Insights into the Relationship Between D- and A-optimal Designs

This paper establishes that the A-optimality criterion can be factored into a D-optimal scale term and a dimensionless sphericity factor dependent on eigenvalue dispersion, thereby explaining performance differences among D-equivalent designs and providing a lightweight method for refining candidate pools.

The Gist

Imagine you are an architect designing a house. Your goal is to make the house as sturdy and useful as possible. In the world of statistics, "designing a house" means planning an experiment to gather data. The "sturdiness" of your experiment is measured by how well you can estimate the effects of different variables (like how much paint, wood, or labor affects the final cost).

This paper introduces a new way to look at two famous rules for building these experimental "houses": the D-criterion and the A-criterion.

Here is the simple breakdown using a few creative analogies.

1. The Two Rules: Volume vs. Shape

For a long time, statisticians have argued about which rule is better.

• The D-Optimal Rule (The "Volume" Rule): This rule tries to make the "room" of your uncertainty as small as possible. Imagine your uncertainty is a balloon. The D-rule tries to shrink the balloon's total volume. If the balloon is tiny, you know a lot.
• The A-Optimal Rule (The "Average Error" Rule): This rule tries to minimize the average error across all your measurements. It cares about how far off your guesses might be on average.

The Problem:
Sometimes, two different experimental designs have the exact same balloon volume (they are tied in the D-rule). But, one design is a perfect sphere, while the other is a long, skinny, weirdly shaped balloon.

• The Sphere is great: your errors are balanced in every direction.
• The Skinny Balloon is bad: you are very precise in some directions but terrible in others.

The D-rule can't tell the difference between the sphere and the skinny balloon because they have the same volume. The A-rule can tell the difference, but until now, nobody had a simple way to explain why they were different.

2. The Big Discovery: Breaking the Equation Down

The authors of this paper found a mathematical "magic trick." They showed that the A-rule is actually just two things multiplied together:

$$\text{A-Rule} = \frac{\text{Volume Factor}}{\text{Shape Factor}}$$

• The Volume Factor: This is just the D-rule (how small the balloon is).
• The Shape Factor (Sphericity): This is a new number they call the Sphericity Index. It measures how "round" or "balanced" your balloon is.

The Analogy:
Think of a basketball team.

• D-Optimality asks: "How many total points did the team score?" (Total volume).
• A-Optimality asks: "What was the average points per player?" (Total volume divided by balance).

If Team A and Team B both scored 100 points total (D-tie), but Team A has five players who each scored 20 points, and Team B has one player who scored 90 points and four who scored 2.5 points... Team A is much more balanced.

The paper says: If you have the same total points (D), the team with the most balanced scoring (Sphericity) will have the best average performance (A).

3. Why This Matters in Real Life

The paper uses this idea to solve three real-world headaches:

A. The "Tie-Breaker"

Sometimes, software gives you two designs that look identical in the D-rule. You don't know which one to pick.

• Old way: Flip a coin.
• New way: Look at the Sphericity Index. Pick the one with the higher score (the rounder balloon). This guarantees you get better predictions and less confusion about your results.

B. The "Infinite Options" Problem

In some complex experiments, there are literally infinite ways to get the perfect D-score. It's like having infinite ways to arrange furniture in a room so the total area is the same.

• The paper shows that even though the "area" is the same, the "flow" of the room (the shape) changes.
• By using the Sphericity Index, you can instantly spot the "perfectly round" arrangement among the infinite options, rather than getting lost in the math.

C. The "Space-Filling" Filter

Sometimes, you want to scatter points evenly across a map (like sprinkling seeds in a garden) without worrying about a specific formula first. This is called a "Space-Filling Design."

• The authors suggest a simple trick: Generate 500 random garden layouts.
• First, keep the ones that look the most evenly scattered (MaxPro).
• Then, among those, pick the one with the best Sphericity (the most balanced "balloon").
• This ensures your garden is not only spread out but also mathematically robust if you later decide to analyze specific plant growth patterns.

