Why Residual Error Choice Matters
A concentration-time model describes a systematic pattern: concentrations rise, peak, and decline according to the chosen structural model and its parameters. Observations do not fall exactly on that pattern. Residual error represents the remaining difference between an observed concentration and its model-predicted mean, conditional on the model’s parameters and any subject-specific effects. Choosing a residual error model specifies how the size of that difference is expected to vary with concentration.
This choice is consequential in bioequivalence analyses. A model’s residual assumptions determine how strongly different observations influence parameter estimation. Since the fitted parameters determine the predicted concentration curves, they can also affect exposure measures such as AUC and Cmax, and their formulation contrasts. A residual model should therefore describe the variability in the data, not be selected merely because it produces a preferred exposure ratio.
The conditional mean \(C_{ij}\) may include subject-specific effects, as appropriate to the fitted model. The residual error model is distinct from the random-effects and covariance structures: it describes observation-level variation remaining after conditioning on those modeled effects.
Three Common Error Models
The following models are stated on the original concentration scale. In each, the residual has conditional mean zero. The parameters \(\sigma_a\) and \(\sigma_p\) are nonnegative; \(\sigma_a\) has concentration units, while \(\sigma_p\) is dimensionless.
Proportional error: \(Y_{ij}=C_{ij}(1+\epsilon_{ij})\), where \(\epsilon_{ij}\sim N(0,\sigma_p^2)\). Equivalently, \(Y_{ij}\mid C_{ij}\sim N(C_{ij},(\sigma_p C_{ij})^2)\). The residual standard deviation is proportional to the predicted concentration.
Combined error: \(Y_{ij}\mid C_{ij}\sim N(C_{ij},\sigma_a^2+(\sigma_p C_{ij})^2)\). The additive and proportional components contribute to the conditional variance.
For the combined model, the variance formula specifies that the two components contribute additively to variance. It does not mean that the errors are necessarily two independently observed quantities. The additive component can matter at low concentrations, while the proportional component can dominate at higher concentrations. The proportional model, as written, has zero variance when \(C_{ij}=0\); that behavior may be unrealistic when concentrations near zero still have measurement variability.
These Gaussian models allow negative observations in principle, especially when the predicted mean is small relative to the residual standard deviation. They do not automatically enforce nonnegative concentrations. A model should be assessed against the measurement scale and data, rather than assumed suitable because its formula is familiar.
Worked Example: Comparing Error Scales at Two Concentrations
Suppose hypothetical fitted values are \(\sigma_a=0.8\) ng/mL and \(\sigma_p=0.15\). Calculate the residual standard deviation under each model when the predicted concentration is 2 ng/mL and 20 ng/mL.
Additive model. The standard deviation is \(0.8\) ng/mL at either prediction.
Proportional model. At 2 ng/mL, the standard deviation is \(0.15(2)=0.30\) ng/mL. At 20 ng/mL, it is \(0.15(20)=3.0\) ng/mL.
Combined model. At 2 ng/mL, the variance is \(0.8^2+(0.15(2))^2=0.64+0.09=0.73\;(\text{ng/mL})^2\), so the standard deviation is \(\sqrt{0.73}=0.8544\) ng/mL. At 20 ng/mL, the variance is \(0.8^2+(0.15(20))^2=0.64+9=9.64\;(\text{ng/mL})^2\), giving a standard deviation of \(\sqrt{9.64}=3.105\) ng/mL.
The additive model assigns the same absolute variability at both concentrations. The proportional model assigns a constant coefficient of variation of 15% when \(C_{ij}>0\). The combined model allows absolute variability to remain appreciable at low concentrations while relative variability contributes more at high concentrations.
How the Error Model Changes Estimation
Suppose, for illustration, that residuals are conditionally independent and normally distributed given the model predictions, with variance \(v_{ij}\). For \(m\) observations, the conditional log-likelihood, apart from no omitted constants, is
This expression shows why the error model affects fitted parameters. A residual contributes a squared-error term divided by its variance. An observation assigned a larger variance has less influence through that term than an equally sized residual assigned a smaller variance. But the likelihood also includes \(\log(v_{ij})\), so choosing a very large variance everywhere is not a way to improve the fit without penalty.
