Why choose the next time using the posterior?
In an adaptive design, a sampling time is chosen after some data have already been observed. The relevant uncertainty is therefore the current posterior uncertainty about the pharmacokinetic parameters, not simply a nominal parameter value or the uncertainty specified before the study began. A time can be informative under one posterior distribution and comparatively uninformative under another.
The decision can be framed as a comparison across feasible candidate times: if the next concentration were measured at each candidate time, how much would that observation be expected to teach us about the parameters? The expectation is over the possible future observations under the current posterior predictive distribution. This is different from selecting a time because its concentration is predicted to be high, or because its sensitivity is large at one particular parameter value.
A local Gaussian calculation
A useful calculation is available when the current posterior for the parameters is approximately Gaussian and the future observation can be approximated by a linear function of them. Let \(\theta\) be a parameter vector with current posterior \(N(\mu,\Sigma)\), and suppose the future observation has the local form \(Y\approx a(t)+g(t)^\mathsf{T}(\theta-\mu)+\epsilon\), where \(\epsilon\sim N(0,r)\) is independent measurement error. The vector \(g(t)\) is the gradient of the predicted observation with respect to \(\theta\), evaluated at the current posterior mean; \(r\) is the observation-error variance on the measurement scale used in this approximation.
Under these assumptions, the variance in the future observation attributable to parameter uncertainty is \(q(t)=g(t)^\mathsf{T}\Sigma g(t)\). The conditional variance of \(Y\) given \(\theta\) is \(r\), while its predictive variance given the current data is \(r+q(t)\). The expected information gain is therefore:
A short derivation makes the role of the assumptions clear. The differential entropy of a univariate normal variable with variance \(v\) is \(\frac{1}{2}\log(2\pi e v)\). Mutual information is the entropy of the predictive observation minus its conditional entropy given the parameters. Substituting the predictive variance \(r+q(t)\) and conditional variance \(r\) gives \(\frac{1}{2}\log\{2\pi e(r+q(t))\}-\frac{1}{2}\log(2\pi e r)=\frac{1}{2}\log(1+q(t)/r)\). Thus the formula compares parameter-driven predictive variation with measurement noise; it does not reward a large concentration on its own.
This expression is exact for the stated linear-Gaussian observation model. For a nonlinear model it is a local approximation if the gradient and Gaussian posterior are used. It is not automatically exact for a nonlinear pharmacokinetic model with a broad or markedly non-Gaussian posterior. In that setting, expected information can instead be estimated by integrating or simulating over the current posterior and future observations.
Worked Example: Choosing a Time With Uncertain Clearance and Volume
Worked Example: Choosing a Time With Uncertain Clearance and Volume
Suppose a hypothetical patient receives a known intravenous bolus dose \(D\), and concentration follows a one-compartment, first-order elimination model: \(C(t)=D V^{-1}\exp(-CL\,t/V)\). Assume the current posterior is summarized on the log-parameter scale by \(\theta=(\log CL,\log V)^\mathsf{T}\), with covariance \(\Sigma=\begin{pmatrix}0.090&0.030\\0.030&0.040\end{pmatrix}\). At the posterior mean, \(CL/V=k=0.25\) per hour. For the log concentration, use a local Gaussian observation model with independent assay error of standard deviation \(0.15\), so \(r=0.15^2=0.0225\). Compare candidate times \(0.5\), \(2\), and \(6\) hours.
The assumptions for this calculation are that the bolus dose is known, the one-compartment model is an adequate local description, and the log-concentration assay error is approximately normal with the specified constant variance. We also use the local linear approximation at the posterior mean. The calculation does not include sample-handling constraints or an assay lower limit.
Taking logs gives \(\log C(t)=\log D-\log V-CL\,t/V\). Its gradient with respect to \((\log CL,\log V)\), evaluated at the posterior mean, is \(g(t)=(-kt,\,kt-1)^\mathsf{T}\). For \(t=0.5\), this is \((-0.125,-0.875)^\mathsf{T}\). Therefore:
The cross term is positive because both gradient components are negative. Thus \(\mathrm{EIG}(0.5)=\frac{1}{2}\log(1+0.03859375/0.0225)\approx0.4994\) nats.
