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Bayesian Calibration and Uncertainty Quantification · Tutorial 638 of 799

Observation Error Models for QSP Calibration

Learn how to formulate and diagnose residual-error likelihoods for QSP observations, and how the chosen error model can alter posterior inference.

Advanced 12 min read

What You'll Learn

  • Write additive, proportional, and combined residual-error likelihoods for model-predicted observations.
  • Distinguish proportional Gaussian error from constant-variance additive error.
  • Include parameter-dependent standard deviations in the likelihood calculation.
  • Compare posterior parameter estimates under alternative residual-error assumptions.
  • Diagnose error-model mismatch using residuals and the scale of the observations.
  • Recognize when these simple likelihoods are insufficient for QSP data.

Why the Observation-Error Model Matters

A QSP model predicts quantities such as drug concentrations or biomarker levels, but measured values rarely coincide exactly with those predictions. The observation-error model describes the difference between a model-predicted mean and a measured value. It is therefore part of the likelihood used for calibration, not a cosmetic choice made after parameter estimation.

The form of the error model determines how discrepancies are weighted. An additive model assigns the same absolute error scale across predictions; a proportional model assigns larger absolute errors to larger predictions; and a combined model allows both. When the error scale depends on the predicted value, the likelihood includes that scale as well as the residual. Ignoring this dependence can change posterior parameter estimates.

Definition: Let \(y_i\) be the observed value at observation condition \(i\), and let \(f_i(\theta)\) be the corresponding model prediction for parameter vector \(\theta\). The residual is \(r_i=y_i-f_i(\theta)\). An observation-error model specifies the distribution of \(y_i\), conditional on \(\theta\), including its mean, variance, and any dependence among observations.

Three Gaussian Residual-Error Models

For the models below, assume observations are conditionally independent given the model parameters and any subject-level effects. Assume also that the QSP model correctly specifies the conditional mean \(f_i(\theta)\), and that the residual distributions are Gaussian with the stated variances. These assumptions may be imperfect; residual diagnostics and scientific knowledge are needed to assess them.

In an additive model, the residual has a constant standard deviation \(\sigma_a\). This describes constant absolute variability in the units of the measurement. In a proportional model, the standard deviation is a fixed fraction of the predicted mean. In a combined model, both an absolute component and a prediction-dependent component contribute to the variance.

Definition: The additive, proportional, and combined Gaussian models are, respectively,
$$ \begin{aligned} y_i\mid\theta &\sim N\!\left(f_i(\theta),\sigma_a^2\right),\\ y_i\mid\theta &\sim N\!\left(f_i(\theta),[\sigma_p f_i(\theta)]^2\right),\\ y_i\mid\theta &\sim N\!\left(f_i(\theta),\sigma_a^2+[\sigma_p f_i(\theta)]^2\right). \end{aligned} $$
Here \(\sigma_a\) has the measurement units of \(y_i\), and \(\sigma_p\) is a dimensionless coefficient of variation. The proportional form assumes positive predictions when used as written.

A proportional Gaussian model specifies a standard deviation proportional to the mean; it is not the same as a lognormal observation model. A lognormal model instead specifies a distribution for the logarithm of a positive measurement and has a different relationship between its parameters and the original-scale mean and variance. The choice should follow the measurement process and the pattern of variability, not merely the fact that the data are positive.

Likelihood and the Scale Term

For any of the Gaussian models, write \(v_i(\theta)\) for the conditional variance. Conditional independence gives the likelihood as a product of normal densities. Its log, apart from a constant common to all parameter values with the same number of observations, is

$$ \ell(\theta) =-\frac{1}{2}\sum_{i=1}^{n} \left[ \frac{(y_i-f_i(\theta))^2}{v_i(\theta)} +\log v_i(\theta) \right]. $$

The first term penalizes residuals relative to their variances. The second term accounts for the width of each normal density. It cannot be dropped when \(v_i(\theta)\) varies with \(\theta\), as in proportional and combined models. Without it, a candidate parameter value could appear preferable simply because its predicted values imply larger error scales.

With prior density \(p(\theta)\), Bayes' rule gives a posterior density proportional to the likelihood times the prior:

$$ p(\theta\mid\mathbf y)\propto \exp\{\ell(\theta)\}\,p(\theta). $$

This expression makes the consequence of the error assumption explicit: changing \(v_i(\theta)\) changes both the standardized residuals and, when the variance varies, the scale term. The resulting posterior can shift, widen, or change shape even when the mechanistic model and prior remain unchanged.

