From a PK Prediction to a Likelihood
A structural pharmacokinetic model predicts a concentration at a specified time, given a patient's dosing history, covariates, and PK parameters. The measured concentration will generally differ from that prediction. To use the measurement in individual inference, we need a probability model for this discrepancy. The resulting likelihood quantifies how compatible the observed concentrations are with a proposed set of individual PK parameters, under the specified residual-error assumptions.
Let \(y_j\) be the measured concentration at time \(t_j\), and let \(f_j(\theta)\) be the structural model's predicted concentration at that time for individual PK parameters \(\theta\). The prediction also depends on the recorded doses and relevant covariates, which are held fixed here. Let \(\psi\) denote residual-error parameters, such as an additive standard deviation or a proportional error coefficient. A single-observation likelihood contribution is the probability density of \(y_j\) conditional on \(\theta\), \(\psi\), and the design information.
The conditional-independence assumption means that, once the model predictions and parameters are specified, the residual at one sampling time gives no further information about the residual at another time. Under that assumption, multiplying the individual densities gives the joint density. Taking logarithms turns the product into a sum: \(\ell(\theta)=\sum_{j=1}^{m}\log p(y_j\mid\theta,\psi)\). This is convenient computationally and helps avoid numerical underflow.
Likelihoods on the Original Concentration Scale
An additive error model assigns a constant residual standard deviation across the prediction range. It is often suitable when measurement or model discrepancy is roughly similar in absolute concentration units, rather than as a percentage of the prediction.
Here \(\sigma_{\mathrm{add}}\) is in concentration units, such as mg/L. The density has reciprocal concentration units. A normal error model on the original scale can assign some probability to negative concentrations; this may be negligible when predictions are large compared with the residual standard deviation, but can be problematic near zero.
Worked Example: Additive Error With Two Concentrations
Suppose a hypothetical patient has measured concentrations \(y_1=9.8\) mg/L and \(y_2=7.2\) mg/L. At the corresponding times, a proposed set of individual parameters predicts \(f_1=10.0\) mg/L and \(f_2=8.0\) mg/L. Assume independent additive normal errors with \(\sigma_{\mathrm{add}}=0.5\) mg/L. This model has a constant standard deviation, and conditional independence is explicitly assumed.
The standardized residuals are \((9.8-10.0)/0.5=-0.4\) and \((7.2-8.0)/0.5=-1.6\). The density contributions are \(p(y_1\mid\theta)=\exp(-0.4^2/2)/(0.5\sqrt{2\pi})\approx0.7365\) per mg/L and \(p(y_2\mid\theta)=\exp(-1.6^2/2)/(0.5\sqrt{2\pi})\approx0.2218\) per mg/L.
Therefore \(L(\theta)=0.7365\times0.2218\approx0.1633\;(\mathrm{mg/L})^{-2}\). The log-likelihood is \(\ell(\theta)=\log(0.7365)+\log(0.2218)\approx-1.812\), with the log of the density carrying the corresponding reference-unit convention. These values describe the density at the observed pair under this parameter proposal; they are not the probability of observing exactly those two values.
With proportional error on the original scale, the residual standard deviation increases with the predicted concentration. A common model is \(y_j=f_j(\theta)(1+\epsilon_j)\), with independent \(\epsilon_j\sim N(0,\sigma_{\mathrm{prop}}^2)\). Equivalently, conditional on \(\theta\), \(y_j\) is normal with mean \(f_j(\theta)\) and standard deviation \(\sigma_{\mathrm{prop}}f_j(\theta)\), assuming positive predictions. The coefficient \(\sigma_{\mathrm{prop}}\) is dimensionless.
Worked Example: Proportional Error on the Original Scale
For a second hypothetical case, suppose the predictions are 10 mg/L and 20 mg/L, and the measured concentrations are 9 mg/L and 21 mg/L. Assume independent proportional normal errors with \(\sigma_{\mathrm{prop}}=0.10\). The conditional standard deviations are \(0.10(10)=1\) mg/L and \(0.10(20)=2\) mg/L, so the model specifies a 10% residual standard deviation at both predictions.
The standardized residuals are \((9-10)/1=-1\) and \((21-20)/2=0.5\). The two density contributions are \(\exp(-1^2/2)/\sqrt{2\pi}\approx0.2420\) per mg/L and \(\exp(-0.5^2/2)/(2\sqrt{2\pi})\approx0.1760\) per mg/L. Hence \(L(\theta)\approx0.2420\times0.1760=0.04259\;(\mathrm{mg/L})^{-2}\), and \(\ell(\theta)\approx-3.156\).
