Average D-Optimal (EW) Design Method

• The average D-optimal design method is a statistical framework that extends classical D-optimality by incorporating parameter uncertainty through expected weighting of information.
• It uses the efficient EW ForLion algorithm to optimize design weights and reduce support points in complex experimental settings with both continuous and discrete factors.
• The method achieves high efficiency and robustness, supporting both sample-based and integral-based approaches to ensure reliable experimental performance.

The average D-optimal design method, often referred to as the EW (Expected–Weighted) D-optimal design framework, extends classical D-optimality to the setting where model parameters are not fixed but instead treated as uncertain and distributed according to either a prior distribution or empirical estimates from pilot data. This approach is particularly relevant in modern experimental design for complex systems with both continuous and discrete factors, offering robust design strategies under parameter uncertainty. Its theoretical underpinnings, efficient computational algorithms, and empirical performance reflect ongoing advances in statistical experimental design, information theory, and optimization.

1. Formal Definition and Problem Setting

Let $𝓧 \subset \mathbb{R}^d$ denote the design region, possibly containing a mixture of continuous and discrete factors. The response model at design point $x \in 𝓧$ is given by a parametric model $M(x,θ)$, where $θ \in Θ \subset \mathbb{R}^p$ represents the unknown parameters. For each $x$, define the per-unit Fisher information as $F(x, θ) \in \mathbb{R}^{p \times p}$. For a design $\xi$ with support $\{x_i\}$ and weights $w_i$ (where $w_i = n_i/n$, $n_i$ units allocated, $\sum w_i = 1$), the total Fisher information for parameter $θ$ is

$M(\xi, θ) = \sum_i w_i F(x_i, θ).$

In the presence of parameter uncertainty, either a prior $Q(dθ)$ on $Θ$ or a set of $B$ bootstrap/pilot estimates $\{\hat θ_j\}_{j=1}^B$ is available.

Define the average (EW) information:

• Integral-based EW: $E[F(x)] = \int_{Θ} F(x,θ) Q(dθ)$,
• Sample-based EW: $$\bar F(x) = \frac{1}{B} \sum_{j=1}^B F(x, \hat θ_j)$$.

The EW (Expected–Weighted) information matrix for design $\xi$ is then

$\bar M(\xi) = \sum_i w_i E[F(x_i)].$

The EW D-optimality criterion is to maximize

$f_{EW}(\xi) = \det \bar M(\xi),$

that is, to find

$\xi^* = \arg\max_{\xi} \det \bar M(\xi).$

2. The EW ForLion Algorithm

The EW ForLion algorithm generalizes traditional D-optimal algorithms to the EW setting, maintaining computational efficiency even for large, mixed-factor design spaces. The optimization is over the set of approximate designs (weights in the probability simplex), with support reduced by merging and grid-rounding as needed for an exact design.

Algorithm outline:

1. Initialization: Start with an initial design $\xi_0 = \{(x_i^{(0)}, w_i^{(0)})\}$ with distinct $x_i$ (minimum separation $\delta$) and full-rank $\bar M_{\xi_0}$.
2. Support Reduction (Merging): While there exists $x_i, x_j$ with $\|x_i - x_j\| < \delta$, attempt to merge to $x̄ = (x_i + x_j)/2$ with weight $w_i + w_j$, subject to maintaining $\det \bar M > 0$.
3. Weight Update (EW lift-one): Iteratively update weights $w_i$ using the EW variant of lift-one, optimizing $\log \det [\sum w_i \bar F_{x_i}]$ until weights converge ($|\Delta w| < \varepsilon$).
4. Pruning: Remove support points with $w_i = 0$.
5. Support Expansion (Addition): Compute the directional sensitivity $$d(x, \xi) = \operatorname{tr}[\, \bar M_{\xi}^{-1} \bar F_x \,]$$ for $x \in 𝓧$. If $\max_x d(x, \xi) > p$, add the maximizer $(x^*, 0)$ to the support and return to step 2. Otherwise, stop.

Optimality condition: By the General Equivalence Theorem, $\xi$ is EW D-optimal if and only if $\max_{x \in 𝓧} d(x, \xi) = p$.

Computational features: The method efficiently handles design spaces with both continuous and discrete factors by exploiting thresholding on proximity ($\delta$ for merging), grid projection for continuous factors, and sparse support from the iterative process.

