Why Concentration Uncertainty Matters
An estimated pharmacodynamic (PD) interaction depends on the concentrations used to describe the two drugs’ exposures. Those concentrations are often not known exactly: they may be estimated from sparse samples, an assay with measurement error, or a pharmacokinetic (PK) model. Treating estimated concentrations as if they were exact can make an interaction estimate look more precise than the data warrant. It can also change the estimate itself when the response model is nonlinear in concentration.
The goal here is to carry concentration uncertainty through to the PD interaction estimate. This is not the same as modeling population variability: variability describes real differences between individuals, whereas uncertainty describes limited knowledge about a concentration, parameter, or estimate. The two can coexist and should not be conflated.
Set Up the Joint Model
Let \(C_{i1}\) and \(C_{i2}\) denote the true concentrations of two drugs for observation \(i\), and let \(W_{ij}\) be the corresponding measured concentration for drug \(j\). Let \(Y_i\) be the PD response. A concentration measurement model specifies how the observed concentration relates to the true concentration. For example, an additive-error model is
Here \(\tau_j\) is the assay-error standard deviation, in the concentration units for drug \(j\). A response model then specifies \(Y_i\) conditional on the true concentrations. For a continuous response with normally distributed residual error, one possible model is
where \(m\) is the specified concentration-response surface and \(\psi\) contains its parameters, including any parameter that represents interaction. These are modeling assumptions, not automatic properties of the data. The error distribution and the form of \(m\) must be appropriate to the assay and response.
When the true concentrations are treated as latent quantities with a distribution \(f_C\), the contribution of observation \(i\) to the likelihood of the measured concentrations and response is
The parameters \(\phi\) describe concentration measurement error, and \(\eta\) describes the concentration distribution when one is part of the model. If concentrations are instead treated as fixed but unknown covariates, they can be included as nuisance parameters and estimated jointly with \(\psi\); that is not the same as assigning them a population distribution. In either approach, the response likelihood is evaluated at plausible true concentrations rather than only at the observed concentration values.
Worked Example: Plug-In Versus Integrated Response Likelihood
Suppose a hypothetical concentration analysis gives a conditional distribution for one true concentration \(C\), given the concentration data, of \(C\mid W\sim N(6,1)\), where concentration is measured in mg/L and the second parameter is the variance. Assume a continuous response follows \(Y\mid C\sim N(C,1)\), with response measured in arbitrary response units. The observed response is \(y=6.6\). Compare the likelihood density using the mean concentration as a plug-in value with the density that integrates over concentration uncertainty.
Plug-in calculation. Fixing \(C\) at its mean, 6 mg/L, gives a response residual of \(6.6-6=0.6\) response units. With response variance 1, the density is
This is a likelihood density, with units inverse to the response unit; it is not a probability of observing exactly 6.6.
Integrated calculation. A normal variable plus an independent normal residual is normal, with means added and variances added. Thus \(Y\mid W\sim N(6,1+1)=N(6,2)\). Its density at 6.6 is
The two likelihood contributions differ because the integrated calculation includes uncertainty about \(C\). In a full interaction analysis, such differences across observations can affect parameter estimates as well as their uncertainty. The calculation assumes the stated normal conditional distributions and independent response residual and concentration uncertainty.
First-Order Propagation With the Delta Method
A useful approximation is available when the interaction estimate is a smooth function of estimated quantities and their joint uncertainty is reasonably summarized by a covariance matrix. Let \(\hat{\boldsymbol q}\) collect those estimates, with covariance matrix \(\Sigma\), and let the target be \(g(\hat{\boldsymbol q})\). A first-order Taylor expansion around the estimates gives
The gradient measures how sensitive the target is to each input. The covariance terms matter: concentration estimates may be correlated because they share PK parameters, sampling times, or assay processes. Ignoring those covariances is justified only when independence is a defensible assumption.
Worked Example: Delta-Method Uncertainty for an Interaction Parameter
Suppose, hypothetically, that a response-scale interaction contrast is estimated as \(\hat{\Delta}=8.0\) response units with standard error \(0.8\). At the concentration pair of interest, the estimated concentrations are \(\hat C_1=4.0\) mg/L and \(\hat C_2=2.0\) mg/L, with standard errors \(0.2\) mg/L and \(0.1\) mg/L. Under the simple bilinear interaction form \(\Delta=\theta C_1C_2\), estimate \(\theta\) and propagate uncertainty. Assume for this calculation that the three estimates are mutually independent and approximately normal.
The interaction-parameter estimate is
For \(g(\Delta,C_1,C_2)=\Delta/(C_1C_2)\), the derivatives at the estimates are \(\partial g/\partial\Delta=1/(C_1C_2)=0.125\), \(\partial g/\partial C_1=-\theta/C_1=-0.25\), and \(\partial g/\partial C_2=-\theta/C_2=-0.50\). The estimated variances are \(0.8^2=0.64\), \(0.2^2=0.04\), and \(0.1^2=0.01\), respectively. Independence makes the covariance terms zero, so
The propagated standard error is \(\sqrt{0.0150}=0.1225\) response units per \((\text{mg/L})^2\). A normal-approximation 95% interval is
Under the approximation and assumptions above, this interval describes values compatible with the data and method at the 95% confidence level; it does not mean there is a 95% probability that the fixed parameter lies in this particular interval. The calculation also shows why concentration uncertainty belongs in the standard error: omitting both concentration terms would give \(SE=0.125(0.8)=0.1000\), smaller than \(0.1225\).
