Robust A- and E-Optimality

Why A- and E-Optimality Need Robust Versions

A design that performs well at one nominal parameter value may perform poorly when the true parameters differ from that value. A- and E-optimality assess different features of the information matrix: A-optimality summarizes parameter variances, while E-optimality focuses on the least-informed direction. Under parameter uncertainty, each criterion must be evaluated across plausible parameter scenarios rather than at just one matrix.

Let \(d\) denote a candidate design, such as a set of sampling times, and let \(\theta_s\) denote parameter scenario \(s\), with scenario weight \(w_s\). Write \(I_s(d)\) for the parameter-specific information matrix under that design and scenario. The weights are nonnegative and sum to one. The matrices may be exact or approximate information matrices, but their interpretation depends on the assumptions behind the method used to calculate them.

The inverse information matrix is used as an approximation to the covariance matrix of an estimator. Accordingly, the trace in A-optimality is the sum of the approximate variances of the parameter estimates. E-optimality instead examines the direction in parameter space that is informed least well. These summaries answer different questions and can favor different designs.

Assumptions and Uncertainty-Aware Criteria

The usual information-based interpretation requires a locally identifiable model and a positive-definite information matrix for each scenario being scored. Under standard regularity conditions and a suitable large-sample approximation, the estimator’s covariance is approximately \(I_s(d)^{-1}\). If an information matrix is singular, at least one parameter combination is not estimable from that design; its inverse does not exist, and the design cannot receive a finite ordinary A-loss. A zero smallest eigenvalue likewise signals failure of E-optimal identifiability.

Parameter-specific matrices must be calculated at each scenario’s parameter values. There are two common ways to aggregate their performance. A scenario-averaged criterion weights performance by the scenario weights. A worst-case criterion protects against the least favorable scenario. These are different robustness choices; neither should be described as equivalent to evaluating the information matrix at an average parameter value.

The average criteria describe performance under the specified scenario weights. The worst-case criteria focus on protection across the listed scenarios, regardless of their weights. In applications, a robust design is therefore not defined by the word “robust” alone: the scenario set and the aggregation rule are part of the design objective.

Why the Criteria Have These Interpretations

The connection between A-optimality and average parameter variance follows directly from the covariance approximation. If \(I_s(d)^{-1}\) has diagonal entries \(v_{1,s},\ldots,v_{p,s}\), then \(\operatorname{tr}(I_s(d)^{-1})=\sum_{j=1}^{p}v_{j,s}\). Thus the A-loss sums the variances of all \(p\) parameter estimates in the chosen parameterization. It is not the variance of every parameter separately, nor is it a measure that automatically gives more importance to one parameter than another.

For E-optimality, let the symmetric information matrix have eigendecomposition \(I=Q\Lambda Q^\mathsf{T}\), with positive eigenvalues \(\lambda_1,\ldots,\lambda_p\). For any unit direction \(v\) in parameter space, the approximate variance in that direction is \(v^\mathsf{T}I^{-1}v\). Expressing \(v\) in the eigenvector basis shows that this variance is a weighted average of the inverse eigenvalues. Its maximum over unit directions is therefore the largest inverse eigenvalue, \(1/\lambda_{\min}(I)\). Maximizing the smallest information eigenvalue consequently reduces the worst directional variance.

This also shows why E-optimality should be interpreted in context: the “direction” is a direction in the parameter coordinates used to form \(I\). If parameters have very different units or scales, the eigenvalues can be dominated by that choice. A-optimality also depends on parameterization because summing variances in incomparable units can be misleading. Standardization or a scientifically justified weighting can help, but it must be specified consistently before comparing designs.

Average Protection and Worst-Case Protection Can Disagree

Averaging and worst-case protection can rank the same candidates differently. This is not a contradiction: the objectives encode different priorities. The following example uses A-losses so the arithmetic is transparent.

Computation, Diagnostics, and Alternatives

A reliable calculation follows a fixed sequence. First specify the candidate designs, parameter scenarios, scenario weights, and parameter scaling. Then calculate the information matrix separately for every design-scenario pair. Check that each matrix is symmetric and positive definite before inverting it or interpreting its eigenvalues. Finally calculate the chosen scenario-level score and aggregate it using the stated average or worst-case rule.

A tempting shortcut is to average the information matrices first and then invert or take an eigenvalue. In general, this does not produce the scenario-averaged A-loss or E-score: inversion and eigenvalue extraction are nonlinear operations. The calculation must follow the stated order of operations.

D-optimality is a useful contrast: it summarizes joint information volume, whereas A- and E-optimality target variance sum and weakest-direction information, respectively. None of the criteria guarantees good performance on every parameter or direction. When one parameter has special scientific importance, an unweighted trace or an unscaled eigenvalue criterion may not express the real objective; state and justify any reweighting or standardization rather than treating it as an innocuous computational choice.

Common Mistakes and Limitations

• Confusing expected information with expected performance. Averaging \(I_s\) and then inverting does not generally equal averaging \(I_s^{-1}\). For scenario-averaged A-optimality, compute each inverse trace first.
• Calling the A-loss an average variance. It is the sum of approximate variances. Dividing by the number of parameters would give their arithmetic mean, but that is not the stated trace criterion.
• Interpreting E-optimality as “best for every parameter.” It protects the least-informed direction in the chosen coordinates. A different direction or parameter scaling can change the result.
• Ignoring a singular scenario. A singular matrix is not a harmless numerical inconvenience: it indicates that the ordinary covariance-based A criterion is undefined and the design lacks information in some direction.
• Changing the robustness rule without noticing. A scenario average and a worst-case criterion can choose different designs, as the worked example showed. Report which aggregation was used.

These criteria also inherit limitations of their information matrices. If an information approximation is unreliable in a parameter region, optimizing its A-loss or E-score does not make the approximation reliable. Robustness only protects against the scenarios and model assumptions included in the calculation.