16 | 2 | -1.00 |
| 40 | 18 | 20 | 2 | -1.00 |
| 40 | 16 | 20 | 2 | -2.00 |
For the first patient, \(e_i=14-12=2\) response units and \(r_i=2/2=1.00\). Applying the same calculation to each row gives the residuals shown. The mean standardized residuals are \( (1.00-0.50)/2=0.25\) at 10 mg, \( (0.50-1.00)/2=-0.25\) at 20 mg, and \( (-1.00-2.00)/2=-1.50\) at 40 mg.
The group means become progressively more negative: in this small hypothetical sample, the model tends to overpredict responses at 40 mg relative to the lower doses. The pattern is worth investigating, but six observations do not establish its stability or cause. Check the individual points, dose assignment and patient composition, and whether the prediction and residual definitions are appropriate. The dose plot does not replace an examination of concentration or predicted effect.
Baseline Response: Mean Pattern or Changing Spread?
Baseline response is often an important predictor of subsequent PD response. Plot residuals against the baseline measurement, keeping the baseline’s role in the model explicit. A residual center that changes with baseline suggests baseline-dependent bias: the model may systematically miss for patients with low or high baseline values. A residual spread that grows with baseline instead points toward baseline-dependent variability. Both patterns can occur together.
The baseline scale and response scale matter. If the model predicts follow-up response, use residuals for that modeled outcome. If the analysis uses change from baseline, the baseline appears in the calculated outcome itself. In that case, plotting change-from-baseline residuals against baseline can create or amplify a relationship through shared quantities, even when the model is not simply failing to account for baseline. Interpret the plot in light of the outcome definition and model rather than assuming every trend is a baseline covariate effect.
Worked Example: Separating Baseline Bias From Baseline-Related Spread
Suppose standardized residuals from a hypothetical follow-up-response model are grouped by baseline response. Each group contains three different patients, with one residual per patient.
| Baseline group (response units) | Standardized residuals | Mean residual | Range |
|---|---|---|---|
| Low: 20–29 | -0.4, 0.0, 0.4 | 0 | 0.8 |
| Middle: 30–39 | -0.8, 0.0, 0.8 | 0 | 1.6 |
| High: 40–49 | -1.6, 0.0, 1.6 | 0 | 3.2 |
The low-baseline mean is \((-0.4+0+0.4)/3=0\); the middle and high means are also zero. Thus, these grouped summaries show no baseline trend in the residual center. The ranges, however, increase from \(0.8\) to \(1.6\) to \(3.2\) standardized-residual units. This is a possible signal of increasing spread, not evidence of increasing mean bias. Because each group has only three patients, the pattern is a prompt to examine the individual observations and the model’s variance assumptions, not a firm finding.
A zero mean within each group does not prove the full conditional mean is correct: a small group can hide patterns, and averaging can conceal subgroups or curvature. Conversely, larger residual spread does not by itself identify why variability increases.
Patient Covariates and Subgroup Patterns
For a categorical covariate, such as a hypothetical disease-severity category, compare residual distributions across its levels. For a continuous covariate, such as body weight, plot residuals against the measured values and consider whether a smooth trend or systematic change in spread is visible. Label groups clearly and show the number of patients contributing to each summary. A group mean alone can hide outliers, skewness, or a mixture of different response patterns.
A subgroup with mostly positive residuals has observed responses that tend to exceed predictions; mostly negative residuals indicate the reverse. If covariate groups also differ in dose, baseline, or sampling times, the residual difference may not be attributable to the covariate itself. Consider whether the apparent pattern persists within comparable dose and baseline ranges, and whether a few influential observations explain it. Do not treat many observations from one patient as many independent confirmations of a patient-level covariate pattern.
Worked Example: Comparing Residuals Across a Patient Covariate
In a hypothetical study, investigators examine residuals by a two-level patient covariate. For illustration, the values below are standardized residuals, one per patient, after fitting the same PD model.
| Covariate group | Residuals | Mean | Calculation |
|---|---|---|---|
| Group A | -0.4, 0.0, 0.2, 0.2 | 0.00 | \((-0.4+0+0.2+0.2)/4=0\) |
| Group B | 0.5, 0.7, 0.9, 1.1 | 0.80 | \((0.5+0.7+0.9+1.1)/4=0.80\) |
Group B’s mean residual is \(0.80\) standardized-residual units higher than Group A’s. Since \(r_i=(y_i-\hat y_i)/SE_i\), the positive Group B residuals indicate that observed responses tend to exceed model predictions in that group. This is a possible subgroup-specific underprediction. Before interpreting it as a covariate-related failure, compare dose and baseline distributions between groups and inspect the points by patient. A covariate pattern that disappears when patients with comparable dose and baseline are considered may reflect the study design or covariate imbalance rather than an independent covariate effect.