4. The Takeaway

The paper is essentially saying: "Don't just look at the size of your uncertainty; look at its shape."

• D-Optimality tells you how big your net is.
• Sphericity tells you if the net has holes in it.

You can have a huge net (good D-score), but if it has giant holes (bad shape), you'll still miss the fish. By combining the two, you get a net that is both big and perfectly woven.

In short: When two designs look the same on paper, use this new "Sphericity" score to find the one that is actually more balanced, reliable, and fair to your data.

Technical Summary

Here is a detailed technical summary of the paper "Insights into the Relationship Between D- and A-optimal Designs" by Andrew T. Karl and Bradley Jones.

1. Problem Statement

In the design of experiments (DoE), particularly for screening, researchers often face a dilemma between D-optimality (maximizing the determinant of the information matrix, $X^\top X$) and A-optimality (minimizing the trace of the covariance matrix, $(X^\top X)^{-1}$).

• The Issue: D-optimality is widely used because it minimizes the volume of the joint confidence ellipsoid. However, D-optimal designs often suffer from "ties" (multiple designs with identical or near-identical D-values).
• The Consequence: These D-ties can exhibit significantly different behaviors regarding coefficient variances, aliasing, and prediction variance (G-criterion). Existing literature notes that D-optimality alone is insufficient to rank screening designs when ties occur, yet a compact theoretical explanation for why A-optimality distinguishes these ties has been lacking.
• Goal: The authors aim to provide a mathematical decomposition that isolates the specific component of A-optimality that D-optimality ignores, thereby explaining the performance gap between D-ties.

2. Methodology

The paper employs a spectral (eigenvalue) analysis of the information matrix $C = X^\top X$ and its inverse (the covariance matrix).

• Definitions:

  • Let $\mu_i$ be the eigenvalues of $C$ (information eigenvalues).
  • Let $\lambda_i = \mu_i^{-1}$ be the eigenvalues of $C^{-1}$ (covariance eigenvalues).
  • D-criterion: $D(X) = (\det C)^{1/p} = \text{GM}(\mu)$ (Geometric Mean of information eigenvalues).
  • A-criterion: $A(X) = \text{tr}(C^{-1}) = \sum \lambda_i = p \cdot \text{AM}(\lambda)$ (Arithmetic Mean of covariance eigenvalues).

• The Core Derivation:
  The authors define a Sphericity Index, $S(C)$, based on the ratio of the geometric mean to the arithmetic mean of the eigenvalues (related to Mauchly's W statistic):
  $$S(C) = \frac{\text{GM}(\lambda)}{\text{AM}(\lambda)} = \frac{(\det C^{-1})^{1/p}}{\text{tr}(C^{-1})/p}$$
  This index is dimensionless, scale-invariant, and bounded $0 < S(C) \leq 1$. It equals 1 if and only if all eigenvalues are equal (perfect sphericity).

• The Factorization:
  By rearranging the definitions, the authors derive the fundamental identity:
  $$A(X) = \frac{p}{D(X)} \cdot \frac{1}{S(C)}$$
  This equation decomposes A-optimality into two distinct components:

  1. Scale Term: $\frac{p}{D(X)}$ (Inverse-D scale).
  2. Shape Penalty: $\frac{1}{S(C)}$ (Dependent solely on spectral balance/eigenvalue dispersion).

3. Key Contributions

A. Theoretical Decomposition

The primary contribution is the proof that A-optimality is simply the inverse of the D-scale multiplied by a sphericity penalty.

• This explains why designs tied on D can differ on A: if $D$ is fixed, $A$ varies entirely based on $S(C)$.
• A higher $S(C)$ (closer to 1) indicates a flatter, more uniform covariance spectrum, which lowers the A-criterion.
• This clarifies that D-optimality controls the volume of the confidence ellipsoid, while A-optimality (via $S$) controls the roundness (isotropy) of that ellipsoid.

