Why Look for Saturation in Residuals?
A formal comparison of linear and saturable elimination can address whether the data support one model over another. Residual diagnostics answer a related but different question: where does a model systematically miss the observed concentrations? This matters because a model can have an acceptable overall fit while consistently underpredicting concentrations in the range where elimination is most saturated.
The central clue is a relationship between residuals and concentration or dose. Saturable elimination removes a smaller fraction of drug per unit time at high concentrations than at low concentrations. If a linear-elimination model is fitted to such data, it may predict that high concentrations decline too quickly. The resulting observations can lie above the model predictions at high concentrations, even if residuals are small at lower concentrations. The exact pattern depends on how the model was fitted, the sampling design, and the residual-error model; no single residual sign is guaranteed in every fitted data set.
Residuals and the Saturation Mechanism
For an individual with volume \(V_i\), Michaelis–Menten elimination alone gives an elimination rate \(V_{\max,i}C/(K_{m,i}+C)\). The concentration equation is therefore
Dividing both sides by \(C\) shows how quickly the log concentration falls. This is a useful justification for looking for concentration-dependent residuals.
As \(C\) increases, the magnitude of this log-slope decreases: the concentration falls more slowly in relative terms. Under linear elimination with clearance \(CL_i\), by contrast, \(d\log C/dt=-CL_i/V_i\), a constant. Thus, if a linear model is calibrated to represent lower concentrations, it may predict too steep a decline while concentrations remain high. Observations then tend to exceed predictions in that region. This is a mechanistic expectation, not a rule that each high-concentration residual must be positive.
In a nonlinear mixed-effects analysis, individual predictions condition on the estimated subject-specific random effects, whereas population predictions use typical parameter values and do not condition on those effects. For diagnosing the structural concentration-time shape, conditional individual residuals are often useful, but they should be interpreted with care: estimated individual effects are informed by the same observations, and sparse data or strong shrinkage can make such residuals appear less variable than they would be with known individual parameters.
Residuals should also be judged on the scale implied by the observation model. If the model has proportional error, a residual of 1 concentration unit means something different at a prediction of 2 than at a prediction of 20. Under a combined additive and proportional error model, one possible standardized residual is
where \(\sigma_{\text{add}}\) is the additive residual standard deviation and \(\sigma_{\text{prop}}\) is the proportional residual coefficient. The denominator is the model-implied residual standard deviation at that prediction. If the error model is different, use its corresponding standard deviation rather than applying this formula automatically.
Worked Example: Standardizing Residuals Across Concentrations
Suppose a fitted linear-elimination model uses a proportional residual error with \(\sigma_{\text{prop}}=0.10\). At one observation, the prediction is \(8.0\) mg/L and the observed concentration is \(10.0\) mg/L. At another, the prediction is \(4.0\) mg/L and the observed concentration is \(4.2\) mg/L. Calculate and interpret the standardized residuals.
For proportional error, the residual standard deviation is \(0.10\hat C\). At the first observation it is \(0.10(8.0)=0.80\) mg/L, so
The model underpredicts the first concentration by \(2.0\) mg/L, or \(2.50\) model-implied residual standard deviations. For the second observation, the residual standard deviation is \(0.10(4.0)=0.40\) mg/L, giving
Both raw residuals are positive, but their standardized magnitudes differ substantially. A plot of raw residuals alone could make the first discrepancy seem large simply because its concentration scale is higher. The standardized residuals account for the specified proportional error, although they do not by themselves establish that saturation is the cause.
What Patterns to Examine
No single residual plot diagnoses saturation on its own. Examine related displays together and ask whether the signs and magnitudes change systematically with concentration, dose, or position in the concentration-time profile.
- Residual versus predicted concentration: A trend, curve, or widening/narrowing pattern indicates that the model may be missing concentration-dependent structure or that the error variance is misspecified. Under a linear model, persistent positive residuals among high predictions can be consistent with underprediction caused by saturation.