Worked Example: Concentration Data Under Three Error Models

Worked Example: Concentration Data Under Three Error Models

Hypothetical data. Suppose two plasma concentrations are observed, \(2.2\) and \(10.5\) nM. Consider two possible parameter values, \(\theta_A\) and \(\theta_B\), with predictions \((2.0,10.0)\) nM and \((2.8,10.8)\) nM, respectively. Assign each parameter value prior probability \(0.5\). Compare the posterior probability of \(\theta_A\) under three specified error models: additive \(\sigma_a=0.5\) nM; proportional \(\sigma_p=0.10\); and combined \(\sigma_a=0.4\) nM, \(\sigma_p=0.08\).

Additive model. The residuals are \((0.2,0.5)\) nM under \(\theta_A\) and \((-0.6,-0.3)\) nM under \(\theta_B\). Since the variance is \(0.5^2=0.25\) nM\(^2\) for each observation, the log-likelihood kernels are

$$ \ell_A=-\frac{0.2^2+0.5^2}{2(0.25)}=-0.5800,\qquad \ell_B=-\frac{(-0.6)^2+(-0.3)^2}{2(0.25)}=-0.9000. $$

With equal prior probabilities, the posterior probability of \(\theta_A\) is

$$ P(\theta_A\mid\mathbf y) =\frac{1}{1+\exp(\ell_B-\ell_A)} =\frac{1}{1+\exp(-0.3200)} \approx 0.5793. $$

Proportional model. The standard deviations under \(\theta_A\) are \(0.2\) and \(1.0\) nM; under \(\theta_B\), they are \(0.28\) and \(1.08\) nM. Including the log-standard-deviation terms, the kernels are

$$ \begin{aligned} \ell_A &=-\frac{1}{2}\left(\frac{0.2^2}{0.2^2}+\frac{0.5^2}{1.0^2}\right) -\log(0.2\times1.0) \approx 0.9844,\\ \ell_B &=-\frac{1}{2}\left(\frac{(-0.6)^2}{0.28^2}+\frac{(-0.3)^2}{1.08^2}\right) -\log(0.28\times1.08) \approx -1.1385. \end{aligned} $$

Thus \(P(\theta_A\mid\mathbf y)=1/[1+\exp(-1.1385-0.9844)]\approx0.8931\). The proportional model favors \(\theta_A\) more strongly because it assigns relatively little absolute variability to the low predicted concentration.

Combined model. Under \(\theta_A\), the two variances are \(0.4^2+(0.08\times2.0)^2=0.1856\) and \(0.4^2+(0.08\times10.0)^2=0.8000\) nM\(^2\). Under \(\theta_B\), they are \(0.210176\) and \(0.906496\) nM\(^2\). Substitution into the log-likelihood gives \(\ell_A\approx0.6897\) and \(\ell_B\approx-0.0771\). Therefore,

$$ P(\theta_A\mid\mathbf y) =\frac{1}{1+\exp(-0.0771-0.6897)} \approx0.6828. $$

In this two-point example, the posterior mean of the numerical label \(\theta\), if \(\theta_A=1\) and \(\theta_B=2\), is \(2-P(\theta_A\mid\mathbf y)\). It is approximately \(1.421\) under the additive model, \(1.107\) under the proportional model, and \(1.317\) under the combined model. The numbers are illustrative, not evidence about a real drug or parameter. They show that posterior estimates can depend materially on the assumed residual-error structure.

Worked Example: A Biomarker Near Its Lower Range

Worked Example: A Biomarker Near Its Lower Range

Hypothetical data. A biomarker assay reports measurements in ng/mL. At two conditions, the observations are \(0.14\) and \(0.62\) ng/mL, while a candidate QSP parameter setting predicts \(0.10\) and \(0.50\) ng/mL. The residuals are \(0.04\) and \(0.12\) ng/mL.

Suppose an additive error model uses \(\sigma_a=0.05\) ng/mL. The standardized residuals are \(0.04/0.05=0.800\) and \(0.12/0.05=2.40\). The high-condition observation is a relatively large discrepancy under this model.

Now suppose the proportional coefficient is \(\sigma_p=0.20\). The standard deviations are \(0.020\) and \(0.100\) ng/mL, so the standardized residuals are \(2.00\) and \(1.20\). The low-condition observation now has the larger standardized discrepancy. A combined model with \(\sigma_a=0.03\) ng/mL and \(\sigma_p=0.15\) gives standard deviations \(\sqrt{0.03^2+(0.15\times0.10)^2}\approx0.03354\) and \(\sqrt{0.03^2+(0.15\times0.50)^2}\approx0.08078\) ng/mL, and standardized residuals approximately \(1.193\) and \(1.486\).

The same raw residuals therefore tell different stories under different assumptions. Near zero, a pure proportional model can imply extremely small absolute errors; a nonzero additive component may be more credible for an assay with a roughly constant absolute measurement component. This example diagnoses relative weighting; it does not establish which error model is scientifically correct.

