Concentration-Dependent Residuals From the Wrong Error Model

Why Residual Spread Can Point to the Error Model

A population pharmacokinetic model can describe the average concentration-time profile reasonably well and still predict observations poorly at some concentration levels. One reason is that the model for residual error does not match how measurement and within-subject variability behave across the concentration range. For example, an additive error model implies roughly constant residual spread in concentration units. If the observed spread instead grows with predicted concentration, residuals may be wider than the model allows at high concentrations and too wide relative to the observations at low concentrations.

The diagnostic question is not simply whether residuals are large. It is whether their spread changes systematically with predicted concentration in a way the specified error model does not allow. First inspect residuals against predictions, then compare candidate error structures, and finally reassess residual behavior under the revised model. A spread pattern is evidence to investigate, not proof that the error model is the only source of misfit.

Three Common Residual-Error Structures

Let \(Y_i\) be an observed concentration and \(F_i\) the model-predicted concentration for observation \(i\). Define the raw residual as \(e_i=Y_i-F_i\). The residual-error model describes the distribution of \(Y_i\) around \(F_i\), conditional on the model and relevant individual information. In the formulations below, error terms have mean zero, are independent across observations conditional on the model, and are independent of each other when a model has more than one component.

Definition: An additive error model specifies \(Y_i=F_i+\epsilon_{add,i}\), with \(\epsilon_{add,i}\sim N(0,\sigma_{add}^2)\). A proportional error model specifies \(Y_i=F_i(1+\epsilon_{prop,i})\), with \(\epsilon_{prop,i}\sim N(0,\sigma_{prop}^2)\). A combined additive and proportional error model specifies \(Y_i=F_i+\epsilon_{add,i}+F_i\epsilon_{prop,i}\), with independent error components having the variances above. These models imply, respectively, constant residual standard deviation, residual standard deviation proportional to \(F_i\), and a standard deviation combining both patterns.

The error terms are often taken as normally distributed for these standard formulations. This is an assumption, not a guarantee about real concentration data. The proportional component is expressed as a fraction of the prediction, so \(\sigma_{prop}\) is dimensionless; \(\sigma_{add}\) has the same units as concentration. A combined model is not automatically preferable: it has more parameters, and the data must support estimating both components.

Why the Spread Patterns Differ

The variance implications follow directly from the model equations. In the additive model, \(e_i=\epsilon_{add,i}\), so the variance does not depend on \(F_i\). In the proportional model, \(e_i=F_i\epsilon_{prop,i}\), so multiplying the error by the prediction multiplies its variance by \(F_i^2\). In the combined model, the two independent, zero-mean components contribute their variances.

\begin{aligned} \operatorname{Var}(e_i\mid F_i)_{\text{additive}}&=\sigma_{add}^2,\\ \operatorname{Var}(e_i\mid F_i)_{\text{proportional}}&=F_i^2\sigma_{prop}^2,\\ \operatorname{Var}(e_i\mid F_i)_{\text{combined}}&=\sigma_{add}^2+F_i^2\sigma_{prop}^2. \end{aligned}

For the combined model, the variance addition is justified by independence: \(\operatorname{Var}(A+B)=\operatorname{Var}(A)+\operatorname{Var}(B)+2\operatorname{Cov}(A,B)\), and independence makes the covariance zero. Taking square roots gives the corresponding standard deviations. A useful standardized residual for this diagnostic is \(e_i\) divided by the model-implied residual standard deviation at \(F_i\). If the variance model is adequate, these standardized residuals should not show a systematic increase or decrease in spread over predictions, although finite samples will fluctuate.

SD(e_i\mid F_i)_{\text{combined}} =\sqrt{\sigma_{add}^2+F_i^2\sigma_{prop}^2}, \qquad r_i^{*}=\frac{e_i}{\sqrt{\sigma_{add}^2+F_i^2\sigma_{prop}^2}}.

This is a diagnostic principle, not a claim that every software package reports this exact residual. Conditional, population, and other standardized residuals can differ in how they account for parameter estimation and random effects. Interpret a plot using the definition of the residual being plotted, and compare it with the variance pattern the fitted model implies.

