Why an Equivalence Theorem Helps
A D-optimal design maximizes the determinant of its information matrix, or equivalently its log determinant, among designs on the allowed design space. Directly comparing every possible design can be difficult: a design may distribute weight over many candidate points, or over a continuum of possible points. The D-optimal equivalence theorem replaces that global comparison with a check at individual candidate points. If no candidate point has sensitivity above the parameter dimension, the proposed design is D-optimal. At points in its support, sensitivity reaches that bound.
This is more than a convenient computational rule. The theorem explains why the bound is the number of parameters and why a violation indicates a feasible way to improve the design. The result applies to the approximate-design information matrices introduced in the earlier material on information matrices for approximate designs.
Set-Up and Definitions
The design space is assumed to be compact, and \(M(x)\) is assumed to be continuous and positive semidefinite on that space. The model dimension \(p\) is fixed, and at least one feasible design has a positive-definite information matrix. The proposed design being tested must itself have positive-definite information; otherwise its determinant is zero and the inverse in the sensitivity definition does not exist. These assumptions make the criterion well-defined and ensure that sensitivity varies continuously across the design space.
The D-optimal criterion is \(\det M(\xi)\), with larger values preferred. Since the logarithm is strictly increasing, maximizing the determinant is equivalent to maximizing \(\log\det M(\xi)\) among designs with positive-definite information. The approximate-design convention treats \(\xi\) as a probability measure, so the information is per design unit; multiplying every design’s information by the same replication count does not change which design maximizes the criterion.
The Equivalence Theorem
The bound applies throughout the design space, including points assigned zero weight. The equality condition describes where the design places weight; it does not say that every point attaining equality must have positive weight. Several points outside the support can also attain the bound.
Why the Bound Is \(p\): A Derivation
The key argument compares \(\xi\) with a design that shifts a small amount of its weight toward one candidate point \(x\). Let \(\delta_x\) denote the design placing all its weight at \(x\), and define the mixture $$ \xi_\varepsilon=(1-\varepsilon)\xi+\varepsilon\delta_x,\qquad 0\leq\varepsilon\leq1. $$ Its information matrix is \(M(\xi_\varepsilon)=(1-\varepsilon)M(\xi)+\varepsilon M(x)\). For a positive-definite matrix \(A(\varepsilon)\), the derivative of its log determinant is \(\operatorname{tr}(A(\varepsilon)^{-1}A'(\varepsilon))\). Applying this at \(\varepsilon=0\) gives $$ \left.\frac{d}{d\varepsilon}\log\det M(\xi_\varepsilon)\right|_{\varepsilon=0} =\operatorname{tr}\!\left(M(\xi)^{-1}[M(x)-M(\xi)]\right) =d(x,\xi)-p. $$ Here \(\operatorname{tr}(M(\xi)^{-1}M(\xi))=\operatorname{tr}(I_p)=p\). If \(d(x,\xi)>p\), the derivative is positive: moving a small amount of design weight toward \(x\) initially increases the log determinant. Therefore a D-optimal design cannot have sensitivity above \(p\) anywhere.
For sufficiency, suppose instead that \(d(x,\xi)\leq p\) for every \(x\). The log determinant is concave on the positive-definite matrices. Its tangent at \(M(\xi)\) therefore bounds its value at any other positive-definite information matrix \(M(\eta)\): $$ \log\det M(\eta)-\log\det M(\xi) \leq \operatorname{tr}\!\left(M(\xi)^{-1}[M(\eta)-M(\xi)]\right) =\int_{\mathcal{X}}d(x,\xi)\,d\eta(x)-p \leq 0. $$ Thus no competing design \(\eta\) has a larger log determinant. A competing design with singular information has determinant zero and cannot outperform the positive determinant of \(\xi\). This proves sufficiency.
