Optimal designs for treatment comparisons represented by graphs

TL;DR

This paper introduces a graph-theoretic approach to designing optimal experiments for comparing treatments, linking eigenvalues of information matrices to graph Laplacians, and providing new insights into optimal treatment allocation.

Contribution

It proposes a novel graph-based representation of treatment comparison systems and establishes connections between eigenvalues of information matrices and graph Laplacians, advancing optimal design theory.

Findings

Eigenvalues of the information matrix are inverses of Laplacian eigenvalues.

Graph representation differs from traditional block design models.

Uniform treatment design is optimal for certain symmetric contrast systems.

Abstract

Consider an experiment consisting of a set of independent trials for comparing a set of treatments. In each trial, one treatment is chosen and the mean response of the trial is equal to the effect of the chosen treatment. We examine the optimal approximate designs for the estimation of a system of treatment contrasts under such model. These approximate treatment designs can be used to provide optimal treatment proportions for designs in more general models with nuisance effects (e.g., time trend, effects of blocks). For any system of pairwise treatment comparisons, we propose to represent such system by a graph. In particular, we represent the treatment designs for these sets of contrasts by the inverses of the vertex weights in the corresponding graph G. We show that then the positive eigenvalues of the information matrix of a treatment design are inverse to the positive eigenvalues of the vertex-weighted Laplacian of G. Note that such representation of treatment designs differs from the well known graph representation of block designs, which are represented by edges. We provide a graph-theoretic interpretation of the D-, A- and E-optimality for estimating sets of pairwise comparisons; as well as some optimality results for both the systems of pairwise comparisons and the general systems of treatment contrasts. Moreover, we provide a class of 'symmetric' systems of treatment contrasts for which the uniform treatment design is optimal with respect to a wide range of optimality criteria.