Why the Criterion Must Match the Inferential Target
An optimal design is optimal with respect to a stated objective, not in every possible sense. One design may estimate a parameter contrast especially precisely while another provides lower average uncertainty across all parameters. A third may protect predictions at the least favorable point in a region. These differences are the reason to choose among alphabetic optimality criteria by first identifying the intended inference.
The criteria considered here use the information matrix for a design, \(M(\xi)\), where \(\xi\) denotes the design. For a nonlinear model, this is typically local information evaluated at specified nominal parameter values. The criteria assess different summaries of the information or of the approximate parameter and prediction uncertainty it implies.
Positive definiteness matters because the inverse information matrix is used to assess uncertainty in every parameter direction. If a candidate information matrix is singular, at least one parameter direction is not locally identifiable from that design, and criteria involving \(M(\xi)^{-1}\) are not defined as ordinary finite variances. Comparisons also lose meaning if candidate designs use different parameterizations or different nominal assumptions.
What the Six Criteria Protect
The six criteria differ in the quantity they reward or control. For \(p\) parameters, D-optimality maximizes information volume, equivalently minimizing the determinant of the approximate covariance matrix. A-optimality minimizes the sum of parameter variances, and E-optimality maximizes the least information eigenvalue. The c-criterion targets one specified linear parameter contrast. G- and I-optimality instead assess prediction variance over a specified region.
- D-optimality: maximize \(\det M(\xi)\), equivalently minimize \(\det\{M(\xi)^{-1}\}\).
- A-optimality: minimize \(\operatorname{tr}\{M(\xi)^{-1}\}\), the sum of the approximate variances of the parameter estimates.
- E-optimality: maximize \(\lambda_{\min}\{M(\xi)\}\), the smallest eigenvalue of the information matrix.
- c-optimality: for a prespecified vector \(c\), minimize \(c^{T}M(\xi)^{-1}c\), the approximate variance of the contrast \(c^{T}\theta\).
- G-optimality: minimize the maximum prediction variance over a prespecified region \(\mathcal{X}\).
- I-optimality: minimize the average prediction variance over \(\mathcal{X}\), using a prespecified probability measure \(\nu\).
For prediction criteria, let \(g(x)\) be the gradient of the model mean at prediction location \(x\) with respect to the parameter vector. The delta method gives approximate prediction variance \(g(x)^T M(\xi)^{-1}g(x)\). Thus G-optimality minimizes the maximum of this quantity over \(x\in\mathcal{X}\), while I-optimality minimizes its integral with respect to \(\nu\). In a linear model, \(g(x)\) is the usual regression vector. For I-optimality, \(\nu\) is commonly normalized to have total probability one, so the criterion is an average rather than an unscaled integral.
These definitions express distinct priorities. D-optimality rewards a small overall covariance volume but does not guarantee that every individual variance is small. A-optimality summarizes the coordinate-wise variances by their sum, effectively giving each parameter coordinate equal weight. E-optimality protects the weakest information direction. c-optimality can focus resources on a scientifically important contrast, and the prediction criteria focus on predictions rather than directly on parameter estimates.
Worked Example: Global Parameter Criteria Can Disagree
Worked Example: Global Parameter Criteria Can Disagree
Suppose two hypothetical designs have information matrices \(M_1=\begin{pmatrix}16&0\\0&1\end{pmatrix}\) and \(M_2=\begin{pmatrix}5&0\\0&5\end{pmatrix}\), for two parameters in a fixed, common parameterization. Both matrices are positive definite, so their inverse matrices give finite approximate variances.
For D-optimality, compare determinants: \(\det(M_1)=16(1)=16\), while \(\det(M_2)=5(5)=25\). Therefore D-optimality favors Design 2.
For A-optimality, calculate the traces of the inverse matrices: \(M_1^{-1}=\begin{pmatrix}1/16&0\\0&1\end{pmatrix}\), so \(\operatorname{tr}(M_1^{-1})=1/16+1=1.0625\). For Design 2, \(M_2^{-1}=\begin{pmatrix}1/5&0\\0&1/5\end{pmatrix}\), giving \(\operatorname{tr}(M_2^{-1})=0.2+0.2=0.4\). A-optimality also favors Design 2.
The smallest information eigenvalues are \(\lambda_{\min}(M_1)=1\) and \(\lambda_{\min}(M_2)=5\), so E-optimality favors Design 2 as well. But consider the contrast \(c^T\theta=\theta_1\), with \(c=(1,0)^T\). Its approximate variances are \(c^TM_1^{-1}c=1/16=0.0625\) and \(c^TM_2^{-1}c=1/5=0.2\). For estimating \(\theta_1\) alone, Design 1 is preferable. The calculation illustrates why the answer depends on the inferential target, even when all criteria are applied to the same valid information matrices.
Worked Example: A Targeted Contrast Versus Global Summaries
Worked Example: A Targeted Contrast Versus Global Summaries
Consider hypothetical information matrices \(M_1=\begin{pmatrix}4&1\\1&2\end{pmatrix}\) and \(M_2=\begin{pmatrix}3&0\\0&3\end{pmatrix}\). Their determinants are \(7\) and \(9\), respectively, so D-optimality favors Design 2. For the targeted contrast \(c^T\theta=\theta_1+\theta_2\), use \(c=(1,1)^T\).
The determinant of \(M_1\) is \(4(2)-1(1)=7\), and hence \(M_1^{-1}=\frac{1}{7}\begin{pmatrix}2&-1\\-1&4\end{pmatrix}\). Substitution gives \(c^TM_1^{-1}c=(2-1-1+4)/7=4/7\approx0.5714\). For Design 2, \(M_2^{-1}=(1/3)I\), so \(c^TM_2^{-1}c=2/3\approx0.6667\). Thus c-optimality for this contrast favors Design 1, despite D-optimality favoring Design 2.