Worked Example: Spread That Grows With Predicted Concentration

Hypothetical data. Suppose observations near predicted concentrations of 2 mg/L have residuals \(-0.2\) and \(0.2\) mg/L, while observations near 20 mg/L have residuals \(-2.0\) and \(2.0\) mg/L. The residuals are centered at zero in both groups, but their magnitude is much greater in the higher-prediction group.

At 2 mg/L, the root mean square residual is \(\sqrt{(0.2^2+0.2^2)/2}=0.200\) mg/L. At 20 mg/L, it is \(\sqrt{(2.0^2+2.0^2)/2}=2.00\) mg/L. The absolute spread has increased tenfold as the prediction increased tenfold. In contrast, the residual-to-prediction magnitudes are \(0.2/2=0.100\) and \(2.0/20=0.100\), or 10% in each group.

This pattern is compatible with proportional variability: the errors are similar in relative size, but not in concentration units. A constant-variance additive model would not represent the tenfold change in spread. It would be reasonable to fit a proportional model and then check residual spread under that model. These few illustrative values do not establish that proportional error is correct; the full concentration range, model fit, and measurement context also matter.

Recognizing Additive and Combined Patterns

If absolute residual spread is roughly stable across predictions, additive error may be plausible. If relative residuals \(e_i/F_i\) have roughly stable spread, proportional error may be plausible. These are visual and descriptive checks, not mechanical rules: predictions close to zero make relative residuals unstable, and a small number of observations in a concentration range gives an imprecise picture of spread.

A combined model is useful when residual spread has both a low-concentration floor and a concentration-dependent component. At low \(F_i\), the additive component can dominate; at sufficiently high \(F_i\), the proportional component can dominate. The transition depends on the fitted components rather than on a universal concentration threshold. If the fitted proportional component is nearly zero, the combined model may not be justified over a simpler additive model; the same reasoning applies to a negligible additive component.

Worked Example: A Proportional Model Misses a Low-Concentration Floor

Hypothetical data. In a low-prediction group with \(F=1\) mg/L, residuals are \(-0.4\) and \(0.4\) mg/L. In a high-prediction group with \(F=10\) mg/L, residuals are \(-0.5\) and \(0.5\) mg/L. The root mean square residuals are 0.400 and 0.500 mg/L, respectively, while the relative residual magnitudes are \(0.4/1=0.400\) and \(0.5/10=0.0500\).

The absolute spreads are similar, but the relative spread is much larger at the lower prediction. A proportional-only model expects absolute spread to grow in proportion to \(F\); this observed pattern does not support that expectation. A combined model could represent the data with illustrative components \(\sigma_{add}=0.4\) mg/L and \(\sigma_{prop}=0.03\). It implies an SD of \(\sqrt{0.4^2+(1\times0.03)^2}=\sqrt{0.1609}=0.401\) mg/L at \(F=1\) mg/L, and \(\sqrt{0.4^2+(10\times0.03)^2}=\sqrt{0.25}=0.500\) mg/L at \(F=10\) mg/L.

Those values match the two illustrative group spreads, but they are not estimates obtained from these four residuals. Actual parameter estimation uses the full data and an appropriate likelihood or estimation procedure. The example shows why a combined structure can be more plausible than proportional error alone when the absolute spread does not approach zero at low predictions.

Assessing a Revised Error Model

After revising the error model, do not conclude that the problem is solved merely because the fitting procedure converged or an objective function improved. Re-examine residuals across the prediction range. Look for both residual location and residual spread: a mean trend can indicate that predictions are systematically too high or too low, whereas changing spread points to variance misspecification. Also check whether a handful of observations, a narrow prediction range, or data near the assay’s quantification limit dominates the apparent pattern.

Worked Example: Checking Standardized Spread Under a Combined Model

Hypothetical data. Suppose a revised model has \(\sigma_{add}=0.4\) mg/L and \(\sigma_{prop}=0.03\). For observations with \(F=1\) mg/L and residuals of \(-0.4\) and \(0.4\) mg/L, the model-implied SD is \(\sqrt{0.4^2+(1\times0.03)^2}=0.401\) mg/L. The standardized residuals are therefore \(-0.4/0.401=-0.998\) and \(0.4/0.401=0.998\).

