Optimal Designs for Discrete Choice Models Via Graph Laplacians

Area of Science:

• Statistics
• Experimental Design
• Graph Theory

Background:

• Information matrix in discrete choice experiments is parameter-dependent, complicating optimal design.
• Nonlinear optimization problems often render optimal design infeasible for arbitrary initial parameters.

Purpose of the Study:

• To develop a computationally feasible method for optimal experimental design in discrete choice models.
• To reduce the complexity of optimal design problems by leveraging graph theory.

Main Methods:

• Connecting discrete choice design theory with Laplacian matrices of undirected graphs.
• Rewriting the D-optimality criterion using Kirchhoff's matrix tree theorem and Laplacian matrices.
• Utilizing the Cayley-Menger determinant of the Farris transform for dual description.
• Applying a gradient descent algorithm for locally D-optimal designs.
• Linking Bradley-Terry models to maximum likelihood estimation for Gaussian graphical models.

Main Results:

• Achieved significant complexity reduction in optimal design problems.
• Enabled the implementation of gradient descent for finding locally D-optimal designs.
• Established a direct link between paired comparison models and Gaussian graphical models.
• Demonstrated the algorithm's performance on real and simulated data.

Conclusions:

• The proposed method offers a feasible and efficient approach to optimal experimental design for discrete choice models.
• The connection to graph theory provides new theoretical insights and practical tools for design optimization.
• The algorithm is applicable to various discrete choice models, including paired comparisons.