A Practical Diagnostic Sequence
Specify which PD observation is being assessed and whether predictions and residuals are conditional or population-level. Keep that choice consistent within the plot.
Use raw residuals when response units are useful and the variance is suitably comparable; use a justified standardized scale when residual variances differ across observations.
Show individual points, identify dose or covariate groups, and report group sizes. Add summaries only as aids to reading the pattern.
Look for a changing residual center as evidence of potential mean bias and for a changing vertical spread as evidence of potential variance misspecification.
Assess dose and covariate imbalance, repeated observations within patients, and whether a small number of points dominate the apparent pattern before proposing a model change.
Assumptions, Alternatives, and Limitations
A useful interpretation assumes that observations and predictions are correctly paired, the residual definition suits the response and variance model, and the plotted variable is measured and coded appropriately. The plot itself does not require normally distributed residuals. However, standard residual magnitudes and summaries can be misleading if the assumed variance model is wrong, and apparent patterns can be unstable in sparse regions of dose or covariate space.
These plots answer different questions from residuals against concentration or predicted effect. Dose can expose structure associated with assigned regimen; baseline can reveal dependence on starting response; patient covariates can reveal subgroup patterns. Covariate-stratified visual predictive checks offer a simulation-based comparison of observed and simulated data distributions, while residual plots emphasize individual prediction errors. Neither approach alone establishes a mechanism. Observed-versus-predicted summaries address a different diagnostic question and should not be substituted for checking residual structure.
A residual plot is descriptive, and visual patterns are affected by sample size, repeated measurements, and covariate overlap. A trend can motivate model evaluation, but it does not prove which term is missing or whether a model change is warranted. Check whether the pattern is reproducible, compatible with the data-generating design, and supported by the model’s scientific purpose. Avoid reading a trend as a causal covariate effect.
Common Mistakes
- Reversing the residual sign. With \(e_i=y_i-\hat y_i\), positive values mean underprediction, not overprediction. State the convention before interpreting a plot.
- Calling every changing spread a mean trend. A fan shape can signal variance structure even when the residual center remains near zero.
- Treating dose as exposure. Assigned dose and realized concentration are not interchangeable; a dose-related pattern does not identify an exposure-response error by itself.
- Attributing group differences directly to a covariate. Dose, baseline, or other characteristics may differ between groups and explain the pattern.
- Overinterpreting a group average. Small groups and repeated observations can make averages unstable or misleading; inspect individual points and patient clustering.
- Ignoring how baseline enters the outcome. A change-from-baseline outcome can share the baseline term used on the horizontal axis, complicating interpretation of a trend.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Using \(e_i=y_i-\hat y_i\), what does a positive residual mean?
- A residual plot has a nearly constant center but increasingly wide scatter as baseline response rises. What pattern does this suggest?
- In a hypothetical dose group, observed response is 9 units, predicted response is 11 units, and the residual SE is 2 units. Calculate the raw and standardized residuals and interpret their signs.
- Two patient-covariate groups have different mean residuals, but they also received different dose distributions. Why is a covariate-specific interpretation uncertain?
- Why can plotting change-from-baseline residuals against baseline require special caution?
Solutions. (1) The observation is above the prediction: \(y_i-\hat y_i>0\), so the model underpredicts that observation. (2) It suggests baseline-dependent residual variability, not necessarily baseline-dependent mean bias. (3) \(e_i=9-11=-2\) response units and \(r_i=-2/2=-1.00\). The observation is 2 response units below prediction, or one residual standard error below it. (4) Dose imbalance could account for some or all of the residual difference; compare groups over similar dose and baseline ranges before attributing the pattern to the covariate. (5) Baseline is part of the change calculation, so the plotted quantities share a term; that mathematical coupling can affect the apparent relationship.