D-Optimality And Generalized Variance in Optimal Design

Why Compare Designs by a Determinant?

A design for estimating several parameters must balance information across the parameter space. A design can provide high precision for one parameter direction and low precision for another; a single diagonal entry of its information matrix does not summarize that trade-off. D-optimality provides a scalar summary: it ranks designs by the determinant of their information matrices. Its geometric interpretation is especially useful: under standard large-sample conditions, maximizing that determinant minimizes the volume of a joint confidence ellipsoid for the parameters.

The comparison is meaningful only when the candidate designs use the same model, parameterization, nominal parameter values when required, and resource budget. In nonlinear models, information is generally local to specified nominal parameter values. As in the earlier material on approximate designs, let a design allocate observations or sampling effort across allowable conditions, and let its information matrix summarize the resulting information about the model parameters.

The matrix should be symmetric and positive definite for a regular, locally identifiable \(p\)-parameter problem. Positive definiteness means that every nonzero parameter direction has positive information. A singular information matrix has determinant zero: at least one direction is unsupported, and the usual local covariance approximation cannot provide a finite-volume ellipsoid for all \(p\) parameters.

Generalized Variance and Ellipsoid Volume

Under regularity conditions, an estimator such as the maximum likelihood estimator is approximately multivariate normal for sufficiently informative data. In particular, its covariance is approximately \(\mathcal I(\xi)^{-1}\). The determinant of a covariance matrix is often called its generalized variance. Unlike a single marginal variance, it accounts for the joint spread and dependence among all parameter estimates.

An approximate \(100(1-\alpha)\%\) Wald confidence ellipsoid for a \(p\)-dimensional parameter vector \(\theta\) is centered at the estimate \(\hat{\theta}\). It contains parameter values whose information-weighted squared distance from that estimate is no greater than the corresponding chi-square quantile.

This is an approximate confidence region, not a statement that a fixed parameter has a probability \(1-\alpha\) of lying in the computed ellipsoid. Its repeated-sampling coverage is approximately \(1-\alpha\) when the model and regularity conditions are appropriate.

To see why its volume depends on the determinant, write \(c=\chi^2_{p,1-\alpha}\) and transform coordinates using \(y=\mathcal I(\xi)^{1/2}(\theta-\hat{\theta})\). The ellipsoid becomes the radius-\(\sqrt c\) ball \(y^\mathsf{T}y\leq c\). The inverse transformation has volume-scaling factor \(\det\{\mathcal I(\xi)^{1/2}\}^{-1}=\det\{\mathcal I(\xi)\}^{-1/2}\). Therefore its volume is the volume of a \(p\)-dimensional radius-\(\sqrt c\) ball multiplied by this factor.

Here \(V_p\) is the volume of the unit ball in \(p\) dimensions. For fixed confidence level and parameter dimension, \(V_p c^{p/2}\) is the same for every candidate. Thus, a larger information determinant means a smaller approximate confidence-ellipsoid volume and a smaller generalized variance. This conclusion concerns the joint region; it does not guarantee that every individual parameter variance is smaller.

Calculating and Interpreting the Criterion

For a two-parameter design with information matrix \(\begin{pmatrix}a&b\\b&d\end{pmatrix}\), the determinant is \(ad-b^2\). For larger matrices, use a reliable determinant calculation and retain enough precision to rank close candidates. Comparisons should use information matrices on the same parameter scale and for the same parameter ordering. The criterion has units that depend on those scales, but a common change of units multiplies every candidate determinant by the same positive factor, so it does not change their ranking.

Replication, Assumptions, and Diagnostics

If a design has per-observation information \(M(\xi)\) and is replicated \(N\) times independently under the same conditions, information adds, giving \(\mathcal I(\xi)=N M(\xi)\). For a \(p \times p\) matrix, multiplying every entry by \(N\) multiplies its determinant by \(N^p\). This makes the sampling budget essential: a design with more observations can have a larger determinant simply because it collected more information, rather than because it allocates a fixed budget better.

The ellipsoid interpretation and determinant ranking rely on the information matrix being appropriate for the model and design. In practice, check that the matrix is symmetric and positive definite, that all compared matrices describe the same parameters and budget, and that the assumed model and nominal parameter values are defensible. A singular matrix signals a missing estimable direction. A nearly singular matrix can yield a very elongated ellipsoid, so a determinant by itself may conceal severe uncertainty along a particular direction. Model misspecification or poor nominal values can also make a precisely computed determinant a poor guide to actual performance.

D-optimality is one summary of design quality, not a universal definition of it. It targets joint confidence-region volume. Other criteria express different priorities: A-optimality focuses on an average of parameter variances, E-optimality on the weakest information direction, and c-optimality on uncertainty in a specified parameter contrast. Those priorities can rank the same candidate designs differently. D-optimality is useful when broad joint precision is the goal, but it need not be best for a single clinical decision or a particular parameter.

Common Pitfalls

• Choosing the matrix with the largest diagonal entries. This ignores off-diagonal terms and the joint geometry. Calculate the determinant rather than comparing selected entries.
• Treating the determinant as an individual-parameter precision measure. It summarizes joint volume; inspect parameter-specific uncertainty when that is the scientific objective.
• Comparing designs with different budgets as if they were equally resourced. Replication scales the determinant by \(N^p\). State and enforce the resource constraint.
• Using the criterion for a singular matrix as though it described a finite ellipsoid. A zero determinant indicates at least one unidentified direction under the local information approximation; resolve the design or model issue before making a full-parameter comparison.