Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

EW D-optimal Designs for Experiments with Mixed Factors

Siting Lin, Yifei Huang, Jie Yang

TL;DR

This work develops and analyzes Expected Weighted D-optimal designs for experiments with mixed discrete and continuous factors under general parametric models. It extends the ForLion framework to EW criteria, providing two practical routes—sample-based and integral-based—plus a rounding method to yield exact designs on user-specified grids. The authors establish theoretical guarantees on the existence and sparsity of EW designs, present efficient computational procedures for multinomial logistic models and GLMs, and demonstrate substantial robustness and efficiency gains in a real paper feeder example and multiple supplementary cases. The approach offers a practical, scalable toolkit for robust experimental design in mixed-factor settings, with software implementation in R.

Abstract

We characterize EW D-optimal designs as robust designs against unknown parameter values for experiments under a general parametric model with discrete and continuous factors. When a pilot study is available, we recommend sample-based EW D-optimal designs for subsequent experiments. Otherwise, we recommend EW D-optimal designs under a prior distribution for model parameters. We propose an EW ForLion algorithm for finding EW D-optimal designs with mixed factors, and justify that the designs found by our algorithm are EW D-optimal. To facilitate potential users in practice, we also develop a rounding algorithm that converts an approximate design with mixed factors to exact designs with prespecified grid points and the total number of experimental units. By applying our algorithms for real experiments under multinomial logistic models or generalized linear models, we show that our designs are highly efficient with respect to locally D-optimal designs and more robust against parameter value misspecifications.

EW D-optimal Designs for Experiments with Mixed Factors

TL;DR

This work develops and analyzes Expected Weighted D-optimal designs for experiments with mixed discrete and continuous factors under general parametric models. It extends the ForLion framework to EW criteria, providing two practical routes—sample-based and integral-based—plus a rounding method to yield exact designs on user-specified grids. The authors establish theoretical guarantees on the existence and sparsity of EW designs, present efficient computational procedures for multinomial logistic models and GLMs, and demonstrate substantial robustness and efficiency gains in a real paper feeder example and multiple supplementary cases. The approach offers a practical, scalable toolkit for robust experimental design in mixed-factor settings, with software implementation in R.

Abstract

We characterize EW D-optimal designs as robust designs against unknown parameter values for experiments under a general parametric model with discrete and continuous factors. When a pilot study is available, we recommend sample-based EW D-optimal designs for subsequent experiments. Otherwise, we recommend EW D-optimal designs under a prior distribution for model parameters. We propose an EW ForLion algorithm for finding EW D-optimal designs with mixed factors, and justify that the designs found by our algorithm are EW D-optimal. To facilitate potential users in practice, we also develop a rounding algorithm that converts an approximate design with mixed factors to exact designs with prespecified grid points and the total number of experimental units. By applying our algorithms for real experiments under multinomial logistic models or generalized linear models, we show that our designs are highly efficient with respect to locally D-optimal designs and more robust against parameter value misspecifications.
Paper Structure (25 sections, 13 theorems, 28 equations, 4 figures, 11 tables, 2 algorithms)

This paper contains 25 sections, 13 theorems, 28 equations, 4 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. EW D-optimal Designs with Mixed Factors
  3. EW D-optimal designs
  4. Characteristics of EW D-optimal designs
  5. Algorithms for EW D-optimal Designs
  6. EW lift-one algorithm for discrete factors only
  7. EW ForLion algorithm and a rounding algorithm
  8. Calculating $E\{{\mathbf F}({\mathbf x}, \boldsymbol{\Theta})\}$ or $\hat{E}\{{\mathbf F}({\mathbf x}, \boldsymbol{\Theta})\}$
  9. First-order derivative of sensitivity function
  10. Application to Paper Feeder Experiment
  11. Conclusion and Discussion
  12. Analytic Solutions for Multinomial Logistic Models
  13. Assumptions and Relevant Results
  14. Proofs of Main Theorems
  15. Model Selection and Design Comparison for Paper Feeder Experiment
  16. ...and 10 more sections

Key Result

Lemma 1

Under Assumptions (A1) and (A2), ${\cal F}_{\rm SEW}({\cal X})$ is convex and compact; while under Assumptions (A1) and (A3), ${\cal F}_{\rm EW}({\cal X})$ is convex and compact.

Figures (4)

  • Figure S.1: Relative efficiency and number of design points of exact designs against relative distance (or merging threshold) for paper feeder experiment
  • Figure S.2: Boxplots of relative efficiencies of designs in Table \ref{['tab:Efficiencies of designs for paper feeder experiment']} with respect to locally D-optimal designs under 100 different sets of parameter values for paper feeder experiment
  • Figure S.3: Relative efficiency comparison among designs for 10,000 sampled parameter sets in the minimizing surface defects experiment
  • Figure S.4: Frequency polygons of objective function values based on 10,000 simulated parameter vectors for electrostatic discharge experiment

Theorems & Definitions (29)

  • Example 1
  • Example 2
  • Example 3
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Corollary 1
  • Example 4
  • ...and 19 more