Finally, the sensitivity averages to \(p\) under the design itself: $$ \int_{\mathcal{X}}d(x,\xi)\,d\xi(x) =\operatorname{tr}\!\left(M(\xi)^{-1}\int_{\mathcal{X}}M(x)\,d\xi(x)\right) =\operatorname{tr}(I_p)=p. $$ At an optimal design, sensitivity is nowhere above \(p\), so its average can equal \(p\) only if it equals \(p\) design-almost everywhere. With continuity, any support point where sensitivity were strictly below \(p\) would have a neighborhood with positive design measure and sensitivity bounded below \(p\), contradicting that average. Hence equality holds at every point in the topological support.
Worked Example: Endpoint Design for a Linear Regression
Consider a two-parameter linear regression on \(\mathcal{X}=[-1,1]\), with elemental information \(M(x)=f(x)f(x)^\mathsf{T}\) and \(f(x)=(1,x)^\mathsf{T}\). The model dimension is \(p=2\). Take the design \(\xi\) that assigns weight \(1/2\) to each endpoint.
The information matrix is $$ M(\xi)=\tfrac12 \begin{pmatrix}1&-1\\-1&1\end{pmatrix} +\tfrac12 \begin{pmatrix}1&1\\1&1\end{pmatrix} = \begin{pmatrix}1&0\\0&1\end{pmatrix}. $$ It is positive definite. Its inverse is also the identity, so at any candidate point \(x\), $$ d(x,\xi)=\operatorname{tr}\!\left(I_2 f(x)f(x)^\mathsf{T}\right) =f(x)^\mathsf{T}f(x)=1+x^2. $$
For all \(x\in[-1,1]\), \(1+x^2\leq2=p\). At \(x=-1\) and \(x=1\), \(d(x,\xi)=2\), so equality holds at both support points. The equivalence theorem establishes that this endpoint design is D-optimal over the full interval, not merely among designs that use the endpoints.
Worked Example: A Three-Point Design That Fails the Bound
Use the same two-parameter regression model, but assign weight \(1/3\) to each of \(-1,0,1\). The design has \(p=2\), and its information matrix is $$ M(\xi)=\tfrac13 \begin{pmatrix}1&-1\\-1&1\end{pmatrix} +\tfrac13 \begin{pmatrix}1&0\\0&0\end{pmatrix} +\tfrac13 \begin{pmatrix}1&1\\1&1\end{pmatrix} = \begin{pmatrix}1&0\\0&2/3\end{pmatrix}. $$ Its determinant is \(2/3\), and its inverse is \(\begin{pmatrix}1&0\\0&3/2\end{pmatrix}\). Thus $$ d(x,\xi)=f(x)^\mathsf{T}M(\xi)^{-1}f(x)=1+\tfrac32x^2. $$
At either endpoint, \(d(x,\xi)=1+3/2=2.5\), which exceeds \(p=2\). The design is therefore not D-optimal. The derivative for shifting weight toward an endpoint is \(d(x,\xi)-p=0.5>0\), confirming that such a shift initially improves the log determinant. As a direct comparison, the equal-weight endpoint design has information \(I_2\) and determinant \(1\), larger than \(2/3\).
Worked Example: Equality on the Support and Strict Inequality Elsewhere
Suppose a finite design space has three candidate points with elemental information matrices $$ M(A)=\begin{pmatrix}2&0\\0&0\end{pmatrix},\qquad M(B)=\begin{pmatrix}0&0\\0&2\end{pmatrix},\qquad M(C)=\begin{pmatrix}1/2&0\\0&1/2\end{pmatrix}. $$ Consider the design \(\xi\) assigning weight \(1/2\) to \(A\), weight \(1/2\) to \(B\), and zero to \(C\). Its support is \(\{A,B\}\), and $$ M(\xi)=\tfrac12M(A)+\tfrac12M(B)=I_2. $$ The information is positive definite and \(p=2\). Since \(M(\xi)^{-1}=I_2\), the sensitivity at \(A\) is \(\operatorname{tr}(M(A))=2\); at \(B\) it is \(\operatorname{tr}(M(B))=2\); and at \(C\) it is \(\operatorname{tr}(M(C))=1\).
