Testing the Maximum by the Mean in Quantitative Group Tests

Group testing, introduced by Dorfman in 1943, increases the efficiency of screening individuals for low prevalence diseases. A wider use of this kind of methodology is restricted by the loss of sensitivity inherent to the mixture of samples. Moreover, as this methodology attains greater cost reduction in the cases of lower prevalence (and, consequently, a higher optimal batch size), the phenomenon of rarefaction is crucial to understand that sensitivity reduction. Suppose, with no loss of generality, that an experimental individual test consists in determining if the amount of substance overpasses some prefixed threshold l. For a pooled sample of size n, the amount of substance of interest is represented by \(\left (Y _{1},\cdots \,,Y _{n}\right )\), with mean \(\overline{Y }_{n}\) and maximum M _n . The goal is to know if any of the individual samples exceeds the threshold l, that is, M _n > l. It is shown that the dependence between \(\overline{Y }_{n}\) and M _n has a crucial role in deciding the use of group testing since a higher dependence corresponds to more information about M _n given by the observed value of \(\overline{Y }_{n}\).