A note on the minimax solution for the two-stage group testing problem

TL;DR

This paper investigates optimal group sizes for two-stage group testing when the probability of infection is unknown, providing minimax and Bayesian solutions with practical recommendations.

Contribution

It introduces minimax and Bayesian methods for determining group sizes in two-stage testing when infection probability is unknown, including explicit solutions and practical guidelines.

Findings

Minimax group size is 8 for unbounded p.

Bayesian strategy with Jeffreys prior suggests size 13.

Recommended group size range is 8 to 13 when p is constrained.

Abstract

Group testing is an active area of current research and has important applications in medicine, biotechnology, genetics, and product testing. There have been recent advances in design and estimation, but the simple Dorfman procedure introduced by R. Dorfman in 1943 is widely used in practice. In many practical situations the exact value of the probability p of being affected is unknown. We present both minimax and Bayesian solutions for the group size problem when p is unknown. For unbounded p we show that the minimax solution for group size is 8, while using a Bayesian strategy with Jeffreys prior results in a group size of 13. We also present solutions when p is bounded from above. For the practitioner we propose strong justification for using a group size of between eight to thirteen when a constraint on p is not incorporated and provide useable code for computing the minimax group size under a constrained p.