Why Plot Residuals Against Exposure and Effect?
A PD model can fit the overall response reasonably well yet miss systematically at particular exposure levels. For example, it may underpredict responses at high concentrations, or its predicted effects may be too high in one part of the response range and too low in another. Plotting residuals against concentration and predicted effect helps reveal these patterns, which can be obscured by a single overall summary of model fit.
The two plots answer related but different questions. A residual-versus-concentration plot asks whether prediction errors change across the exposure range. A residual-versus-predicted-effect plot asks whether errors change across the range of effects represented by the fitted model. Neither plot identifies a unique cause: a pattern can suggest a problem with the exposure–effect relationship, but it does not by itself establish which model component is wrong.
The concentration should be defined consistently with the model and the question: for example, the concentration measure used to drive the PD model, at the time of the response observation. The plotted residual must also be appropriate to the response scale. If residual variability changes with the mean, a variance-standardized residual may be more interpretable than a raw residual. For example, \(r_{ij}=e_{ij}/\hat\sigma_{ij}\), where \(\hat\sigma_{ij}\) is the model-estimated residual standard deviation for that observation. Residuals with different definitions, scales, or conditioning levels should not be mixed in one plot without a clear reason.
What a Pattern Means
Under an adequate conditional mean model, residuals should not have a systematic mean trend over the relevant predictor range. To see why, write the true conditional mean response at concentration \(C\) as \(m(C)=E(Y\mid C)\), and the model prediction as \(\hat m(C)\). Then:
If the model's conditional mean is correct, \(m(C)=\hat m(C)\), so this conditional expectation is zero. Thus, a sustained positive trend in residuals as concentration increases indicates that observed responses tend to exceed predictions in that region. A sustained negative trend indicates that predictions tend to exceed observations. This is a diagnostic justification, not a claim that every individual residual should be near zero.
A residual-versus-predicted-effect plot can expose a different feature. If residuals are mostly negative at low and high predicted effects but positive in the middle, the model may be missing curvature or otherwise describing the exposure–effect shape poorly. A roughly straight upward or downward trend can indicate a systematic mismatch across the prediction range. The pattern is evidence to investigate, not proof of a specific functional form or biological mechanism.
A changing vertical spread is not the same as a changing mean. Residuals may remain centered near zero while becoming more variable at higher concentrations. That pattern points toward concentration-dependent variability or a scale issue, rather than necessarily indicating systematic underprediction. Inspect both the center and the spread of the residual cloud.
Worked Examples
Worked Example: Increasing Underprediction at Higher Concentrations
In a hypothetical continuous-response analysis, six observations have the following concentration values and raw residuals. Concentration is in mg/L and residuals are in response units.
| Concentration (mg/L) | Residual (response units) |
|---|---|
| 1 | −0.2 |
| 2 | −0.1 |
| 4 | 0.1 |
| 5 | 0.2 |
| 8 | 0.6 |
| 10 | 0.7 |
To summarize the pattern descriptively, calculate the average residual in three concentration ranges. At low concentration, the mean is \((-0.2-0.1)/2=-0.15\) response units. In the middle range it is \((0.1+0.2)/2=0.15\) response units. At high concentration it is \((0.6+0.7)/2=0.65\) response units.
The residuals rise across the concentration range, moving from mostly negative to clearly positive. Because residual equals observed minus predicted response, the fitted model overpredicts on average at low concentration and underpredicts at high concentration. This pattern is consistent with concentration-dependent bias and may motivate checking the exposure–effect shape. Six points are not enough to establish a mechanism or select a replacement model; the plot identifies a feature to investigate.
Worked Example: Curvature Across Predicted Effect
Suppose a hypothetical PD model produces the following predicted effects and raw residuals. Both are on the same response scale.
| Predicted effect (response units) | Residual (response units) |
|---|---|
| 1 | −0.2 |
| 2 | −0.1 |
| 4 | 0.3 |
| 6 | 0.4 |
| 8 | −0.2 |
| 10 | −0.4 |
The mean residual for the two lowest predictions is \((-0.2-0.1)/2=-0.15\) response units. For the middle predictions it is \((0.3+0.4)/2=0.35\) response units. For the two highest predictions it is \((-0.2-0.4)/2=-0.30\) response units.
The residuals are negative at both ends and positive in the middle. The model therefore tends to predict too much at the low and high ends, and too little in the middle. This curved pattern suggests that the predicted-effect range is not being represented adequately by the fitted mean relationship. It does not, on its own, show whether the issue is the exposure–effect function, another model component, or the limited observations in these ranges.
Worked Example: Separating Spread from Bias
Consider a hypothetical dataset with four residuals. At low concentration, the residuals are −0.2 and 0.2 response units; at high concentration, they are −1.0 and 1.0 response units. The average residual at each concentration level is zero:
The high-concentration residuals have a wider spread, but these observations do not show a mean shift toward underprediction or overprediction. If the fitted residual standard deviations are 0.2 response units at low concentration and 1.0 response unit at high concentration, the standardized residuals at each level are \((-1,1)\): for example, \(-1.0/1.0=-1\) and \(1.0/1.0=1\) at high concentration.
