Optimal Dorfman Group Testing For Symmetric Distributions

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 nonoverlapping 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 indentically 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.Comment: 20 pages w/o references, 2 figure

Testes conjuntos:extensões da teoria de Dorfman

Tese de doutoramento, Estatística e Investigação Operacional (Probabilidades e Estatística), Universidade de Lisboa, Faculdade de Ciências, 2013As análises conjuntas ao sangue, propostas por Dorfman durante a segunda grande Guerra, permitiram uma gestão mais eficiente de recursos na deteção dos infetados com sífilis no exército americano, e tornaram-se um paradigma, que se pode tornar mais realista e aplicável se considerarmos que os testes de diagnóstico são sujeitos a erros de classificação. No âmbito deste trabalho estendemos os conceitos de sensibilidade e especificidade para a realização de testes conjuntos, adotando a proposta de Santos, Pestana e Martins (2012) para modelar a sensibilidade e a especificidade, que tem em linha de conta o problema da diluição e consequente rarefação. Analisamos, via simulação, o comportamento de alguns estimadores para a taxa de prevalência baseados em testes conjuntos, quer na ausência quer na presença de erros de classificação. Para os testes quantitativos discretos estendemos os cálculos da sensibilidade e da especificidade para o modelo de Poisson a populações mais dispersas, nomeadamente binomiais negativas, dando especial relevo ao caso mais tratável de população geométrica. Por fim, no que toca a testes quantitativos contínuos, é investigada a informação da média sobre o máximo (ou sobre o mínimo) da amostra a fim de, com base num resultado conjunto, decidir os casos em que a amostra conjunta é classificada como suspeita de conter um ou mais infetados.The composite sampling proposed by Dorfman during the SecondWorldWar led to a more efficient management of resources in the detection of the infected with syphilis in the U.S. Army, and became a paradigm, which could be more realistic and enforceable considering that diagnostic tests are subjected to classification errors. In this work we extended the concepts of sensitivity and specificity for the performance of compound tests, adopting the Santos, Pestana and Martins (2012) proposal to model the sensitivity and specificity, which takes into account the dilution and consequent rarefaction problem. We analyse, through simulation, the behaviour of some estimators for the prevalence rate based on compound tests, let it be the absence or in the presence of errors of classification For quantitative discrete tests we extended the calculations of sensitivity and specificity for the Poisson model to more dispersed populations, namely negative binomial models, with more detailed analysis of geometric populations. Finally, concerning the continuous quantitative tests, we investigated the information that the sample mean can provide about the maximum (or the minimum) of the sample, based on a joint result, and its bearing on identifying composite samples which eventually include infected individuals