B. Extension to Kiefer's $\Phi$-Class

The authors generalize this scale/shape separation to Kiefer's broader class of optimality criteria ($\Phi_r$).

• They define a Sphericity Profile $S_r(C) = \Phi_r(C) / \Phi_0(C)$.
• This creates a continuous bridge from D-optimality ($r=0$) to A-optimality ($r=-1$) and E-optimality ($r=-\infty$), showing that the "shape" component is a universal concept across these criteria.

C. Practical Application: Space-Filling Post-Screen

The paper proposes a lightweight workflow for Space-Filling Designs (e.g., MaxPro designs) where no specific linear model is initially assumed.

• Method: Generate a pool of space-filling candidates (e.g., using Fast Flexible Filling).
• Selection: Instead of optimizing A directly, use a composite score:
  $$\text{Score}(\xi) = \frac{\text{MaxPro}(\xi)}{S(\xi)}$$
• Logic: MaxPro ensures good geometric coverage (spacing), while $S(\xi)$ (computed under a working model) acts as a secondary filter to select designs with balanced spectral properties, avoiding candidates that are space-filling but have poor prediction variance in specific directions.

4. Results and Illustrations

The authors validate their theory using three specific examples:

1. Augmentation Example (Figure 1):

   • Comparing a D-optimal vs. A-optimal augmentation of a seed design.
   • Result: The A-optimal design had a slightly lower D-efficiency (23.03 vs. 24.25) but a significantly higher sphericity (0.575 vs. 0.464).
   • Outcome: The A-optimal design yielded a "rounder" confidence ellipse and better prediction variance (G-efficiency), demonstrating that the small sacrifice in D-scale was worth the gain in spectral balance.

2. Published D-Tie (Jones et al., 2021):

   • Analyzed a case where A-optimal and D-optimal designs were an exact tie in D.
   • Result: The A-optimal design had a higher sphericity index (0.973 vs. 0.945).
   • Outcome: The A-optimal design distributed covariance more evenly, resulting in a 20% better G-efficiency (40.00 vs. 57.14, where lower is better for G, but the table shows Aeff/Geff values; note: the text states the A-optimal has "markedly better Geff"). The normalized eigenvalue spectrum was flatter for the A-optimal design.

3. Infinite D-Ties (Stallrich et al., 2023):

   • Examined a scenario with infinitely many D-optimal designs.
   • Result: Only one design maximized $S(C)$.
   • Outcome: The unique A-optimal design had the flattest spectrum. Other D-optimal variations concentrated mass in specific covariance directions, leading to significantly worse prediction variance (G-efficiency) despite having identical D-values.

4. Space-Filling Pool (Section 5):

   • In a pool of 500 FFF candidates, MaxPro and Sphericity showed a weak correlation (-0.16).
   • Outcome: Selecting candidates based on the composite score $\text{MaxPro}/S$ successfully identified designs with superior prediction standard deviation contours and lower coefficient correlations compared to those selected by MaxPro alone.

5. Significance

• Diagnostic Clarity: The paper provides a rigorous mathematical reason why D-optimality is insufficient for screening when ties occur. It isolates "spectral balance" as the critical missing factor.
• Interpretability: The sphericity index $S$ is directly available in software like JMP (calculated as $A_{eff} / D_{eff}$), making it an immediately actionable metric for practitioners.
• Workflow Improvement: The proposed "post-screen" methodology allows practitioners to retain the robustness of space-filling designs while ensuring they are also optimal for specific working models, without needing complex global optimization algorithms.
• Theoretical Unification: By linking A-optimality to Kiefer's $\Phi$-class via sphericity profiles, the paper unifies the understanding of various optimality criteria, showing they differ primarily in how they penalize spectral imbalance.

In summary, the paper argues that D-optimality controls the size of the uncertainty region, while Sphericity controls its shape. For screening and prediction tasks, a design with a slightly smaller volume but a more spherical shape (higher $S$) is often superior to a design with a larger volume but a distorted shape.