A-Optimality And Average Parameter Variance in Optimal Design

What A-Optimality Measures

A design can provide information about several model parameters at once, but it may not estimate each parameter equally precisely. A-optimality evaluates a design by the sum of the marginal variances of the parameter estimates, or equivalently their average when the number of parameters is fixed. It therefore rewards designs that keep overall parameter uncertainty small on the parameter scale being used.

The inverse information matrix is central to this criterion. The interpretation of its diagonal entries requires care: they approximate the variances of an asymptotically efficient estimator under suitable conditions, but they are not automatically the covariance matrix of every estimator. For a general regular estimator, inverse information is a lower bound rather than a guaranteed attained covariance.

Definition: Let \(M(\xi)\) be the per-observation information matrix for design \(\xi\), evaluated at the specified nominal parameter values, and suppose it is finite and positive definite. The A-criterion is \(\Phi_A(\xi)=\operatorname{tr}\{M(\xi)^{-1}\}\). An A-optimal design minimizes \(\Phi_A(\xi)\) over the allowed designs. If the model has \(p\) parameters, \(\Phi_A(\xi)/p\) is the average of the diagonal entries of \(M(\xi)^{-1}\). The information matrix and criterion are understood on a specified parameter scale.

For a design with \(n\) independent observations and per-observation information \(M(\xi)\), the total information is \(I_n(\xi)=nM(\xi)\), when the observations follow that design’s information contributions. Its inverse is \(I_n(\xi)^{-1}=M(\xi)^{-1}/n\). Thus the corresponding total-information criterion is \(\operatorname{tr}\{I_n(\xi)^{-1}\}=\Phi_A(\xi)/n\). At fixed \(n\), minimizing either criterion gives the same design; the total-information version also reflects the reduction in variance associated with increasing sample size.

Why the Trace Is an Average-Variance Criterion

Suppose the parameter vector is \(\theta=(\theta_1,\ldots,\theta_p)^T\), and an estimator is \(\hat{\theta}\). Under suitable regularity conditions, a suitable asymptotically efficient estimator, such as a regular maximum-likelihood estimator under additional conditions, has covariance approximately equal to the inverse total Fisher information. In that case, the diagonal entries of \(I_n(\xi)^{-1}\) approximate the marginal variances of the estimated parameters. The trace adds those diagonal entries.

\operatorname{tr}\{\operatorname{Cov}(\hat{\theta})\} =\sum_{j=1}^{p}\operatorname{Var}(\hat{\theta}_j) \ \approx\ \operatorname{tr}\{I_n(\xi)^{-1}\} =\frac{1}{n}\operatorname{tr}\{M(\xi)^{-1}\}.

Dividing by \(p\) gives the average marginal variance on the chosen scale. The square root of that average is sometimes a useful summary, but minimizing the average variance and minimizing its square root give the same ranking because the square root is increasing. The criterion does not add the off-diagonal covariances themselves. Those covariances can still matter for uncertainty in combinations of parameters; A-optimality’s particular summary is the sum of the marginal variances.

The approximation in the display is not a statement that all estimators attain the information bound. Regularity and nonsingular information make the inverse-information bound relevant, but equality as an approximate covariance requires an estimator with suitable efficiency properties. An estimator with greater asymptotic variance can have a larger trace than the inverse-information trace.

Worked Design Comparisons

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

Suppose two hypothetical candidate designs have the following per-observation information matrices for two parameters:

M_A= \begin{pmatrix} 100&0\\ 0&0.2 \end{pmatrix}, \qquad M_B= \begin{pmatrix} 4&0\\ 0&4 \end{pmatrix}.

Both matrices are positive definite, so they can be inverted. For design A,

M_A^{-1}= \begin{pmatrix} 0.01&0\\ 0&5 \end{pmatrix}, \qquad \Phi_A(A)=0.01+5=5.01.

For design B,

M_B^{-1}= \begin{pmatrix} 0.25&0\\ 0&0.25 \end{pmatrix}, \qquad \Phi_A(B)=0.25+0.25=0.50.

A-optimality selects design B because \(0.50<5.01\). For comparison, the determinants are \(\det(M_A)=100(0.2)=20\) and \(\det(M_B)=4(4)=16\). The determinant criterion favors design A, while the trace-of-inverse criterion favors design B. Design A has very high information for one parameter but little for the other; its large marginal variance for the second parameter dominates the sum. This is a hypothetical comparison, and it does not establish that either design is best under criteria other than the ones calculated.

Worked Example: Translating an A-Criterion Into Approximate Variances

Consider a hypothetical design with per-observation information

M= \begin{pmatrix} 4&1\\ 1&2 \end{pmatrix},

and \(n=20\) independent observations contributing this information. The determinant is \(4(2)-1(1)=7\), so the matrix is nonsingular. Its inverse is

M^{-1}=\frac{1}{7} \begin{pmatrix} 2&-1\\ -1&4 \end{pmatrix}.

The per-observation A-criterion is \(\Phi_A=2/7+4/7=6/7\approx0.8571\). Total information is \(20M\), so

(20M)^{-1} =\frac{1}{20}M^{-1} =\begin{pmatrix} 0.01429&-0.007143\\ -0.007143&0.02857 \end{pmatrix}.

If the estimator is asymptotically efficient and the information approximation is appropriate, the two diagonal entries approximate the marginal variances of its parameter estimates, in the squared units of their respective parameters. Their sum is \(0.01429+0.02857=0.04286\). The average marginal variance is \(0.04286/2=0.02143\), equivalently \((6/7)/(20\cdot2)\). The negative off-diagonal entry represents an approximate negative covariance; it is not included in the A-criterion’s sum.

