1. What Is a Residual Error Model?
In pharmacokinetic modeling, the structural model describes the systematic behavior of drug concentrations over time. Even when the structural model is appropriate, observed concentrations will not lie exactly on the model-predicted curve.
The difference between an observation and its corresponding model prediction is represented through a residual error model, also called an observation model.
A residual error model describes the remaining discrepancy between observed concentrations and the concentrations predicted by the structural PK model.
2. The Observation Model
Let \(C_i\) denote the observed concentration for observation \(i\), and let \(F_i\) denote the concentration predicted by the structural PK model.
A general observation model can be written conceptually as:
For an additive model, this becomes:
The residual term \(\epsilon_i\) represents unexplained variability between the observed concentration and the model prediction.
The important point is that residual variability is not the same thing as between-subject variability. Between-subject variability describes differences in underlying PK parameters among individuals. Residual variability describes observation-level discrepancies that remain after accounting for the structural model and modeled subject-level variability.
| Component | What it describes | Typical example |
|---|---|---|
| Structural model | Systematic concentration-time behavior | Clearance, volume, absorption rate |
| Interindividual variability | Differences between individuals in PK parameters | One patient has higher CL than another |
| Residual error | Observation-level unexplained variability | Assay variation or model discrepancy |
3. The Additive Error Model
The additive error model assumes that the magnitude of the residual error is approximately constant across the concentration range.
Here, \(\sigma\) is the residual standard deviation. The model therefore assumes that the standard deviation of the observation error does not depend on the magnitude of the predicted concentration.
What does "additive" mean?
Suppose the residual standard deviation is \(1\) mg/L. Under an additive model, a prediction of \(2\) mg/L and a prediction of \(20\) mg/L would both have approximately the same residual standard deviation of \(1\) mg/L.
| Predicted concentration | Approximate residual SD | Interpretation |
|---|---|---|
| 2 mg/L | 1 mg/L | Error is large relative to the prediction |
| 10 mg/L | 1 mg/L | Error is moderate relative to the prediction |
| 20 mg/L | 1 mg/L | Error is relatively small |
Thus, an additive error model implies a constant absolute error scale, not a constant relative error scale.
4. The Proportional Error Model
The proportional error model assumes that residual variability increases with the magnitude of the predicted concentration.
Equivalently, the residual component can be expressed as a standard deviation proportional to the prediction:
The key feature is that the coefficient of variation is approximately constant.
What does "proportional" mean?
Suppose \(\sigma=0.20\). A prediction of \(5\) mg/L would have an approximate residual SD of \(1\) mg/L, while a prediction of \(20\) mg/L would have an approximate residual SD of \(4\) mg/L.
| Predicted concentration | Approximate residual SD | Relative error scale |
|---|---|---|
| 5 mg/L | 1 mg/L | 20% |
| 10 mg/L | 2 mg/L | 20% |
| 20 mg/L | 4 mg/L | 20% |
This makes proportional error particularly useful when measurement variability is naturally expressed as a percentage of concentration rather than as a fixed concentration amount.
5. The Combined Additive + Proportional Error Model
Many PK datasets show a pattern in which variability is not well described by either a purely additive or purely proportional model across the entire concentration range.
A combined error model allows both components to contribute:
where the proportional component is represented by \(F_i\epsilon_{1i}\), while \(\epsilon_{2i}\) represents an additive component.
A common formulation is:
Under the usual independence assumption, the conditional variance is approximately:
Therefore, the residual standard deviation changes with the predicted concentration rather than remaining completely constant or increasing strictly in direct proportion.
Conceptual residual standard deviation as a function of predicted concentration. Additive error is approximately constant; proportional error increases with prediction; combined error incorporates both patterns.
6. Additive vs. Proportional vs. Combined Error
| Feature | Additive | Proportional | Combined |
|---|---|---|---|
| Basic form | \(C=F+\epsilon\) | \(C=F(1+\epsilon)\) | \(C=F+F\epsilon_1+\epsilon_2\) |
| Absolute error | Approximately constant | Increases with \(F\) | Can have both components |
| Relative error | Increases as concentration decreases | Approximately constant | Can change across concentration range |
| Low concentrations | Can be useful | May become problematic | Can accommodate additive noise |
| High concentrations | May underestimate variability | Can be useful | Can accommodate proportional variability |
| Main assumption | Constant absolute SD | Constant relative SD | Both absolute and relative components |
These are not simply three levels of model complexity. They represent different assumptions about how observation variability behaves across the concentration range.
7. Understanding the Variance of Each Error Model
One of the clearest ways to understand residual error models is to examine how their variance changes with the model prediction.
