Choosing Robustness Criteria for PK Sampling

Why the Criterion Must Match the Scientific Question

A sampling schedule can be robust in one sense and poor in another. A schedule may keep the overall uncertainty region small while leaving one parameter direction imprecise; another may protect the least-informed direction but sacrifice precision elsewhere. The choice between robust D-, A-, and E-optimality is therefore not a contest for one universally best design. It is a choice about what kind of precision matters scientifically, and how performance should be protected when PK parameters are uncertain.

The comparison below uses a one-compartment intravenous bolus model and a finite set of plausible parameter scenarios. It uses a worst-case aggregation across scenarios for all three criteria, so that differences arise from the precision objective rather than from mixing different definitions of robustness. Other aggregations, such as averaging performance over scenarios, answer a different question.

Model, Information, and Robust Criteria

Suppose concentration follows a one-compartment model after an intravenous bolus dose \(D\):

Here \(CL\) is clearance, \(V\) is volume of distribution, and \(t\) is time after dosing. For the worked calculations, assume measured log concentrations satisfy \(z_j=\log C(t_j)+\epsilon_j\), where the errors are independent and \(\epsilon_j\sim N(0,\sigma^2)\) with known, constant variance. Assume the dose is known, the model is correctly specified, sampling times are fixed in advance, and there are no censoring or missing-data complications. This is a local-information calculation: the Fisher information is evaluated at a specified parameter value or scenario.

Use \(\theta=(\log CL,\log V)\) as the parameter vector. This gives dimensionless parameter coordinates and makes information-based comparisons easier to interpret than comparisons in mixed physical units. Define \(k=CL/V\). The gradient of the expected log concentration is

For a schedule \(d\) with times \(t_1,\ldots,t_m\), the local Fisher information matrix under a scenario \(s\) is

This expression assumes independent observations with the stated common variance. Correlated errors or a different residual model require a corresponding information calculation. A nonsingular \(F_s(d)\) is required for the inverse-information precision summaries below; singularity means at least one parameter combination is not locally identifiable from that schedule in that scenario.

The scenario set represents the parameter values at which the candidate schedule is evaluated. The worst-case aggregation makes the design’s score depend on its least favorable scenario under the relevant measure. Robustness here is therefore conditional on the chosen scenarios; it does not guarantee good performance outside them.

What the Three Criteria Protect

Under the usual local normal approximation, the covariance of an estimator is approximated by \(F_s(d)^{-1}\). This connects the criteria to different aspects of uncertainty. D-optimality uses the determinant of information: in two parameters, \(\det F\) summarizes the area of the information ellipse, and larger values correspond to a smaller approximate joint confidence-ellipsoid volume. Taking the logarithm is convenient because it preserves rankings and turns products of eigenvalues into sums.

A-optimality uses the trace of the inverse information matrix. For the unweighted criterion, this is the sum of the approximate variances of the coordinates in \(\theta\). It favors reducing total marginal variance, but it does not require that the single weakest direction be as precise as possible. A weighted version, \(\operatorname{tr}\{W F^{-1}\}\), can assign more importance to selected parameter combinations when the weight matrix \(W\) is defined in advance.

E-optimality focuses on the smallest eigenvalue of information. Since the largest eigenvalue of \(F^{-1}\) is \(1/\lambda_{\min}(F)\), maximizing the smallest information eigenvalue protects the least-informed linear combination of the parameter coordinates. This is useful when a weakly identified direction is especially concerning, but it can give less attention to improving already well-informed directions.

A short derivation makes the distinction between D and E explicit. Let the positive eigenvalues of a two-parameter information matrix be \(\lambda_1\) and \(\lambda_2\). Then \(\det(F)=\lambda_1\lambda_2\), while \(\lambda_{\min}(F)=\min(\lambda_1,\lambda_2)\). Thus determinant rewards the product of both information eigenvalues, whereas E-optimality rewards only the smaller one. The trace of inverse information is \(1/\lambda_1+1/\lambda_2\), which penalizes large variance contributions from either direction. These summaries can rank schedules differently because they are not interchangeable measures of precision.

Comparing Robust Scores

The next hypothetical comparison uses information matrices already expressed in the same scaled coordinates. The diagonal matrices simplify the arithmetic; they are illustrative local information summaries, not claims about particular sampling times. Each candidate is evaluated in two scenarios, and the robust score is calculated over the less favorable scenario for that criterion.

Let the Estimand Set the Precision Objective

A scientific question may focus on one parameter rather than on the joint precision of all model parameters. For a linear combination \(c^{\mathsf T}\theta\), the approximate variance is \(c^{\mathsf T}F^{-1}c\). If the target is specifically \(\log CL\), then \(c=(1,0)^{\mathsf T}\); if the target is \(\log V\), then \(c=(0,1)^{\mathsf T}\). A targeted variance criterion can be more directly aligned with the question than D-, A-, or E-optimality.

Diagnostics, Scaling, and Limitations

Before interpreting a ranking, check that every candidate is evaluated using the same residual-error assumptions, parameter coordinates, scenario set, and treatment of nuisance parameters. Inspect eigenvalues or condition numbers for near-singularity: a very small eigenvalue indicates a poorly informed direction, and the inverse may then be numerically unstable. Also verify that any reported determinant is positive and that its logarithm is defined. If the information is singular in any scenario, inverse-based A-optimality is undefined there; in practice, the design fails to provide finite local variance in at least one direction.

Parameter scaling is especially important for A- and E-optimality. Changing the units or rescaling a parameter changes the numerical trace of \(F^{-1}\) and the eigenvalues of \(F\), and can therefore change their rankings. Log parameters provide a consistent dimensionless representation for the example, but they do not make all possible parameterizations equivalent. D-optimal rankings are invariant to a fixed nonsingular linear reparameterization up to a common determinant multiplier, which cancels when comparing designs in the same coordinates. A and E do not generally have that invariance. If a weighted A criterion or standardized E criterion is used, state the scaling or weights explicitly and use them consistently across scenarios and candidate schedules.

The scenario-based worst-case scores also have limitations. A finite scenario set can miss parameter combinations where a design performs poorly, and a single extreme scenario can dominate a worst-case score. These criteria do not account for scenario probabilities; an expected-utility design would be appropriate for a probability-weighted objective instead. Nor do local Fisher-information criteria guarantee finite-sample estimator performance when the model is strongly nonlinear, the information approximation is poor, or the likelihood is irregular. The information approximation and model diagnostics remain relevant.

A common mistake is to say that D-optimality is automatically best because it uses the whole information matrix, or to call E-optimality best because it protects the weakest direction. Neither conclusion follows without a scientific objective. The correction is to state what uncertainty matters: joint uncertainty volume, aggregate parameter variance, the least-informed combination, or a specific estimand. Then use the corresponding criterion and verify that its scaling and robustness aggregation match the intended interpretation.