For context, A-optimality favors Design 2: the traces are \(6/7\approx0.8571\) for Design 1 and \(2/3\approx0.6667\) for Design 2. The smallest eigenvalue of \(M_1\) is \((6-\sqrt{8})/2\approx1.586\), compared with \(3\) for \(M_2\), so E-optimality favors Design 2. The contrast-specific conclusion is not a contradiction: Design 1 happens to deliver lower approximate uncertainty for the chosen sum, while Design 2 performs better according to these broader summaries.
Prediction Targets: Worst Case or Average
G- and I-optimality both require a prediction region, but they answer different questions about it. G-optimality asks how large prediction variance can become at its worst point. I-optimality asks how large it is on average under a chosen distribution of prediction locations. The region and, for I-optimality, the averaging measure are part of the scientific objective; changing them can change which design is preferred.
Worked Example: Worst-Case and Average Prediction Variance
Suppose two hypothetical designs have approximate covariance matrices \(C_1=M_1^{-1}=\begin{pmatrix}0.1&0\\0&0.9\end{pmatrix}\) and \(C_2=M_2^{-1}=\begin{pmatrix}0.35&0\\0&0.3\end{pmatrix}\) for the intercept and slope of a linear prediction model. Consider predictions at \(x\in[-1,1]\), with regression vector \(f(x)=(1,x)^T\). Both covariance matrices are positive definite.
The prediction variance is \(f(x)^TC_if(x)\). For Design 1 it is \(0.1+0.9x^2\); for Design 2 it is \(0.35+0.3x^2\). Each is largest at \(x=-1\) or \(x=1\), giving worst-case variances of \(1.0\) and \(0.65\), respectively. G-optimality therefore favors Design 2.
Now let \(\nu\) be the uniform probability distribution on \([-1,1]\). Since the mean of \(x^2\) under this distribution is \(\frac{1}{2}\int_{-1}^{1}x^2\,dx=1/3\), the average prediction variances are \(0.1+0.9/3=0.4\) for Design 1 and \(0.35+0.3/3=0.45\) for Design 2. I-optimality favors Design 1. It accepts a higher worst-case variance in return for lower average variance over the specified region.
A Practical Selection and Checking Process
Decide whether the priority is joint parameter estimation, average parameter precision, protection against a weak parameter direction, one contrast, or predictions across a region.
For c-optimality, state the contrast vector \(c\). For G-optimality, state the prediction region. For I-optimality, state both the region and its averaging measure.
Confirm that candidate designs use the same model, nominal parameter values, parameterization, and constraints, and verify that each matrix is positive definite before using its inverse.
Report which objective it optimizes and what that objective leaves unprotected. A design that is best for one contrast need not be best for overall precision or prediction across the region.
A useful diagnostic is to inspect whether the criterion’s inputs actually reflect the planned inference. A c-vector that omits the scientifically important contrast, a prediction region that excludes likely use, or an I-optimal averaging measure that overweights irrelevant locations can produce a mathematically correct but practically unhelpful design. Also check eigenvalues and matrix conditioning: a very small eigenvalue signals a weak direction, and numerical instability can make computed inverses and criterion rankings unreliable.
The main alternatives are not interchangeable shortcuts. D-optimality is suited to joint parameter precision summarized by covariance volume; A-optimality to the sum of coordinate variances; E-optimality to guarding the least-informed direction; c-optimality to a particular linear combination; G-optimality to controlling the worst prediction variance; and I-optimality to average prediction precision under a specified use distribution. If a study has several important objectives, choosing one criterion necessarily prioritizes one summary. A single optimum cannot be claimed to be universally best.
Common Mistakes and Limitations
- Calling a design “best” without naming the criterion. State the target and criterion; otherwise the claim is incomplete.
- Assuming D-optimality makes every variance smallest. It minimizes a determinant, not each diagonal element or every contrast variance.
- Confusing worst-case and average prediction. G-optimality uses a maximum over the region; I-optimality averages according to a measure.
- Using a prediction criterion without specifying its region. The range of locations being protected is part of the question, not a cosmetic detail.
- Ignoring local and asymptotic approximations. Information-based rankings rely on the model and nominal values used to construct information, and approximate covariance need not equal finite-sample performance.
These criteria are summaries of a model-based information calculation, not guarantees of realized performance. Misspecification, parameter values far from the nominal point, practical constraints, or an unsuitable prediction region can weaken the relevance of the result. The appropriate criterion is therefore the one whose inferential target and assumptions are defensible for the study—not the one with the most appealing label.
Check Your Understanding
Answer each question by identifying the objective and showing the key calculation or reasoning.
- A design has a larger determinant of information than a competitor. Which criterion does that directly favor, and what does it not establish?
- For a prespecified contrast \(c^T\theta\), what quantity should be compared across designs?
- A design has lower average prediction variance over the planned use region, but a higher maximum variance there. Which of G- and I-optimality favors it?
- Why must candidate designs be evaluated under the same parameterization and model assumptions?
Solutions. (1) It favors D-optimality, which maximizes \(\det M(\xi)\); it does not establish that every parameter variance or contrast variance is lower. (2) Compare \(c^TM(\xi)^{-1}c\); the smaller value means lower approximate variance for that contrast. (3) I-optimality favors the design because it minimizes average prediction variance under the stated measure; G-optimality instead favors the design with the lower maximum. (4) The criteria summarize information and its implied uncertainty, so changing the parameterization, model, or nominal assumptions changes the quantities being compared and may change their rankings.