Additive model
The variance is constant regardless of \(F_i\).
Proportional model
The variance increases quadratically with the predicted concentration.
Combined model
The combined model therefore has a constant variance component plus a concentration-dependent component.
8. Why the Same Error Can Look Different at Different Concentrations
Consider an absolute residual of \(1\) mg/L.
| Observed or predicted concentration | Absolute error | Relative error |
|---|---|---|
| 2 mg/L | 1 mg/L | 50% |
| 5 mg/L | 1 mg/L | 20% |
| 10 mg/L | 1 mg/L | 10% |
| 20 mg/L | 1 mg/L | 5% |
The same absolute measurement error therefore represents a very different percentage of the concentration depending on where the observation lies in the concentration range.
This is one reason residual error models should be evaluated against the concentration range and measurement process rather than selected solely because one model has a simpler equation.
9. How Log Transformation Changes the Error Model
Another common approach is to model concentrations on the logarithmic scale.
Suppose:
Exponentiating both sides gives:
This creates a multiplicative error structure on the original concentration scale.
Log-scale modeling can therefore be useful when variability is naturally related to relative rather than absolute concentration differences. It also ensures that the back-transformed prediction remains positive.
10. How Should You Choose an Error Model?
Error-model selection should be guided by the observed data, the assay or measurement process, the concentration range, and model diagnostics.
- Start with scientific knowledge. Consider whether measurement error is expected to be approximately constant in absolute units or proportional to concentration.
- Inspect the concentration range. A narrow range may make different residual models difficult to distinguish.
- Fit plausible candidate models. Compare additive, proportional, combined, or log-scale alternatives when scientifically reasonable.
- Examine residual diagnostics. Look for trends in residuals versus predictions, time, or other relevant variables.
- Evaluate parameter estimates. Check whether the residual parameters are plausible and reasonably estimated.
- Consider objective-function or likelihood-based comparisons where appropriate. Statistical comparison can support, but should not replace, examination of model adequacy.
- Prefer an adequate and interpretable model. Additional complexity should have a modeling purpose.
11. What Do Residual Diagnostics Tell Us?
Residual diagnostics are particularly important because a residual error model can appear mathematically reasonable while still leaving systematic patterns in the data.
Residuals versus predictions
A plot of residuals against predictions can reveal whether residual variability changes systematically with concentration.
For example, a funnel-shaped pattern may indicate that variability increases as predicted concentrations increase, suggesting that a purely additive error model may not adequately describe the data.
Conditional weighted residuals
Population PK analyses often use residual diagnostics such as conditional weighted residuals to examine whether residuals are approximately centered around zero and whether systematic trends remain.
Residuals versus time
Patterns over time can indicate that the structural PK model is not adequately capturing some feature of the concentration-time profile. Not every residual pattern should be "fixed" by changing the error model.
12. Worked Example: Comparing Additive and Proportional Error
Suppose a population PK model predicts a concentration of 10 mg/L.
Step 1: Additive error
Assume the additive residual standard deviation is \(2\) mg/L.
The model therefore implies a residual SD of \(2\) mg/L at a prediction of \(10\) mg/L.
Step 2: Proportional error
Now suppose the proportional residual standard deviation is represented by a coefficient of variation of \(20\%\), or \(\sigma_{\mathrm{prop}}=0.20\).
At \(10\) mg/L, the additive and proportional models happen to produce the same residual SD.
Step 3: Compare them at 2 mg/L
Under the additive model:
Under the proportional model:
The models now make very different assumptions. The additive model retains a \(2\) mg/L residual SD, whereas the proportional model predicts a much smaller \(0.4\) mg/L residual SD.
Step 4: Compare them at 20 mg/L
Under the additive model:
Under the proportional model:
The proportional model now allows twice as much absolute variability as the additive model.
Step 5: Interpretation
| Prediction | Additive SD | Proportional SD |
|---|---|---|
| 2 mg/L | 2 mg/L | 0.4 mg/L |
| 10 mg/L | 2 mg/L | 2 mg/L |
| 20 mg/L | 2 mg/L | 4 mg/L |
This example shows why the two models can behave similarly over one part of the concentration range while making very different assumptions at the extremes.
13. Worked Example: Understanding a Combined Error Model
Suppose a combined error model has:
The conditional variance is:
At \(F=2\) mg/L
At \(F=10\) mg/L
At \(F=20\) mg/L
The additive component provides a baseline amount of variability, while the proportional component becomes increasingly important as the predicted concentration increases.