Worked Through the Likelihood Weighting
Worked Example: Standardizing Residuals Under Different Models
Consider two hypothetical observations with predicted concentrations of 1 and 10 ng/mL and residuals of \(+0.6\) and \(+2.0\) ng/mL, respectively. Compare their standardized residuals under an additive model with \(\sigma_a=0.5\) ng/mL, a proportional model with \(\sigma_p=0.20\), and a combined model with \(\sigma_a=0.4\) ng/mL and \(\sigma_p=0.15\). Assume the same conditional normal error framework for each candidate.
A standardized residual is \(r_{ij}=e_{ij}/\sqrt{v_{ij}}\). Under the additive model, both standard deviations are 0.5 ng/mL. Thus the standardized residuals are \(0.6/0.5=1.20\) and \(2.0/0.5=4.00\).
Under the proportional model, the standard deviations are \(0.20(1)=0.20\) ng/mL and \(0.20(10)=2.0\) ng/mL. The standardized residuals are \(0.6/0.20=3.00\) and \(2.0/2.0=1.00\).
Under the combined model, the first variance is \(0.4^2+(0.15(1))^2=0.1825\;(\text{ng/mL})^2\), giving a standard deviation of \(\sqrt{0.1825}=0.4272\) ng/mL. The second variance is \(0.4^2+(0.15(10))^2=2.41\;(\text{ng/mL})^2\), giving a standard deviation of \(\sqrt{2.41}=1.552\) ng/mL. The standardized residuals are therefore \(0.6/0.4272=1.404\) and \(2.0/1.552=1.289\).
The same raw residuals can look quite different after accounting for the assumed variance. In this example, an additive model treats the high-concentration residual as relatively extreme, while a proportional model treats the low-concentration residual as relatively extreme. This is why residual diagnostics should examine both raw residuals and residuals scaled by the model’s predicted standard deviation.
The likelihood expression also provides a short justification for the weighting interpretation. With fixed \(v_{ij}\), the part of \(-2\ell\) that depends on the predictions is \(\sum_{i,j}(Y_{ij}-C_{ij})^2/v_{ij}\). Minimizing this quantity gives less weight to observations with larger residual variance. When the variance depends on \(C_{ij}\), however, both the squared-residual term and the \(\log(v_{ij})\) term change as the model changes. Estimation is therefore not simply least squares with weights calculated once from observed concentrations.
Choosing and Checking a Candidate Model
A useful comparison keeps the structural model, random-effects structure, covariance assumptions, and analysis dataset the same while changing the residual error specification. Otherwise, a difference in fit cannot be attributed cleanly to the residual model. Under the conditional Gaussian assumptions above, candidates can be compared with their maximized log-likelihoods and information criteria. For example, \(\mathrm{AIC}=-2\ell+2k\), where \(k\) is the number of estimated parameters in the likelihood being compared. A smaller AIC is a relative preference among the candidates considered; it is not proof that the selected model is true or that a bioequivalence conclusion is correct.
Diagnostics should be consistent with the proposed variance pattern. Plot raw residuals against predicted concentration and time, and examine standardized residuals, using the model-implied standard deviation. A suitable model should not leave a clear increase or decrease in standardized residual spread across predicted concentrations. Also inspect whether residuals have time patterns, outliers, or dependence that the assumed residual model does not explain. Such patterns can signal problems beyond the choice among these three variance formulas.
A proportional model can be appealing if residual spread grows with concentration, but its vanishing variance at a zero prediction and strong relative weighting near zero may be problematic. An additive model may be reasonable when absolute error is roughly stable, but can overemphasize high concentrations if their variability is mainly relative. The combined model is more flexible, but adds a parameter and can be poorly supported when the data provide little information about one of its components. More flexibility is not automatically better.
| Model | Standard deviation at prediction \(C\) | Pattern it represents | Potential concern |
|---|---|---|---|
| Additive | \(\sigma_a\) | Constant absolute spread | May not represent variability that grows with concentration |
| Proportional | \(\sigma_p C\) | Constant relative spread for positive \(C\) | Variance approaches zero as \(C\) approaches zero |
| Combined | \(\sqrt{\sigma_a^2+(\sigma_p C)^2}\) | Both low-concentration absolute and concentration-dependent components | Extra flexibility and an additional parameter to estimate |
Effects on Exposure Contrasts
In a model-based bioequivalence analysis, each candidate residual model can produce different fitted parameters and therefore different predicted concentration-time curves. Applying the same exposure calculation to those curves can then produce different AUC or Cmax estimates and formulation ratios. The residual model does not directly define the exposure metric; its effect is through fitting the concentration model. A materially different exposure contrast across plausible residual models is a reason to investigate sensitivity, not a license to select the model with the most favorable result.