Scale Is Part of the Criterion

A trace adds variances, so its numerical value depends on the units and scaling of the parameters. This is not a defect in matrix algebra; it is a consequence of asking for an unweighted sum of variances. If one parameter is recorded in a unit that makes its numerical variance much larger, that parameter can dominate the sum. Consequently, A-optimality is meaningful only after specifying the parameter scale or an appropriate weighting.

Worked Example: Changing Units Can Change the Preferred Design

Suppose two hypothetical designs have inverse information matrices, on the original parameter scale,

M_A^{-1}= \begin{pmatrix} 0.01&0\\ 0&1 \end{pmatrix}, \qquad M_B^{-1}= \begin{pmatrix} 1&0\\ 0&0.01 \end{pmatrix}.

On this scale, both designs have the same A-criterion: \(0.01+1=1.01\). Now report the parameters as \(\phi=(10\theta_1,\theta_2)^T\). The covariance on the new scale is obtained by multiplying the old covariance on the left and right by \(C=\operatorname{diag}(10,1)\) and \(C^T\). The transformed inverse-information matrices are therefore

CM_A^{-1}C^T= \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}, \qquad CM_B^{-1}C^T= \begin{pmatrix} 100&0\\ 0&0.01 \end{pmatrix}.

The new traces are \(2\) for design A and \(100.01\) for design B, so design A is preferred on this scale. If instead the second parameter were multiplied by 10, design B would be preferred: the transformed traces would be \(100.01\) and \(2\), respectively. The designs’ information has not changed; the unweighted sum of variances has changed because the parameter coordinates have changed. In applications, parameters may be rescaled to scientifically interpretable or dimensionless units, or a weighting may be specified to express relative importance.

Interpretation: A-optimality minimizes a sum of marginal variances on a declared parameter scale. Rescaling parameters can change the criterion and its minimizing design. State the parameterization, units or standardization, and any weighting whenever an A-optimal design is reported.

Assumptions, Diagnostics, and Limitations

The criterion is based on information matrices for a specified statistical model and design. In nonlinear models, the information may depend on nominal parameter values; the criterion then compares designs locally at those values. Its variance interpretation also relies on an appropriate large-sample information approximation and, for approximate equality with inverse information, an asymptotically efficient estimator. It does not account automatically for model misspecification, estimator bias, finite-sample behavior, or uncertainty in the nominal parameters.

Before interpreting a calculated trace, check that the information matrix is finite, symmetric as expected, and positive definite. A singular matrix has no ordinary inverse, so the standard A-criterion is undefined; a nearly singular matrix may produce a very large trace and unstable numerical results. Inspecting the diagonal of the inverse helps identify which parameter variance drives a large criterion. Also check that candidate designs are compared at the same parameter scale, sample size convention, nominal values, and model assumptions.

A-optimality is one choice among design objectives. D-optimality instead compares determinants and therefore summarizes generalized variance; as the worked comparison showed, it can select a different design. A-optimality focuses on the sum of marginal variances, not the determinant, a single selected parameter, or the worst direction of uncertainty. If the scientific goal is not well represented by an unweighted average across all parameters, a different criterion or an explicitly weighted A-criterion may be more appropriate.

How to Apply the Criterion

Specify the comparison.
Fix the model, parameter scale, nominal values when needed, sample size convention, and allowed design set.

Form the information matrix.
Calculate the per-observation or total information for each candidate design using the same convention.

Check invertibility.
Confirm that each matrix being scored is positive definite. If it is singular, the ordinary A-criterion cannot be evaluated.

Calculate and interpret the trace.
Invert each matrix and sum its diagonal entries. Identify whether a particular marginal variance dominates the result.

Report the scope of the conclusion.
State which design minimizes the criterion on the specified scale and under the assumed model; do not treat this as a guarantee of finite-sample estimator performance.

Common Pitfall

A frequent error is to say, “The inverse information is the estimator’s covariance, so the smallest trace guarantees the smallest realized estimation error.” The first part is too broad, and the conclusion overstates what the design calculation establishes. For a general regular estimator, inverse information is a lower bound, not necessarily its covariance. The approximate covariance equality requires suitable efficiency and regularity conditions. In addition, the trace is an expected-uncertainty summary on a chosen scale, not a guarantee about any particular data set or realized estimate. A defensible conclusion is that the design minimizes the trace of inverse information among the candidates considered, with an average-variance interpretation when the stated approximation is justified.

Check Your Understanding

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

For a three-parameter design with \(M^{-1}\) diagonal entries \(0.2\), \(0.5\), and \(1.3\), what are the A-criterion and the average diagonal entry?
If a fixed design is replicated so that the total number of independent observations doubles, what happens to its total-information A-criterion?
Why can a design with a smaller determinant have a smaller A-criterion than a competing design?
Does regularity alone justify treating inverse information as the covariance of any estimator? Explain.
What must be checked before computing the ordinary A-criterion, and why can parameter units affect the result?

Solutions. (1) The trace is \(0.2+0.5+1.3=2.0\), and the average diagonal entry is \(2.0/3\approx0.6667\). (2) Total information doubles, so its inverse and the total-information A-criterion are halved: \((2I_n)^{-1}=I_n^{-1}/2\). (3) The determinant summarizes a different feature of the information matrix; the A-criterion sums the diagonal entries of its inverse, so uneven information can lead the criteria to rank designs differently. (4) No. Inverse information is a lower bound for a general regular estimator; approximate equality as covariance requires a suitable asymptotically efficient estimator and appropriate conditions. (5) Check that the information matrix is finite and positive definite so it has an ordinary inverse. The trace adds variances, whose numerical values depend on parameter units; changing scale can change the sum and the preferred design.