14. Common Error-Modeling Mistakes
Mistake 1: Confusing residual error with IIV
Residual error describes observation-level variability. Interindividual variability describes differences in PK parameters among individuals. They represent different levels of the model.
Mistake 2: Assuming proportional error means constant absolute error
Under proportional error, the absolute residual SD increases with the predicted concentration. What remains approximately constant is the relative error scale.
Mistake 3: Automatically choosing combined error
A combined model is flexible, but flexibility alone is not evidence that it is appropriate. If the data support a simpler error structure, unnecessary complexity may not provide a meaningful benefit.
Mistake 4: Treating residual patterns as purely an error-model problem
A trend in residuals can result from misspecified clearance, volume, absorption, covariates, time structure, or other aspects of the structural model. Changing the residual error model may not solve the underlying problem.
Mistake 5: Ignoring the concentration scale
Error behavior should be interpreted on the scale on which the observation model is defined. An error model on the original concentration scale has different implications from one defined after logarithmic transformation.
Mistake 6: Assuming one error model works for every assay
Residual variability depends partly on how observations are generated and measured. The appropriate model can therefore differ among compounds, assays, laboratories, concentration ranges, and study designs.
15. How Error Models Appear in Population PK
In population PK software, residual error models are typically specified alongside the structural model and interindividual variability model.
A simplified hierarchy can be represented as:
For an individual subject \(i\), the structural model may depend on individual-specific parameters such as:
Those parameters can incorporate interindividual variability. The residual error model is then applied to the resulting prediction.
| Model layer | Question |
|---|---|
| Structural model | What concentration-time behavior is expected? |
| IIV model | How do PK parameters differ among individuals? |
| Covariate model | Can observed characteristics explain some parameter differences? |
| Residual error model | How do observations vary around the individual predictions? |
This separation is fundamental to interpreting population PK models correctly.
16. Reading Residual Patterns in Practice
Consider several conceptual patterns that might appear in a residual-versus-prediction plot.
| Pattern | Possible interpretation |
|---|---|
| Approximately constant spread | An additive error model may be plausible. |
| Spread increases with prediction | A proportional component may be needed. |
| Large spread at low concentrations plus increasing spread at high concentrations | A combined model may be plausible. |
| Systematic positive or negative trend | Potential structural-model or covariate misspecification should be investigated. |
| Strong asymmetry or non-normal residual behavior | The assumed residual distribution or observation scale may require reconsideration. |
These patterns are diagnostic clues rather than automatic decisions. A residual plot should be interpreted alongside the study design, assay characteristics, structural model, and other diagnostics.
17. A Practical Error-Modeling Workflow
- Understand the measurement process. Determine whether assay variability is expected to be approximately absolute, proportional, or mixed.
- Inspect the concentration range. Very wide ranges can make the distinction between additive and proportional variability especially important.
- Fit a scientifically plausible starting model. Do not begin with maximum complexity automatically.
- Inspect residual diagnostics. Examine residual behavior versus predictions and time.
- Compare alternative error structures. Consider additive, proportional, combined, and log-scale formulations where appropriate.
- Check parameter precision and plausibility. A complicated residual model should still be identifiable and interpretable.
- Evaluate the overall model. Residual error should be considered together with structural model adequacy, IIV, and covariate relationships.
- Select the simplest adequate formulation. The final model should represent the observation process sufficiently well for the scientific purpose.
18. Key Takeaways
- The residual error model describes unexplained variability between observed concentrations and model predictions.
- An additive error model assumes approximately constant absolute residual variability.
- A proportional error model assumes that absolute variability increases with predicted concentration while relative variability remains approximately constant.
- A combined additive + proportional model allows both a baseline absolute component and a concentration-dependent component.
- The variance of a combined model can be represented as the sum of additive and proportional variance components under the usual independence assumption.
- Residual error is distinct from interindividual variability: residual error describes observation-level discrepancies, whereas IIV describes differences in PK parameters among individuals.
- Log-scale models create multiplicative error behavior on the original concentration scale.
- Residual-versus-prediction diagnostics can help identify whether variability changes across the concentration range.
- Systematic residual patterns are not necessarily solved by changing the error model; structural-model and covariate misspecification should also be considered.
- The most appropriate error model depends on the measurement process, concentration range, data, diagnostics, and scientific purpose.
- A more complex combined error model is not automatically preferable to a simpler model.
Where to Go Next
A natural progression is to study interindividual variability in population PK, followed by interoccasion variability, residual unexplained variability models, and the relationship between fixed effects, random effects, and residual error.
These concepts together provide the foundation for understanding how a population PK model separates typical population behavior, between-subject differences, and observation-level variability.