The proportional model has larger absolute error spread at the higher prediction, even though the relative spread is the same. Because a normal distribution can also produce negative values, this model can be physically implausible when its coefficient of variation is large or predictions are low.
A combined error model allows both a roughly constant absolute component and a component that grows with predicted concentration. One standard specification uses independent additive and proportional normal components: \(y_j=f_j(\theta)+\epsilon_{\mathrm{add},j}+f_j(\theta)\epsilon_{\mathrm{prop},j}\). If the components are independent, centered normal variables with standard deviations \(\sigma_{\mathrm{add}}\) and \(\sigma_{\mathrm{prop}}\), their sum is normal with variance \(\sigma_{\mathrm{add}}^2+\sigma_{\mathrm{prop}}^2f_j(\theta)^2\). Thus the likelihood is normal with mean \(f_j(\theta)\) and that variance. This derivation depends on independence of the two error components as well as their normality.
Worked Example: Combined Additive and Proportional Error
Consider hypothetical predictions of 2 and 8 mg/L, with observations of 2.4 and 7.0 mg/L. Let \(\sigma_{\mathrm{add}}=0.3\) mg/L and \(\sigma_{\mathrm{prop}}=0.10\), and assume the independent normal components described above. The conditional variances are \(0.3^2+(0.10\times2)^2=0.13\;(\mathrm{mg/L})^2\) and \(0.3^2+(0.10\times8)^2=0.73\;(\mathrm{mg/L})^2\). The corresponding standard deviations are approximately 0.3606 and 0.8544 mg/L.
The standardized residuals are \(0.4/\sqrt{0.13}\approx1.109\) and \(-1/\sqrt{0.73}\approx-1.170\). Using the normal density, the contributions are \(\exp(-1.109^2/2)/(\sqrt{0.13}\sqrt{2\pi})\approx0.5980\) per mg/L and \(\exp(-1.170^2/2)/(\sqrt{0.73}\sqrt{2\pi})\approx0.2354\) per mg/L. Their product is \(L(\theta)\approx0.1408\;(\mathrm{mg/L})^{-2}\), with \(\ell(\theta)\approx-1.961\).
The additive component matters more relative to the proportional component at the lower prediction; at the higher prediction, the proportional component contributes more to the variance. This model is appropriate only if that variance pattern is a reasonable representation of the residuals.
Log-Scale Errors and the Change of Variables
A different model assumes errors are normal after taking logarithms. It requires positive observed and predicted concentrations. Write \(\log y_j=\log f_j(\theta)+\eta_j\), with independent \(\eta_j\sim N(0,\sigma_{\log}^2)\). This is a lognormal model on the original scale. The error is additive on the log scale, but multiplicative on the concentration scale.
To obtain the density for the measured concentration \(y\), rather than for its logarithm, use the change-of-variables rule. If \(Z=\log Y\), then \(dZ/dY=1/Y\); consequently \(p_Y(y)=p_Z(\log y)/y\) for \(y>0\). Substituting the normal density of \(\log Y\) gives the lognormal density below. The factor \(1/y\) is essential: the density of \(\log y\) is not itself the density of \(y\).
Under this model, \(f(\theta)\) is the conditional median of \(Y\), not its arithmetic mean. The mean is \(f(\theta)\exp(\sigma_{\log}^2/2)\), because the mean of a lognormal variable exceeds its median when \(\sigma_{\log}>0\). The parameter \(\sigma_{\log}\) is dimensionless and describes spread on the log scale.
Worked Example: Converting a Log-Scale Density
Suppose a hypothetical observation is \(y=12.2\) mg/L, the structural prediction is \(f(\theta)=10.0\) mg/L, and \(\sigma_{\log}=0.20\). The log residual is \(\log(12.2/10.0)=\log(1.22)\approx0.1989\), so its standardized value is approximately \(0.9943\). The density of the transformed observation \(\log y\) is \(\exp(-0.9943^2/2)/(0.20\sqrt{2\pi})\approx1.217\).