At \(2\) hours, \(g(2)=(-0.5,-0.5)^\mathsf{T}\), so \(q(2)=0.090(0.5^2)+2(0.030)(0.5)(0.5)+0.040(0.5^2)=0.0475\). Hence \(\mathrm{EIG}(2)=\frac{1}{2}\log(1+0.0475/0.0225)\approx0.5675\) nats.
At \(6\) hours, \(g(6)=(-1.5,0.5)^\mathsf{T}\). Here the cross term is negative because the gradient components have opposite signs: \(q(6)=0.090(1.5^2)-2(0.030)(1.5)(0.5)+0.040(0.5^2)=0.1675\). Thus \(\mathrm{EIG}(6)=\frac{1}{2}\log(1+0.1675/0.0225)\approx1.0668\) nats.
Among these three candidates, \(6\) hours maximizes the local Gaussian expected information gain. The conclusion is limited to the candidates and assumptions specified: it does not establish that \(6\) hours is optimal over every possible time, nor that a late concentration will be measurable. The covariance matters as well as the gradient; omitting its correlation would change the calculation.
Information is not the only possible objective
Expected information gain is a principled criterion when the aim is to learn about the parameters. A practical design may instead value information while also accounting for the burden or feasibility of collecting a sample. Such a decision requires a utility that puts these considerations on a common scale. For example, one might define \(U(t)=\mathrm{EIG}(t)-c(t)\), where \(c(t)\) is a pre-specified cost expressed in the same utility units. The cost should represent a real design consideration rather than being chosen after seeing which time wins.
Worked Example: Choosing by Information Minus Sampling Burden
Use the expected information gains from the previous example, but suppose a hypothetical design team has specified burden costs of \(0\), \(0.10\), and \(0.80\) utility units for samples at \(0.5\), \(2\), and \(6\) hours, respectively. Assume one utility unit is defined to be one nat of information value. Calculate \(U(t)=\mathrm{EIG}(t)-c(t)\).
For \(0.5\) hours, \(U(0.5)=0.4994-0=0.4994\). For \(2\) hours, \(U(2)=0.5675-0.10=0.4675\). For \(6\) hours, \(U(6)=1.0668-0.80=0.2668\). The largest net utility is at \(0.5\) hours, even though \(6\) hours has the largest expected information gain.
This choice follows from the stated utility scale and costs, not from a universal preference for early samples. If the costs or the value assigned to information change, the preferred time can change. A utility analysis makes the trade-off explicit; it does not make an arbitrary or poorly justified cost model reliable.
How a different posterior can change the choice
Adaptation has a direct Bayesian interpretation: use the posterior currently available at the decision point to evaluate the candidates. That posterior may differ from the one used in an earlier planning calculation. In the local Gaussian expression, changes in posterior covariance \(\Sigma\) can change the ranking even if the model sensitivities \(g(t)\) are unchanged. The posterior mean can also alter the sensitivities in a nonlinear model.
Worked Example: A Posterior With Greater Volume Uncertainty
Suppose a hypothetical interim analysis has produced the current posterior covariance \(\Sigma=\begin{pmatrix}0.020&0\\0&0.120\end{pmatrix}\) for \((\log CL,\log V)\). Keep the same posterior mean elimination rate \(k=0.25\) per hour, candidate times, and observation-error variance \(r=0.0225\). This posterior has more uncertainty in log volume than in log clearance.
At \(0.5\) hours, \(g=(-0.125,-0.875)^\mathsf{T}\), giving \(q(0.5)=0.020(0.125^2)+0.120(0.875^2)=0.0921875\). Therefore, \(\mathrm{EIG}(0.5)=\frac{1}{2}\log(1+0.0921875/0.0225)\approx0.8143\) nats.