3. Conversion from Approximate to Exact Designs

Practical experiments require integer allocations. The rounding algorithm maps the continuous solution (weights) to an exact design on a prescribed grid:

1. Merge design points within radius $\delta_r$ sharing the same levels for discrete factors.
2. Round each continuous coordinate to the nearest multiple of the grid step size $L_j$.
3. Allocate $n_i = \lfloor n w_i \rfloor$ units, with the remaining $r = n - \sum \lfloor n w_i \rfloor$ units distributed one at a time to the support points with largest $d(x_i, \xi) - p$ (maximal incremental gain in $\log \det \bar M$).
4. Discard support points with $n_i = 0$.

Theoretical performance: The resulting loss in efficiency relative to the optimal approximate design is $O(1/n)$ in $\log \det$, with observed efficiencies typically $95\%$–$99\%$.

4. Sample-Based vs. Integral-Based EW Designs

Average-case D-optimal designs hinge on whether parameter uncertainty is encoded as a subjective prior (integral-based) or as an empirical distribution from pilot or bootstrap estimates (sample-based):

• Sample-Based EW (SEW): Bootstrap or otherwise generate $B$ estimates $\hat θ_j$ from pilot data, compute $\bar F(x)$, and use these in the main experiment’s design. Empirically, SEW offers computational efficiency while closely tracking fully Bayesian D-optimal designs.
• Integral-Based EW: Integrate $F(x, θ)$ over a prior $Q$ (e.g., via Monte Carlo draws). Recommended when pilot data are unavailable and prior beliefs can be reliably quantified.

Guideline: Use SEW when sufficient pilot data are available; otherwise, employ integral-based EW with a well-specified prior.

5. Empirical Performance and Illustrative Examples

The EW D-optimal framework outperforms grid-based and locally optimal alternatives in both robustness and efficiency:

• Paper-feeder experiment (multinomial logistic, 8 discrete + 1 continuous): EW ForLion achieves $\sim98.5\%$ efficiency, with support size reduced from $183$ to $38$, and median relative efficiency in repeated parameter draws of $\sim0.936$ versus local D-optimal designs.
• Surface defects minimization (5 continuous + 1 discrete): Prior-based EW design yields much more robust performance than multiple grid-based alternatives, with efficiency concentrated near 1 across $10,000$ parameter draws.
• Electrostatic-discharge (binary logistic, 4 discrete + 1 continuous): Over $10,000$ draws, EW ForLion outperforms both local D-optimal and PSO approaches in mean and median log-determinant.
• 3-factor GLM (logistic, continuous domain): EW ForLion maintains high efficiency $(\sim99\%)$ relative to alternative approaches as the design is rounded.

General theoretical results: Under mild regularity (compact design space, continuity and boundedness of information), EW D-optimal designs exist with number of support points bounded by $p(p+1)/2$.

6. Robustness and Practical Implications

The average D-optimal design framework offers robust performance against parameter misspecification:

• Performance in expectation: All guarantees are with respect to the expected log-determinant over the parameter distribution.
• Robustness: EW designs consistently reduce the worst-case shortfall across the uncertainty in $θ$—outperforming approaches such as grid-based designs or those optimized for nominal parameter values.
• Scalability: The EW ForLion algorithm and rounding procedures scale to large, mixed-type design spaces, and the process is fully implemented in open-source R code.

A table summarizing design modes is provided below:

Mode	Data Requirement	Typical Use Case
Sample-based EW	Moderate pilot data	Sequential experimental campaigns
Integral-based EW	Prior distribution	First-stage or expert-driven research

7. Connections to Related Methods and Theoretical Guarantees

The EW D-optimal method fits within a broader class of robust experimental design techniques:

• Local D-optimality: Traditional criterion, not robust to parameter uncertainty.
• Bayesian D-optimality: Maximizes average information, computationally intensive for high-dimensional or complex models.
• EW D-optimality: Provides a computationally tractable surrogate to full Bayesian design, especially for mixed discrete-continuous domains.

Under general compactness and regularity, EW D-optimal designs satisfy the optimality system $\max_x d(x,\xi) = p$, and the associated rounding procedure achieves exact integer designs with negligible efficiency loss.

Summary: The average (EW) D-optimal design method provides a principled, robust, and computationally efficient framework for optimal experimental design under parameter uncertainty, unifying empirical and prior-based modes of information aggregation, and supported by theoretical optimality guarantees and practical algorithms for exact design construction (Lin et al., 1 May 2025).