The bound holds at all three candidate points. Equality holds at both support points, while the unused point has strict inequality. The theorem establishes D-optimality over this three-point design space. Strict inequality at \(C\) is consistent with optimality: it means that shifting weight toward \(C\) is not a locally improving direction.
Applying the Test Carefully
For a finite candidate set, the test is direct: calculate \(M(\xi)\), verify that it is positive definite, calculate its inverse, and evaluate \(d(x,\xi)\) at every candidate point. The candidate with the largest value determines whether the bound is met. For a continuous design space, the same condition applies everywhere. Checking only a coarse grid provides evidence, not a proof, unless a separate argument establishes that the maximum over the continuous space occurs at the checked points.
If a candidate point has sensitivity above \(p\), the derivative calculation identifies the immediate interpretation: adding a small amount of weight at that point increases the log determinant to first order. This is a certificate that the proposed design is not optimal, not necessarily a complete recipe for the globally optimal weights. Conversely, a sensitivity bound that is met everywhere, together with the stated assumptions, is a global certificate of D-optimality.
Common Mistakes and Limitations
- Checking only the support. Equality on the support is necessary, but not enough to establish optimality. The bound must also hold at every point outside the support.
- Reversing the inequality. A sensitivity greater than \(p\) signals a possible improvement by shifting weight toward that point; it is not evidence that the point should be excluded.
- Using the wrong dimension. The bound is \(p\), the dimension of the information matrix being optimized. It is not the number of support points or the number of observations.
- Defining support only by positive point weights. That works for discrete designs but fails for continuous design measures. The general definition uses neighborhoods with positive design measure; for a discrete design it reduces to the points with positive weights.
- Treating a grid check as a continuous-space proof. A grid can miss a narrow region where sensitivity exceeds \(p\). A continuous-domain maximum needs to be checked or bounded over the whole domain.
- Applying the result to singular information. If \(M(\xi)\) is singular, the inverse and sensitivity used by the theorem are unavailable. A positive-definite feasible design is required for the stated characterization.
The equivalence theorem is specific to the D criterion and its log-determinant geometry. A-optimality, E-optimality, and prediction-based criteria assess other features of information and have their own optimality conditions; satisfying the D-optimal sensitivity bound does not establish optimality for those alternatives. The theorem is also an approximate-design result. An approximate design may prescribe fractional weights, and its optimality does not by itself prove that a rounded, finite exact allocation is optimal.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For a three-parameter model, what sensitivity bound must a D-optimal design satisfy at every candidate point?
- A proposed design has positive-definite information and a candidate point has sensitivity \(p+0.02\). What does the theorem imply, and what is the directional derivative of log determinant for shifting weight toward that point?
- Why is equality of sensitivity to \(p\) on every support point not, by itself, a sufficient optimality check?
- A continuous design assigns zero measure to each individual point in an interval. Does that mean its support is empty? State the appropriate support definition.
- For \(p=2\), suppose the sensitivity is at most \(2\) everywhere and equals \(2\) at every support point. Under the theorem’s assumptions, what conclusion follows?
Solutions. (1) The bound is \(d(x,\xi)\leq3\) for every candidate point. (2) The design is not D-optimal because the bound is violated. The directional derivative is \(d(x,\xi)-p=(p+0.02)-p=0.02>0\), so a small shift of weight toward that point initially increases log determinant. (3) The bound must also hold at every point outside the support; an omitted point with sensitivity above \(p\) would show the design is improvable. (4) No. The support is the set of points every neighborhood of which has positive design measure, so a continuous uniform design on an interval has that interval as its support. (5) The design is D-optimal on the stated design space. The global bound gives sufficiency, and equality on the support is the corresponding support condition.