At \(2\) hours, \(q(2)=0.020(0.5^2)+0.120(0.5^2)=0.0350\), so \(\mathrm{EIG}(2)=\frac{1}{2}\log(1+0.0350/0.0225)\approx0.4691\) nats. At \(6\) hours, \(q(6)=0.020(1.5^2)+0.120(0.5^2)=0.0750\), and \(\mathrm{EIG}(6)=\frac{1}{2}\log(1+0.0750/0.0225)\approx0.7332\) nats.
For this posterior, \(0.5\) hours is best among the candidates, whereas the first example favored \(6\) hours. The short-time log concentration is more sensitive to log volume here, and the posterior assigns substantial uncertainty to that parameter. This illustrates why an adaptive choice should evaluate candidates using the current joint posterior rather than carry forward a time selected for a different uncertainty pattern.
Diagnostics, alternatives, and limitations
Before implementing an information-based choice, check whether its inputs and approximation are credible. In particular, inspect whether the candidate times produce plausible predicted concentrations, whether assay error is approximately described on the chosen scale, and whether the posterior covariance represents the dependence between clearance and volume. If the local linear approximation is poor, compare it with posterior simulation: draw parameters from the current posterior, simulate possible observations at each candidate time, and estimate the expected posterior-to-prior KL divergence. Large differences between the local calculation and simulation warn that curvature, skewness, or other nonlinearity matters.
Other design criteria answer different questions. A nominal design assesses performance at a selected parameter value; robust criteria such as maximin design focus on performance across specified uncertainty sets or distributions. Expected information gain averages over the current posterior and values information through a posterior-to-prior divergence. These criteria need not select the same time because their objectives differ. A utility criterion adds specified operational or decision consequences, while a pure information criterion does not. The appropriate choice depends on the scientific aim and the constraints, not on which criterion produces the most attractive numerical score.
- Measurement limits: A candidate may have high theoretical information but predict concentrations near or below the assay’s quantification limit. The observation model must represent censoring or non-detection if these outcomes are plausible.
- Approximation failure: A broad, multimodal, or strongly skewed posterior may not be represented adequately by a single Gaussian covariance and local gradient.
- Model misspecification: If the one-compartment model or error model is inadequate, maximizing its calculated information can select a time that is not useful under the actual data-generating process.
- Feasibility: Candidate times must respect clinical, logistical, and sampling constraints. Optimizing over infeasible times does not produce an implementable design.
A common mistake is to choose the time with the largest absolute gradient or the largest predicted concentration and call it the most informative. The correction is to calculate how parameter uncertainty projects onto the observation, \(g(t)^\mathsf{T}\Sigma g(t)\), and compare that quantity with observation noise; for a broader design objective, evaluate the specified utility instead. A high sensitivity can be unhelpful if uncertainty in that direction is small, parameters are correlated, or measurement noise dominates.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In the local Gaussian expression, what do \(q(t)\) and \(r\) represent, and why does increasing \(r\) generally lower expected information gain?
- For a candidate time with \(q(t)=0.030\) and \(r=0.020\), calculate the expected information gain in nats.
- Why can a sampling time with a large model gradient still have low expected information gain?
- In the worked utility example, why was the selected time different from the time with the largest expected information gain?
- Name one reason the local Gaussian formula may be inadequate for a nonlinear pharmacokinetic model.
Solutions. (1) \(q(t)=g(t)^\mathsf{T}\Sigma g(t)\) is the predictive variance attributable to parameter uncertainty, while \(r\) is observation-error variance. Larger \(r\) makes the observation noisier relative to parameter-driven variation, reducing \(\frac{1}{2}\log(1+q/r)\). (2) \(\mathrm{EIG}=\frac{1}{2}\log(1+0.030/0.020)=\frac{1}{2}\log(2.5)\approx0.4581\) nats. (3) The posterior may have little uncertainty in the direction of that gradient, parameter covariance may offset its contribution, or measurement noise may be large. (4) The burden cost at \(6\) hours reduced its net utility below that at \(0.5\) hours, despite its greater information gain. (5) For example, a broad or skewed posterior can make a local linear approximation unreliable; posterior simulation or a more suitable nonlinear calculation may then be needed.