That is not yet the density per mg/L. Applying the Jacobian gives \(p(y\mid\theta)\approx1.217/12.2=0.09974\) per mg/L (using unrounded values in the division). The likelihood contribution for this observed concentration on the original scale is therefore approximately \(0.09974\;(\mathrm{mg/L})^{-1}\). Omitting \(1/y\) would report a density on the transformed scale and would not give the likelihood density for the original concentration.
Choosing and Checking an Error Model
The residual model determines how deviations at different concentration levels contribute to the likelihood. Additive error treats the same absolute deviation similarly across the prediction range. Proportional error scales the standard deviation with the prediction. Combined error permits both patterns, while lognormal error describes multiplicative variation and ensures positive modelled concentrations. These models are not interchangeable: they can assign different likelihoods to the same observed data and parameter values.
| Model | Conditional standard deviation | Typical use and limitation |
|---|---|---|
| Additive normal | \(\sigma_{\mathrm{add}}\) | Constant absolute spread; may allow negative concentrations. |
| Proportional normal | \(\sigma_{\mathrm{prop}}f_j(\theta)\) | Spread grows with prediction; may allow negative concentrations. |
| Combined normal | \(\sqrt{\sigma_{\mathrm{add}}^2+\sigma_{\mathrm{prop}}^2f_j(\theta)^2}\) | Allows both absolute and prediction-dependent spread; adds parameters to specify or estimate. |
| Lognormal | Spread is specified on the log scale by \(\sigma_{\log}\) | Multiplicative variation and positive concentrations; the prediction is the median, not the mean. |
Residual diagnostics should be considered alongside the scientific context. Under the chosen model, standardized residuals should not show an obvious trend in magnitude with predicted concentration if the variance specification is adequate. Patterns over time or sampling order, clusters of unusually large residuals, or systematic underprediction in one concentration range can indicate that assumptions are failing. Such patterns may reflect an unsuitable residual model, a structural-model mismatch, assay effects, or dependence among observations; a likelihood calculation alone does not distinguish these causes.
Conditional independence is also substantive, not just a convenient algebraic step. If residuals are correlated, the product of marginal densities is not the correct joint likelihood; a joint error distribution must represent that dependence. Likewise, if residual variance parameters are estimated or depend on the predictions, their contributions to the density must be retained. In a normal likelihood, for example, the term involving the log standard deviation cannot be dropped when that standard deviation varies with the parameters being compared.
- Common mistake: treating a density as a point probability. A continuous concentration has probability zero of equaling one exact value. The likelihood is a density evaluated at the observation; its numerical value depends on the measurement units.
- Common mistake: using log-scale normal densities directly for concentrations. The density for \(y\) requires the \(1/y\) Jacobian. Without it, the calculation is on the wrong scale.
- Common mistake: assuming “proportional” means equal absolute errors. With proportional normal error, the standard deviation is \(\sigma_{\mathrm{prop}}f_j(\theta)\), so it changes with the prediction.
- Common mistake: omitting variance terms when comparing proposals. If a model's variance depends on its prediction, the density's normalization as well as its squared-residual term affects the likelihood.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Under an additive normal model, what are the units of \(\sigma_{\mathrm{add}}\) and of a single-observation density?
- A proportional normal model has prediction \(f=5\) mg/L and \(\sigma_{\mathrm{prop}}=0.08\). What is its conditional residual standard deviation?
- Why does a lognormal likelihood for an observed concentration include a factor \(1/y\)?
- For a combined model with \(\sigma_{\mathrm{add}}=0.4\) mg/L, \(\sigma_{\mathrm{prop}}=0.05\), and \(f=6\) mg/L, calculate the conditional standard deviation.
- Two residuals at successive sampling times appear strongly related even after conditioning on the predictions. Which likelihood assumption should be questioned?
Solutions. (1) \(\sigma_{\mathrm{add}}\) has concentration units, such as mg/L; the density has reciprocal units, such as \((\mathrm{mg/L})^{-1}\). (2) The standard deviation is \(0.08(5)=0.40\) mg/L. (3) The transformation \(z=\log y\) has derivative \(dz/dy=1/y\), so converting the density from log concentration to concentration requires the Jacobian. (4) The variance is \(0.4^2+(0.05\times6)^2=0.16+0.09=0.25\;(\mathrm{mg/L})^2\), giving standard deviation \(\sqrt{0.25}=0.50\) mg/L. (5) Conditional independence of residuals should be questioned; a product of marginal densities may not represent their joint distribution if the residuals are correlated.