Optimal Dorfman Group Testing for Symmetric Distributions

TL;DR

This paper extends Dorfman's group testing protocol to models with exchangeable, correlated specimen statuses, providing a new characterization, efficient solution methods, and empirical validation with COVID-19 data.

Contribution

It introduces a novel exchangeable distribution framework for group testing, characterizes it with a function q, and reduces the partitioning problem to an efficiently solvable integer partitioning problem.

Findings

Efficient solution for the partitioning problem under exchangeable models.

Application to COVID-19 data explains high empirical testing efficiency.

Framework captures positive correlations among specimens from the same community.

Abstract

We study Dorfman's classical group testing protocol in a novel setting where individual specimen statuses are modeled as exchangeable random variables. We are motivated by infectious disease screening. In that case, specimens which arrive together for testing often originate from the same community and so their statuses may exhibit positive correlation. Dorfman's protocol screens a population of n specimens for a binary trait by partitioning it into non-overlapping groups, testing these, and only individually retesting the specimens of each positive group. The partition is chosen to minimize the expected number of tests under a probabilistic model of specimen statuses. We relax the typical assumption that these are independent and identically distributed and instead model them as exchangeable random variables. In this case, their joint distribution is symmetric in the sense that it is invariant under permutations. We give a characterization of such distributions in terms of a function q where q(h) is the marginal probability that any group of size h tests negative. We use this interpretable representation to show that the set partitioning problem arising in Dorfman's protocol can be reduced to an integer partitioning problem and efficiently solved. We apply these tools to an empirical dataset from the COVID-19 pandemic. The methodology helps explain the unexpectedly high empirical efficiency reported by the original investigators.