[Paper Review] Optimal Designs for Two-Level Factorial Experiments with Binary Response

Abhyuday Mandal, Jie Yang+1 more|arXiv (Cornell University)|Mar 8, 2010

Optimal Experimental Design Methods3 citations

TL;DR

This paper develops locally D-optimal designs for two-level factorial experiments with binary responses using generalized linear models. It derives analytic solutions for special cases in 2² designs and demonstrates that the uniform design is highly efficient and maximin D-optimal for general 2ᵏ experiments, offering robustness and practical utility in screening applications.

ABSTRACT

We consider the problem of obtaining locally D-optimal designs for factorial experiments with qualitative factors at two levels each with binary response. Our focus is primarily on the 2^2 experiment. In this paper, we derive analytic results for some special cases and indicate how to handle the general case. The performance of the uniform design in examined and we show that this design is highly efficient in general. For the general 2^k case we show that the uniform design has a maximin property.

Motivation & Objective

• To establish optimal design theory for two-level factorial experiments with binary responses, where traditional linear models are inapplicable.
• To address the challenge of D-optimality in generalized linear models with binary outcomes, particularly when the information matrix depends on unknown parameters.
• To evaluate the performance and robustness of the uniform design across various configurations of the 2² and 2ᵏ factorial experiments.
• To provide analytic solutions for special cases and computational strategies (e.g., cylindrical algebraic decomposition) for general cases.
• To demonstrate that the uniform design achieves a maximin D-optimality property in the general 2ᵏ setting, ensuring robustness under parameter uncertainty.

Proposed method

• Uses the local optimality approach of Chernoff (1953), replacing unknown parameters with assumed values to define the D-optimality criterion.
• Applies the D-criterion to maximize the determinant of the asymptotic information matrix, derived from the Fisher information in generalized linear models.
• Employs cylindrical algebraic decomposition (CAD) to solve the non-convex optimization problem for general 2² designs where analytic solutions are not available.
• Considers common link functions (logit, probit, log-log, complementary log-log) in the generalized linear model framework.
• Analyzes the uniform design by evaluating its D-efficiency and robustness across parameter configurations using the maximin criterion.
• Derives bounds on the D-optimality criterion using inequalities (e.g., AM-GM) and proves optimality conditions via Lagrangian and perturbation analysis.

Experimental results

Research questions

• RQ1 What are the analytic forms of locally D-optimal designs for 2² factorial experiments with binary responses under main-effects models?
• RQ2 How does the performance of the uniform design compare to other designs in terms of D-efficiency across different parameter configurations?
• RQ3 Can the uniform design be shown to be maximin D-optimal in the general 2ᵏ factorial setting?
• RQ4 What conditions on the parameter vector ensure that the uniform design is optimal or near-optimal?
• RQ5 How can the optimal design be computed when analytic solutions are not available, particularly in the 2² case?

Key findings

• For the 2² design with main effects, the uniform design achieves a D-efficiency of at least 1 - (3/4) × 2^{1/3} ≈ 0.442 when the main effect variance is balanced.
• When the main effect variance v₁ ≥ v₂ + v₃ + v₄, the optimal design is non-uniform with p₁ = 0 and p₂ = p₃ = p₄ = 1/3, achieving L = v₁/27.
• The uniform design is maximin D-optimal for general 2ᵏ experiments, meaning it maximizes the lower bound of the D-criterion across all parameter values.
• The uniform design is highly efficient in general, with robust performance even when the true parameter values deviate from assumed values.
• When v₁ = v₂ + v₃ + v₄, the D-efficiency of the uniform design is exactly 1 - (3/4) × 2^{1/3} ≈ 0.442, and this value is the minimum achievable under the maximin criterion.
• Perturbation analysis shows that if v₁ < v₂ + v₃ + v₄, a small increase in p₁ and proportional decrease in p₂, p₃, p₄ can improve the D-criterion, indicating that uniformity is not optimal in this regime.