At \(F=10\) mg/L, the implied SD is \(\sqrt{0.4^2+(10\times0.03)^2}=0.500\) mg/L. Residuals of \(-0.5\) and \(0.5\) mg/L give standardized residuals of \(-1.00\) and \(1.00\). The raw spread rises from about 0.4 to 0.5 mg/L, as the combined model allows, while the standardized magnitudes are similar across the two groups.

This is the desired qualitative check: the revised variance model accounts for the change in raw spread without leaving an obvious concentration-related change in standardized spread. Two observations per group cannot validate a model, and no diagnostic plot proves the error distribution is correct. In practice, review the complete residual display, the adequacy of the model-predicted mean profile, and whether the estimated error components are sufficiently supported by the data.

Comparison, Diagnostics, and Limitations

Error model	Implied residual SD	Pattern it can represent	Potential warning sign
Additive	\(\sigma_{add}\)	Approximately constant spread in concentration units	Residual spread grows or shrinks systematically with \(F\)
Proportional	\(F\sigma_{prop}\)	Spread increasing in proportion to predicted concentration	Substantial low-concentration spread remains when \(F\) is small
Combined	\(\sqrt{\sigma_{add}^2+F^2\sigma_{prop}^2}\)	A low-level additive component plus concentration-dependent spread	One component is poorly identified, or standardized spread still trends

The models are alternatives for the residual variance, not substitutes for checking the structural prediction. A misspecified absorption or disposition pattern can also produce residual patterns over time or concentration. Use the earlier material on reading residual patterns to localize model failure to distinguish a variance trend from systematic error in the predicted profile. Likewise, an apparent trend can be influenced by assay behavior, data handling, or an unusual subset of observations. Do not attribute every funnel-shaped plot to a residual-error model without checking those possibilities.

Candidate error models can be compared using likelihood-based criteria when they are fitted to the same observations with compatible likelihoods. A better numerical fit alone is not enough: a more flexible model can improve fit while estimating an unnecessary or weakly identified component. Consider parameter uncertainty, residual plots, and whether the estimated pattern is plausible for the measurement process. The normal formulations also have limitations, including potential incompatibility with strongly skewed or censored observations; changing the error model does not automatically resolve those issues.

Common pitfall: “The residuals are larger at high concentrations, so the mean PK model must be wrong.” Larger raw residuals alone do not establish a mean-structure error. If residuals remain centered near zero and their spread grows in proportion to prediction, a proportional variance model may be more appropriate. Conversely, a residual trend in the mean, an unexplained time pattern, or persistent concentration-dependent spread after standardization calls for further investigation rather than simply adding another error component.

The central comparison is between observed spread and the spread permitted by the assumed error model. Additive, proportional, and combined structures make different predictions about that relationship. A revised model is more credible when its implied variance matches residual behavior across the usable concentration range and does so without obscuring other systematic lack of fit.

Check Your Understanding

Answer each question, then compare with the solutions below.

An additive model is fitted, and raw residual spread rises with predicted concentration while relative residual spread is roughly stable. Which alternative error structure is suggested, and why?
A proportional model predicts an SD of 0.06 mg/L at \(F=2\) mg/L when \(\sigma_{prop}=0.03\). If an additive component of 0.4 mg/L is added independently, what is the combined SD?
Under a combined model, \(\sigma_{add}=0.3\) mg/L, \(\sigma_{prop}=0.04\), and \(F=5\) mg/L. What is the model-implied residual SD?
After revising an error model, raw residual spread still rises with prediction, but standardized residual spread is roughly stable and residuals remain centered near zero. Does the raw trend alone show that the revision failed?

Solutions. (1) A proportional model is suggested because stable \(e_i/F_i\) spread indicates roughly constant relative error; an additive model instead implies constant spread in concentration units. (2) The combined SD is \(\sqrt{0.4^2+0.06^2}=\sqrt{0.1636}=0.404\) mg/L. (3) The combined SD is \(\sqrt{0.3^2+(5\times0.04)^2}=\sqrt{0.09+0.04}=\sqrt{0.13}=0.361\) mg/L. (4) No. A combined or proportional model can allow raw spread to rise with prediction. Stable standardized spread and residuals centered near zero are consistent with the revised variance pattern, although the full diagnostic evidence and model assumptions still require review.