- Residual versus time, separately by dose: At a dose that produces high concentrations, positive residuals early in the decline followed by residuals nearer zero as concentrations fall can match the expected signature of a linear model that declines too rapidly at high concentrations. Look for replication across subjects, not a pattern created by one profile.
- Residual versus dose: If higher dose groups show increasingly positive residuals under a linear model, that supports a dose-related lack of fit. Dose is not itself the mechanism: differences in sampling, formulation, adherence, or subject composition can also generate dose-associated patterns.
- Residual spread and distribution: Residuals centered near zero with no systematic trend and a spread compatible with the error model are more consistent with random scatter. A funnel shape may indicate a variance problem rather than saturation in the mean concentration-time curve.
A useful diagnostic question is whether a pattern is ordered and repeatable. A run of residuals with the same sign in a concentration band, a smooth trend across predicted concentrations, or similar high-dose patterns across subjects is more concerning than a few isolated large residuals without a consistent direction. Standardized residuals near zero are expected on average under a correctly specified model; they are not required to alternate signs or fall within a fixed range in every small data set.
Worked Example: A Dose- and Concentration-Related Pattern
Consider hypothetical standardized residuals from a linear-elimination model. The residuals are conditional on individual predictions and use the fitted residual-error scale.
| Dose group | Concentration range | Standardized residuals |
|---|---|---|
| Low dose | Lower | -0.4, 0.2, 0.6 |
| High dose | Highest | 1.7, 2.1, 1.9 |
| High dose | Lower, later samples | 0.3, -0.2, 0.5 |
The low-dose residuals are mixed in sign and modest in magnitude. At the high dose, all three residuals in the highest concentration range are positive and near or above 2 standardized units. Later, as concentrations enter the lower range, the residuals return closer to zero and are mixed in sign. This ordered pattern is consistent with the linear model underpredicting at high concentrations and is mechanistically compatible with saturable elimination.
The data do not prove saturation. There are only three residuals in each group, and a dose-related design or a poorly estimated error model could contribute. The pattern should prompt examination of the fitted saturable model and of other diagnostics, rather than a conclusion based on the table alone.
Compare Residuals Before and After Fitting Saturable Elimination
A diagnostic claim is strongest when the proposed model change removes the specific pattern it was meant to address, without creating a new one. If the linear model shows high-concentration underprediction and the saturable model reduces that trend while leaving residuals roughly centered across concentration and dose, the residual evidence is coherent with saturation. If the same concentration-related trend remains, the saturable model has not resolved that feature of the fit. Residual improvement is not sufficient evidence by itself to select a model; it complements the formal model comparison and assessment of parameter plausibility.
Worked Example: Checking Whether the Pattern Remains
Suppose a hypothetical high-dose profile has observed concentrations of \(15.0\), \(10.0\), \(5.5\), and \(3.1\) mg/L. A linear model predicts \(12.0\), \(8.0\), \(5.0\), and \(3.0\) mg/L at the same times. A saturable model predicts \(14.5\), \(10.2\), \(5.4\), and \(3.2\) mg/L. Assume, for this illustration, an additive residual standard deviation of \(1.0\) mg/L for both models. Compute the residuals and standardized residuals.
For the linear model, observed minus predicted concentrations are \(3.0\), \(2.0\), \(0.5\), and \(0.1\) mg/L. Since the residual standard deviation is \(1.0\) mg/L at every point, the standardized residuals are \(3.0\), \(2.0\), \(0.5\), and \(0.1\). The first two, highest-concentration observations are underpredicted, while the later, lower-concentration observations are close to prediction.
For the saturable model, the residuals are \(15.0-14.5=0.5\), \(10.0-10.2=-0.2\), \(5.5-5.4=0.1\), and \(3.1-3.2=-0.1\) mg/L. Dividing each by \(1.0\) mg/L gives standardized residuals \(0.5\), \(-0.2\), \(0.1\), and \(-0.1\). In this example, the systematic early positive pattern largely disappears.