Simulation-Based Propagation
Simulation is useful when the target is nonlinear, the uncertainty is asymmetric, or a first-order approximation is questionable. The essential requirement is to draw inputs from their joint uncertainty distribution. If the estimates are correlated, drawing each concentration independently destroys that dependence and can misstate the uncertainty of the interaction estimate.
Draw concentration values, PK parameters, and response-model parameters together from a joint likelihood approximation, a parametric bootstrap, or another justified uncertainty distribution.
For each draw, evaluate the same interaction estimand using that draw’s concentrations and model parameters.
Report a point summary and an interval or distribution summary appropriate to the estimand, while checking that the simulation has enough draws for stable results.
Worked Example: Propagating Paired Concentration Draws
Suppose a hypothetical analysis yields five paired draws of concentrations for an exposure condition, and the response-scale contrast is held at \(\Delta=8.0\) response units for illustration. Each row is one joint draw; the values are not independently paired after sampling.
| Draw | \(C_1\) (mg/L) | \(C_2\) (mg/L) | \(\Delta/(C_1C_2)\) [response units per (mg/L)\(^2\)] |
|---|---|---|---|
| 1 | 3.8 | 1.9 | 8/(3.8 × 1.9) = 1.108 |
| 2 | 4.2 | 2.1 | 8/(4.2 × 2.1) = 0.9070 |
| 3 | 3.9 | 2.2 | 8/(3.9 × 2.2) = 0.9324 |
| 4 | 4.1 | 1.8 | 8/(4.1 × 1.8) = 1.084 |
| 5 | 4.0 | 2.0 | 8/(4.0 × 2.0) = 1.000 |
The mean of these five transformed estimates is \((1.1080+0.9070+0.9324+1.0840+1.0000)/5=1.006\), using the unrounded values in the calculations. Their sample standard deviation is approximately \(0.08908\). These summaries describe only the five illustrative draws. Five draws are far too few to support a stable uncertainty interval; a practical analysis would use many more draws and assess Monte Carlo stability. A full analysis would also draw \(\Delta\), and any uncertain response parameters, rather than holding them fixed as this arithmetic demonstration does.
Choosing an Approach and Checking It
| Approach | What it does | When it is useful | Main caution |
|---|---|---|---|
| Plug-in | Uses estimated concentrations as if known | Exploratory calculation when concentration uncertainty is negligible | Usually omits concentration uncertainty and may distort nonlinear estimates |
| Delta method | Uses a local linear approximation and covariance matrix | Smooth estimands with modest, approximately symmetric uncertainty | Can be unreliable near boundaries or with skewed, strongly nonlinear transformations |
| Joint likelihood | Estimates concentration and response components together | When concentration and response data can be represented in a coherent generative model | Depends on correct measurement, concentration, and response models and may be computationally demanding |
| Simulation propagation | Recalculates the estimand across joint uncertainty draws | Nonlinear estimands or uncertainty distributions not well summarized by a normal approximation | Requires valid joint draws and enough simulation to control Monte Carlo error |
Before interpreting an interaction estimate, check that concentration units and time alignment are consistent, the uncertainty model reflects the available assay or PK information, and correlations among estimated concentrations have not been discarded. For simulation, inspect whether the estimated mean, standard deviation, and interval endpoints stabilize as the number of draws increases. For a delta-method result, compare its standard error with a simulation-based result when the transformation is strongly nonlinear or concentrations are close to zero.
Warning signs include implausible or negative concentrations under a model that requires positive values, a highly skewed distribution of the transformed interaction estimate, large changes under small model perturbations, and confidence intervals that depend heavily on one or two extreme draws. These signs do not by themselves identify the cause, but they indicate that a local normal approximation or assumed concentration distribution may be inadequate. Propagation cannot repair a poorly identified interaction model or uninformative sampling design; it only carries the uncertainty represented by the chosen model and data.
Common Mistake and Key Takeaway
A common mistake is to treat a concentration estimate as an observed truth, then report the response-model standard error as if it covered all uncertainty. For example, calculating uncertainty in \(\hat\Delta\) while holding both \(C_1\) and \(C_2\) fixed ignores the concentration contributions shown in the delta-method example. The correction is to include their variances and covariances, or to propagate joint draws through the estimand. The result is only as defensible as the underlying error model, concentration information, and PD model.
Check Your Understanding
Answer each question, then compare with the solutions.
- In one sentence, distinguish uncertainty in an individual’s concentration estimate from population variability in concentration.
- If two concentration estimates have positive covariance, what additional term appears in the delta-method variance of \(g(C_1,C_2)\)?
- For \(g=\Delta/(C_1C_2)\), calculate the plug-in estimate when \(\Delta=6\) response units, \(C_1=3\) mg/L, and \(C_2=2\) mg/L.
- Why can independently drawing \(C_1\) and \(C_2\) give an incorrect propagated uncertainty when their estimation errors are correlated?
Solutions. (1) Individual-level uncertainty is lack of knowledge about a particular person’s concentration; population variability is real concentration differences among people. (2) The term is \(2(\partial g/\partial C_1)(\partial g/\partial C_2)\operatorname{Cov}(C_1,C_2)\), evaluated at the estimates. (3) \(6/[(3)(2)]=1.00\) response unit per \((\text{mg/L})^2\). (4) Independent draws remove the covariance and change the joint distribution of the product \(C_1C_2\), potentially changing the variability and shape of the interaction-estimate distribution.