The raw plot signals a larger absolute error spread at high concentration, while the standardized plot shows comparable deviations relative to the fitted variability. This distinction matters: a funnel-shaped cloud should not automatically be described as concentration-dependent bias. The residual means and residual spreads answer different diagnostic questions.
Practical Reading and Failure Modes
A useful inspection starts with the zero-residual reference line and the full range of the horizontal axis. Look for a sustained trend, curvature, a change in spread, and regions with few observations. A smooth curve can help reveal broad structure, but it should be interpreted alongside the individual points; a few influential observations can pull a smoother away from the pattern of the rest. Binned means can summarize local direction, but their values depend on the bin boundaries and should not replace inspection of the data.
Record whether the vertical axis is raw, standardized, or another defined residual, and state the concentration measure and response scale.
Check whether residuals remain roughly centered near zero across concentration or predicted effect, rather than systematically changing sign or magnitude.
Distinguish a wider cloud from a shifted cloud. If variability is modeled as changing, inspect residuals standardized using that variance model as well as the raw scale where useful.
Be cautious at concentration or predicted-effect extremes with very few observations. A trend there may be unstable or driven by a small number of points.
The residual type also matters. Population-level and individual-level predictions answer different questions, so their residual patterns need not be identical. Use a consistent prediction level and residual definition for the diagnostic question. If the residuals are conditional on estimated individual parameters, remember that those estimates were informed by the same responses; a visually quiet plot is not independent confirmation of model adequacy.
Repeated observations from the same individual may also contribute correlated residuals. A concentration plot can display that structure without showing why it occurs, and observations should not be treated as independent evidence merely because they appear as separate points. The earlier material on residuals versus time addresses a different ordering of the observations; concentration and predicted effect do not replace the time axis when temporal structure is the question.
Comparison, Limitations, and Common Mistakes
| Diagnostic view | Primary question | Important limitation |
|---|---|---|
| Residual versus concentration | Does prediction error vary systematically over exposure? | A pattern does not identify which model component caused it. |
| Residual versus predicted effect | Does prediction error vary across the model's predicted response range? | The fitted effect is model-derived, so the plot is not independent of model fitting. |
| Residual versus time | Does prediction error vary with observation time? | A time pattern and an exposure pattern answer different questions. |
These plots are complementary to checks based on predictions and simulations, not substitutes for them. A visible residual trend can reveal a local mismatch that an aggregate measure conceals. Conversely, a plot without an obvious trend cannot prove the model is correct, especially when exposure coverage is sparse, residuals are correlated, or the chosen residual scale is inappropriate. Use the plots to locate and characterize a concern, then evaluate possible explanations with suitable model assessment.
- Mistake: treating every positive residual as underprediction at all concentrations. A single positive value says the observed response exceeds the prediction for that observation. Concentration-dependent underprediction requires a systematic pattern, not an isolated point.
- Mistake: calling a wider high-concentration cloud a mean bias. Wider spread indicates greater variability; bias concerns a shifted conditional mean. Examine both separately.
- Mistake: assuming a trend proves the exposure–effect function is wrong. It is a clue, not a diagnosis with a unique cause. Check residual scale, prediction level, data support, and other model features before drawing a conclusion.
- Mistake: reading an empty exposure region as evidence of good fit. Few or no observations provide little information about model performance there. Describe the lack of support rather than extrapolating a reassuring pattern.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A residual is defined as observed minus predicted response. What does a sustained positive residual trend at higher concentrations suggest?
- Residuals are centered near zero at both low and high concentration, but their spread is larger at high concentration. Is this evidence of mean underprediction? Explain.
- Residuals against predicted effect average −0.2 response units at low predictions, +0.3 in the middle, and −0.1 at high predictions. Describe the pattern and a cautious interpretation.
- Why should a concentration range with very few observations be interpreted cautiously?
- At a particular observation, \(Y_{ij}=7.4\) response units and \(\hat E_{ij}=8.0\) response units. Calculate the raw residual and interpret its sign.
Solutions. (1) Observed responses tend to exceed predictions at higher concentrations, consistent with high-concentration underprediction; the plot does not establish its cause. (2) No. The residual center remains near zero, so the stated feature is increased spread, not mean underprediction. (3) The residual pattern is negative at low predictions, positive in the middle, and negative at high predictions. It suggests possible curvature or a mismatch across the predicted-effect range, but is not specific to one cause. (4) A few points may dominate the apparent pattern, and the model's behavior in a sparsely observed region is weakly supported. (5) \(e_{ij}=Y_{ij}-\hat E_{ij}=7.4-8.0=-0.6\) response units. The model overpredicts this observation by 0.6 response units.