Local and Robust Fisher Information Designs

Why a Design Can Depend on the Parameter Values

A sampling schedule is often evaluated by the precision it is expected to provide for the model parameters. In a nonlinear model, however, the predicted observations and their sensitivities to those parameters vary with the parameter values themselves. Consequently, a schedule that is highly informative at one nominal value may be less useful elsewhere in a plausible parameter region.

This tutorial compares local designs, evaluated at a single specified parameter vector, with robust designs that account for multiple plausible vectors. The distinction matters when prior knowledge is limited, estimates are uncertain, or the planned study must perform acceptably across a range of possible pharmacokinetic profiles.

Information at a Parameter Value

Let \(\xi\) denote a candidate design, such as the observation times assigned to subjects, and let \(\psi\) denote the model parameters. In an NLME model, \(\psi\) may include population parameters, random-effect covariance parameters, and residual-error parameters. The marginal Fisher information for a design is the expected information in the observed data after accounting for the random effects. It is evaluated under the assumed model and parameter vector.

For a target parameter vector of dimension \(p\), a common criterion is D-optimality: maximize \(\log\det I(\xi;\psi)\), when the relevant information matrix is positive definite. Larger values correspond locally to a smaller volume for the approximate joint confidence ellipsoid for the target parameters. If nuisance parameters must also be estimated, the target information should be adjusted for them, for example through the Schur complement described in the earlier material on adjusting design information for nuisance parameters.

The information matrix is conditional on more than the structural model. It also depends on the assumed random-effect distribution, residual-error model, covariate distribution, and design implementation. In an NLME calculation, one should use the appropriate marginal information for the planned population, rather than treating each subject’s latent random effects as known. The criteria below compare designs only when these modeling choices and the target parameterization are held fixed.

A Pharmacokinetic Sensitivity Example

Consider a one-compartment IV-bolus model with dose \(D\), clearance \(CL\), volume \(V\), and concentration mean

The sensitivities to log clearance and log volume reveal why information changes across parameter scenarios. Writing \(k=CL/V\), differentiation gives

These are local derivatives: they describe how the model mean changes for small parameter changes around the value at which they are evaluated. The expected information also depends on the residual and random-effect model. In the next example, the derivatives are used to show scenario dependence; the matrix-ranking example uses illustrative marginal information matrices from a hypothetical NLME calculation.

Local and Robust Criteria

A local D-optimal design at \(\psi_0\) maximizes \(\log\det I(\xi;\psi_0)\). It is appropriate when the nominal vector is sufficiently well supported and the design question is explicitly about performance near that vector. It can be efficient, but it does not guarantee good performance far from \(\psi_0\).

For plausible scenarios \(\psi_s\) with weights \(w_s\), where \(w_s\geq 0\) and \(\sum_{s=1}^S w_s=1\), one robust criterion is the weighted average of the scenario-specific log-determinants:

Maximizing this criterion favors designs with good average log-determinant performance under the specified scenario weights. A different choice is the maximin criterion:

Maximizing the minimum protects the least favorable listed scenario. It does not protect against parameter values omitted from the scenario set. The weights and scenarios are part of the design specification, not quantities that the criterion determines automatically.

A tempting alternative is to average the information matrices and then take their log-determinant. This is generally not the same as averaging the log-determinants:

The left side scores average scenario-specific log-volume performance. The right side first combines information matrices and then scores the combined matrix. They answer different questions; the latter should not be described as the expected scenario-specific log-determinant.

Coordinate Choices and Scenario Comparisons

Information criteria depend on parameter coordinates. Under a fixed nonsingular linear change of coordinates, the information matrix is transformed by a fixed Jacobian. Its log-determinant changes by the same additive constant at every scenario, so rankings among designs are preserved when the same target parameters and transformation are used throughout.

That conclusion does not generally extend to a nonlinear transformation. If \(\eta=f(\theta)\) and \(J(\eta)=\partial\theta/\partial\eta\), then \[ I_{\eta}=J(\eta)^{T}I_{\theta}J(\eta), \qquad \log\det I_{\eta}=\log\det I_{\theta}+2\log|\det J(\eta)|. \] When the Jacobian depends on the parameter value, the added term can vary across scenarios. Therefore, scenario-based robust rankings, especially maximin rankings, need not be preserved by a nonlinear reparameterization.

Implementation Checks, Alternatives, and Limitations

A practical calculation begins by stating the target parameters, information matrix, scenario set, scenario weights (if used), and criterion. Then compute the information separately at every scenario and every candidate design. Keep the parameterization, residual model, random-effect assumptions, covariate distribution, and target set consistent across those calculations.

Several diagnostics help expose unstable or misleading rankings:

• Check matrix validity. Information matrices should be symmetric and positive semidefinite, and a log-determinant criterion requires positive definiteness for the target information. A singular matrix signals an uninformative direction or a numerical problem; do not silently compare its determinant as if it were positive.
• Inspect scenario-specific results. A good average can conceal a poor scenario. Report the individual log-determinants and the minimum as well as the selected aggregate criterion.
• Vary plausible inputs. Recalculate with defensible changes to scenario values or weights. Large ranking changes indicate that the decision depends strongly on uncertain design assumptions.
• Verify the model calculation. Compare numerical sensitivities with analytic derivatives where available, and check that the integration over random effects and the residual-error model match the planned analysis.

Local optimization is the main alternative when a well-supported nominal vector is the explicit design target. Weighted-average and maximin criteria are alternatives when uncertainty across scenarios matters, with different trade-offs: the average accepts weaker performance in some scenarios in exchange for stronger performance elsewhere, whereas maximin prioritizes the weakest listed case. These criteria do not remove model uncertainty, and a finite scenario set cannot guarantee performance between or beyond its listed values. Simulation under plausible parameter distributions can provide a complementary assessment, but its conclusions still depend on the distribution and model assumptions used.

A common mistake is to call a design “robust” merely because it performs well on average. The correction is to name the criterion: weighted-average performance is not worst-case protection. Another mistake is to assume that a nonlinear reparameterization adds the same constant to every scenario’s log-determinant. That is valid for a fixed nonsingular linear transformation, but a parameter-dependent Jacobian can alter scenario-based rankings.