Worked Example: Comparing Illustrative Exposure Ratios
Suppose three residual models are fitted to the same hypothetical concentration dataset, using the same structural model and random-effects specification. Assume the reported AIC values below count all estimated parameters and are based on the same observations and likelihood scale.
| Residual model | Number of parameters \(k\) | \(-2\ell\) | AIC | Model-derived AUC GMR |
|---|---|---|---|---|
| Additive | 8 | 112 | 128 | 1.040 |
| Proportional | 8 | 101 | 117 | 1.030 |
| Combined | 9 | 98 | 116 | 1.020 |
For the additive model, \(\mathrm{AIC}=112+2(8)=128\). For the proportional model, \(\mathrm{AIC}=101+2(8)=117\). For the combined model, \(\mathrm{AIC}=98+2(9)=116\). Among these candidates, the combined model has the lowest AIC, by 1 unit relative to the proportional model; this is a small difference and does not establish that the combined model is uniquely correct.
The AUC geometric mean ratios (GMRs) are 1.040, 1.030, and 1.020. Their respective interpretations are that the model-derived test-to-reference AUC ratios are 1.040, 1.030, and 1.020. The additive-to-combined difference in ratio is \(1.040-1.020=0.020\), or 2 percentage points on the ratio scale. These illustrative point estimates alone do not establish bioequivalence: an equivalence assessment also requires the applicable inferential procedure and limits. The example shows why the error model should be reported and why sensitivity of exposure contrasts to that choice matters.
Common Mistakes and Limitations
- Choosing by the exposure ratio. A residual model should not be chosen because it moves a GMR toward a desired value. Compare models using defensible likelihood and diagnostic evidence, then report how the exposure contrast behaves across plausible choices.
- Inspecting only raw residuals. A residual of 1 ng/mL has a different scale of surprise when the model predicts 2 ng/mL than when it predicts 20 ng/mL under a proportional error model. Examine standardized residuals as well as raw residuals.
- Assuming a lower AIC proves correctness. AIC ranks candidate models relative to one another under the chosen likelihood and dataset. It does not verify the assumptions, exclude unconsidered models, or certify equivalence.
- Ignoring conditional independence. The formulas above assume the residuals are conditionally independent as specified. Remaining serial patterns or other dependence can invalidate that assumption; changing the variance function alone may not resolve the problem.
- Treating the combined model as universally safest. It may capture both absolute and relative variation, but its extra flexibility can be weakly supported or lead to unstable component estimates. Simpler models can be preferable when their assumptions are adequate.
These models are not the only possible descriptions of concentration variability. If a Gaussian model on the original scale gives implausible negative concentrations or persistent non-normal residual patterns, alternatives may be considered, but their likelihood and interpretation must match the data scale. The error model should be judged alongside the structural and random-effects assumptions, not used to compensate for a misspecified concentration-time pattern.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Under a proportional model with \(\sigma_p=0.12\), what is the residual standard deviation when the predicted concentration is 5 ng/mL?
- A combined model has \(\sigma_a=0.6\) ng/mL and \(\sigma_p=0.10\). Calculate its residual standard deviation at a predicted concentration of 4 ng/mL.
- Why can two observations with equal raw residuals have different standardized residuals under a proportional model?
- Two models have \(-2\ell=90\), \(k=7\), and \(-2\ell=86\), \(k=9\), respectively. Calculate both AIC values and state what the comparison does and does not establish.
- If plausible residual models give noticeably different model-derived AUC GMRs, what is an appropriate interpretation?
Solutions. (1) The standard deviation is \(\sigma_p C=0.12(5)=0.60\) ng/mL. (2) The variance is \(0.6^2+(0.10(4))^2=0.36+0.16=0.52\;(\text{ng/mL})^2\), so the standard deviation is \(\sqrt{0.52}=0.7211\) ng/mL. (3) Standardization divides each residual by a standard deviation that depends on its predicted concentration, so equal raw residuals need not be equally large relative to expected variability. (4) The AIC values are \(90+2(7)=104\) and \(86+2(9)=104\). They tie on AIC among these candidates; this does not prove either model correct or settle whether bioequivalence is established. (5) The exposure contrast is sensitive to residual-model choice. Report the variation across plausible models and investigate the diagnostics; do not select a model solely to favor a particular contrast.