A-Optimal Designs for Average Parameter Precision

Why Average Parameter Precision Needs Its Own Criterion

A candidate NLME sampling schedule can be informative in several ways. It may improve precision across parameters fairly evenly, or sharply improve one parameter while leaving another imprecise. A design objective must therefore say what “good precision” means for the parameters of interest. A-optimality answers one specific question: among the parameters being targeted, which design gives the smallest sum of their approximate marginal variances?

The criterion is not a universal measure of design quality. It is a decision rule for a specified parameter set, information matrix, and parameter scale. The same schedules can be ranked differently if the target parameters or their relative priorities change.

Definition: Let $I(\xi)$ be the Fisher information matrix for a $p$-dimensional target parameter vector under design $\xi$. If $I(\xi)$ is nonsingular, the unweighted A-optimality criterion is $$ \Phi_A(\xi)=\operatorname{tr}\!\left\{I(\xi)^{-1}\right\}. $$ An A-optimal design minimizes $\Phi_A(\xi)$ over the candidate designs. Under the usual local, large-sample approximation, $I(\xi)^{-1}$ approximates the covariance matrix of the estimator of the target parameters, so $\Phi_A$ is the sum of their marginal variances.

In NLME design calculations, $I(\xi)$ is the information for the population-level target parameter set under the proposed sampling schedule and the assumed model. The information may be calculated using the marginal likelihood and the design methods introduced in the earlier material on Fisher information for population pharmacokinetic designs. When independent subjects contribute information under a common design, their expected information contributions add. The criterion is then evaluated using the resulting information matrix for the planned study.

The interpretation as an estimator covariance relies on regularity and approximation conditions: the model is treated as correctly specified at the parameter values used for the calculation; the target parameters are locally identifiable; the information matrix is finite and positive definite; and the estimator is in a regime where the usual asymptotic covariance approximation is useful. In a nonlinear mixed-effects model, the result is generally local to the assumed parameter values and model. It is not a guarantee that a finite study will attain the predicted precision.

Why the Trace Represents Marginal Variances

Write the approximate covariance matrix as $C(\xi)=I(\xi)^{-1}$. Its diagonal elements $C_{jj}(\xi)$ are the approximate marginal variances of the estimated target parameters. The trace of a square matrix is the sum of its diagonal elements, so

\Phi_A(\xi) =\operatorname{tr}\{C(\xi)\} =\sum_{j=1}^{p} C_{jj}(\xi) \approx \sum_{j=1}^{p}\operatorname{Var}(\hat{\theta}_j).

Thus, minimizing the trace minimizes the sum of marginal variances, not the sum of standard errors. Dividing by $p$ gives the average marginal variance and does not change the ranking when every candidate has the same target parameter set. Covariances between different parameter estimates are not added directly to the trace. However, they can affect the diagonal elements of $I^{-1}$, so they can still affect the A-score.

For a two-parameter information matrix with entries $a,b,c$,

I= \begin{pmatrix} a & c\\ c & b \end{pmatrix}, \qquad I^{-1}= \frac{1}{ab-c^2} \begin{pmatrix} b & -c\\ -c & a \end{pmatrix}, \qquad \Phi_A=\frac{a+b}{ab-c^2}.

This expression requires $ab-c^2>0$, as well as positive diagonal information for a positive-definite matrix. It shows why a large information value for one parameter alone does not ensure a good A-score: the determinant and information for the other direction also matter. As $ab-c^2$ approaches zero, the matrix approaches singularity and the criterion can become very large.

Worked Example: Reading an A-Score as Marginal Variances

Suppose a hypothetical NLME calculation targets two dimensionless parameters, $\theta_1$ and $\theta_2$, and gives the information matrix $$ I= \begin{pmatrix} 100 & 0\\ 0 & 25 \end{pmatrix}. $$ Calculate its A-score and interpret it.

The inverse is $$ I^{-1}= \begin{pmatrix} 1/100 & 0\\ 0 & 1/25 \end{pmatrix} = \begin{pmatrix} 0.0100 & 0\\ 0 & 0.0400 \end{pmatrix}. $$ The approximate marginal variances are therefore $0.0100$ for $\theta_1$ and $0.0400$ for $\theta_2$, in squared units of their respective parameter scales. The A-score is $$ \Phi_A=0.0100+0.0400=0.0500. $$ The average of the two marginal variances is $0.0500/2=0.0250$. The calculation does not say that the two parameters are equally precise: the second variance is four times the first. The score simply adds their variances.

Selecting Between Candidate Designs

A comparison is meaningful only when each candidate uses the same model assumptions and the same target parameter vector in the same parameterization. The information matrix for each candidate must be positive definite. Once these checks are met, compute its inverse, add the diagonal elements, and select the smallest score if average marginal variance is the chosen objective.

Worked Example: A-Optimality and D-Optimality Can Disagree

Consider two hypothetical sampling designs for the same two dimensionless target parameters. Their calculated information matrices are $$ I_A= \begin{pmatrix} 100 & 0\\ 0 & 10 \end{pmatrix}, \qquad I_B= \begin{pmatrix} 30 & 0\\ 0 & 30 \end{pmatrix}. $$ Both matrices are positive definite, so both candidate scores are defined.

For design A, $$ I_A^{-1}= \begin{pmatrix} 0.0100 & 0\\ 0 & 0.1000 \end{pmatrix}, \qquad \Phi_A(A)=0.0100+0.1000=0.1100. $$ For design B, $$ I_B^{-1}= \begin{pmatrix} 1/30 & 0\\ 0 & 1/30 \end{pmatrix}, \qquad \Phi_A(B)=\frac{1}{30}+\frac{1}{30}=0.0666667. $$ Because $0.0666667<0.1100$, design B is preferred by unweighted A-optimality. Its two variances are balanced and their sum is smaller.