This is a useful diagnostic contrast, not a model-selection proof. The calculation assumes the stated additive error scale is appropriate, and one profile cannot establish population-level behavior. Other subjects and dose levels must be examined, alongside the formal comparison and parameter estimates.
Assumptions, Alternatives, and Failure Modes
Residual interpretation presumes that the prediction corresponds to the observation being diagnosed, the residual-error model is reasonably specified, and the recorded dose and sampling time are correct. For conditional individual residuals, it also matters how individual random effects were estimated. If the observation model is proportional but residuals are judged on an additive scale, apparent concentration-related spread may be an error-scale artifact. Conversely, a poor structural model may be mistaken for an error-model problem if only residual spread is examined.
Residual diagnostics complement the main alternatives rather than replacing them. A linear-elimination model is a simpler structural description; the saturable model represents a concentration-dependent elimination rate. Formal testing or information-based comparison evaluates overall support under the fitted models, while residual plots localize where a model fails. Population-level or individual prediction plots provide a broader view of fit, but can make a systematic signed residual trend less obvious. No alternative diagnostic removes the need to check that the design contains informative observations across relevant concentration ranges.
Several failure modes deserve particular attention:
- Using population predictions without considering subject effects. Differences between a subject and the typical parameter values can create apparent residual structure. Separate between-subject variation from within-subject lack of fit where the diagnostic permits.
- Calling any positive high-dose residual “saturation.” A few positive values can occur by chance. Look for a coherent trend across subjects, doses, or time and verify the error scale.
- Ignoring sample size and leverage. A small number of high-concentration observations may dominate visual interpretation or parameter fitting. A smooth-looking pattern in a sparse range is weak evidence.
- Expecting residuals to be perfectly random. A correctly specified model does not force a particular sign sequence in a finite sample. Judge the overall pattern and its magnitude relative to the residual variability.
- Treating a time pattern as uniquely diagnostic. Residuals that vary over time can have more than one explanation. Residual plots alone cannot establish why a time-related trend occurs; investigate the structural and design context before attributing it to saturation.
The practical conclusion is deliberately measured: a repeatable concentration- or dose-related residual pattern can make saturation a plausible explanation and identify where a linear model is inadequate. Its absence does not prove that elimination is linear, especially when the data contain few observations in the concentration range where saturation would be visible.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A linear model has residuals mostly near zero at low concentrations and repeatedly positive at the highest concentrations across several dose groups. What does this pattern suggest, and what does it not prove?
- With proportional residual error, why can raw residuals be misleading when comparing low and high predictions?
- A high-dose profile has positive residuals at early, high concentrations and mixed residuals later at lower concentrations. Explain why this is compatible with saturation under a linear model.
- After fitting a saturable model, the high-concentration trend remains. What should be concluded about the diagnostic evidence?
- Why might conditional individual residuals look less variable than expected?
Solutions. (1) The repeated positive high-concentration residuals suggest that the linear model underpredicts in the range where saturation may matter. They do not prove saturation, because chance, error-model misspecification, design features, or other structural discrepancies may contribute. (2) Under proportional error, residual variability increases with prediction, so the same raw residual can represent a large standardized discrepancy at low concentration and a small one at high concentration; compare residuals on the model-implied standardized scale. (3) Saturation reduces the relative elimination rate at high concentrations, so a linear model may predict too rapid a decline there; as concentration falls, the discrepancy can diminish. This pattern is compatible with saturation but is not unique to it. (4) The saturable model has not removed that aspect of the lack of fit. Reassess the model, error assumptions, data, and competing explanations rather than claiming the residual evidence supports an adequate saturable fit. (5) Individual random effects are estimated from the same observations used to calculate residuals. Conditioning on those fitted effects can absorb part of the observed variation, particularly when individual parameters are weakly informed or shrinkage is substantial.