Worked Example: Why the Likelihood Needs Its Scale Term

Worked Example: Why the Likelihood Needs Its Scale Term

Hypothetical data. One concentration measurement is \(1.1\) mg/L. Consider predictions \(1.0\) mg/L and \(2.0\) mg/L with equal prior probabilities under a proportional Gaussian model with \(\sigma_p=0.20\). The corresponding standard deviations are \(0.20\) and \(0.40\) mg/L.

The log-likelihood kernels, omitting the common normalizing constant, are

$$ \ell_1=-\frac{1}{2}\left(\frac{0.1}{0.20}\right)^2-\log(0.20) \approx1.4844, \qquad \ell_2=-\frac{1}{2}\left(\frac{-0.9}{0.40}\right)^2-\log(0.40) \approx-1.6150. $$

The posterior probability of the first prediction is \(1/[1+\exp(\ell_2-\ell_1)]\approx0.9568\). If the log-scale terms were incorrectly omitted, the kernels would be \(-0.1250\) and \(-2.5313\), giving a posterior probability of only about \(0.9173\). That difference arises because the omitted term changes with the prediction-dependent standard deviation.

Diagnostics, Alternatives, and Limitations

Residual plots should be interpreted against the assumed error scale. Under an additive model, plot residuals against predictions and check whether their absolute spread is roughly stable. Under a proportional model, check whether relative residuals, such as \(r_i/f_i(\theta)\), have roughly stable spread where predictions are positive. For a combined model, examine residuals standardized by the fitted standard deviations. In each case, patterns, outliers, or systematic shifts may indicate a misspecified mean model, error structure, or dependence assumption; a plot alone does not identify which explanation is correct.

For repeated observations within a subject, conditional independence may be implausible. A shared subject effect or correlated residual process can induce dependence even after conditioning on population parameters. Hierarchical priors for between-subject variation address one source of subject heterogeneity, but do not automatically specify within-subject residual correlation. Likewise, the additive, proportional, and combined forms assume Gaussian residuals. Skewness, heavy tails, assay saturation, or measurements reported only as bounds may require different observation models.

The main alternatives are not interchangeable defaults. Additive error is interpretable when absolute measurement uncertainty is nearly constant. Proportional error is suitable when relative uncertainty is nearly constant over the relevant range. Combined error can represent a nonzero absolute floor plus increasing variability, but its two components may be difficult to distinguish without observations spanning a useful range or replicated measurements. A flexible model is not necessarily a better model: weak data information can leave its error parameters poorly determined or trade them off against mechanistic parameters.

1
Inspect the measurement scale.
Use units and assay knowledge to ask whether absolute or relative deviations are plausibly stable.
2
Specify the conditional distribution.
State the mean, variance, and independence or dependence assumptions before constructing the likelihood.
3
Compare plausible assumptions.
Check posterior sensitivity to reasonable additive, proportional, or combined specifications rather than selecting a form solely because it produces a preferred estimate.
4
Check the fitted residual pattern.
Look for remaining structure and assess whether the assumed error scale matches the observed spread across predictions and subjects.

A common mistake is to choose the error model with the smallest sum of squared raw residuals. That criterion does not account for different variances across observations, and it omits the likelihood scale term when the variance depends on the parameters. The correction is to evaluate the full likelihood under each explicitly specified model, combine it with the prior, and assess whether the assumptions and resulting residual behavior are credible. Posterior differences are conditional on those assumptions; they do not show that the error model caused a biological effect.

Key takeaway: An observation-error model determines how each discrepancy contributes to calibration. State its variance in measurement units, retain all parameter-dependent likelihood terms, and examine whether the implied residual scale is plausible for both concentration and biomarker data.

Check Your Understanding

Answer each question, then compare with the brief solutions.

  1. A concentration prediction is \(5\) mg/L and the proportional coefficient is \(0.12\). What standard deviation does the proportional model assign?
  2. For a combined model with \(\sigma_a=0.2\) units, \(\sigma_p=0.1\), and prediction \(3\) units, calculate the residual standard deviation.
  3. Why must the log-variance term remain in a proportional-model likelihood when comparing parameter values?
  4. Residual spread increases with the predicted biomarker level, but appears to approach a nonzero absolute spread near zero. Which of the three models is a plausible candidate, and what should be checked next?

Solutions. (1) The standard deviation is \(0.12\times5=0.600\) mg/L. (2) The variance is \(0.2^2+(0.1\times3)^2=0.13\) units\(^2\), so the standard deviation is \(\sqrt{0.13}\approx0.3606\) units. (3) The standard deviation depends on the predicted value, so its normal-density scale changes with the parameter; omitting the log-variance term changes relative likelihoods. (4) A combined model is a plausible candidate because it includes both an absolute component and a prediction-dependent component. Check residuals standardized by the combined scale, and use assay knowledge and available data to assess whether both components are identifiable and credible.

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