For comparison, the determinants are $\det(I_A)=1000$ and $\det(I_B)=900$. A D-optimal comparison, which favors the larger determinant for this same parameter set, would prefer design A. These rankings are not contradictory: the criteria summarize information differently. A-optimality targets the sum of marginal variances; D-optimality targets the determinant of the information matrix. The appropriate ranking depends on the design objective, not on a claim that one criterion is always superior.

Parameter Scale and Priorities

An unweighted trace adds variances, so the parameterization matters. If one parameter is measured in a scale thousands of times larger than another, its numerical variance can dominate the sum even when that is not the scientific priority. For a defensible unweighted comparison, use the same meaningful parameter scales across candidates. A common approach is to express targets in dimensionless, clinically interpretable scaled coordinates before calculating the criterion.

If some parameters deserve greater emphasis, use a specified positive-semidefinite weight matrix $W$ and compare $$ \Phi_{A,W}(\xi)=\operatorname{tr}\{W I(\xi)^{-1}\}. $$ For diagonal $W$, the diagonal weights multiply the corresponding marginal variances. The weights must be chosen before examining which design wins and must reflect the stated objective. They do not make an unsuitable scale harmless: weights and parameter scales should be reported together so the score can be interpreted.

Worked Example: A Priority Weight Changes the Design Choice

Suppose two hypothetical designs target two parameters already expressed on prespecified dimensionless scales. Their approximate covariance matrices are $$ C_A=I_A^{-1}= \begin{pmatrix} 0.0400 & 0\\ 0 & 0.0900 \end{pmatrix}, \qquad C_B=I_B^{-1}= \begin{pmatrix} 0.0700 & 0\\ 0 & 0.0400 \end{pmatrix}. $$ The unweighted scores are $\Phi_A(A)=0.0400+0.0900=0.1300$ and $\Phi_A(B)=0.0700+0.0400=0.1100$, so unweighted A-optimality selects design B.

Now suppose the design objective gives four times as much weight to the first parameter, using $W=\operatorname{diag}(4,1)$. Then $$ \Phi_{A,W}(A)=4(0.0400)+1(0.0900)=0.2500, $$ while $$ \Phi_{A,W}(B)=4(0.0700)+1(0.0400)=0.3200. $$ The weighted criterion selects design A. This is not a computational inconsistency: design A has the smaller variance for the parameter given greater priority. The ranking changes because the objective changed, and the priority weight should therefore be justified in the study context.

Checks, Limitations, and Common Pitfalls

Before acting on an A-optimal ranking, inspect more than the final scalar score. Check that the target set is identical across candidates, that each information matrix is positive definite, and that the calculated inverse and score are numerically stable. Examine the individual diagonal entries as well as their sum: a favorable total can conceal one target parameter with much poorer precision. Also check that the information calculation uses the intended NLME model, design assumptions, and parameter values. Because the information is model-based, a ranking may change if those assumptions change.

A frequent mistake is to call the design with the largest information entry, or the smallest variance for one parameter, “A-optimal.” A-optimality is defined by the sum of the target marginal variances, so one component cannot determine the ranking by itself. Another mistake is comparing traces from different parameterizations as if they were automatically comparable. A numerical trace has squared parameter units and depends on scale; state the target scales and any weights. Finally, a nonsingular matrix is necessary but not a guarantee of reliable finite-sample estimation: the inverse-information covariance is an approximation, and model misspecification or weak practical information can make it optimistic.

The main alternative already introduced, D-optimality, ranks designs by the determinant of information and can favor a different balance of parameter precision. A-optimality is useful when a sum or average of marginal variances matches the scientific objective. Neither criterion removes the need to define the target parameters and inspect their component-wise precision. A design with an undefined A-score has singular information for the target set and cannot be ranked by this criterion without changing the target set or design.

Key takeaway: A-optimality minimizes $\operatorname{tr}\{I(\xi)^{-1}\}$, the sum of approximate marginal variances for a specified target parameter vector. Its ranking is meaningful only with a common model, nonsingular information matrices, and clearly stated parameter scales and priorities.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

If a two-parameter design has $I^{-1}=\operatorname{diag}(0.04,0.09)$, what is its unweighted A-score and its average marginal variance?
Two designs have A-scores $0.08$ and $0.11$ for the same target parameters. Which does unweighted A-optimality select, and what does that selection mean?
Why is a singular information matrix a failure for calculating the ordinary A-criterion?
For $I=\begin{pmatrix} 20&0\\0&5\end{pmatrix}$, calculate the A-score.
Why might a weighting matrix change the design ranking?

Solutions. (1) The score is $0.04+0.09=0.13$; the average marginal variance is $0.13/2=0.065$. (2) Select the design with score $0.08$, because it has the smaller sum of approximate marginal variances for that target set; it is not necessarily best under another objective. (3) A singular matrix has no ordinary inverse, so $\operatorname{tr}(I^{-1})$ is undefined; it also indicates that at least one target direction is not locally identified by the information. (4) The inverse is $\operatorname{diag}(1/20,1/5)=\operatorname{diag}(0.05,0.20)$, giving an A-score of $0.25$. (5) A weighting matrix changes how much each marginal variance contributes to the criterion, so a design that is better for a prioritized parameter can outrank one with a smaller unweighted sum.