Maximally efficient two-stage screening.

In DNA library screening, blood testing, and monoclonal antibody generation, significant savings in the number of assays can be realized by employing group sampling. Practical considerations often limit the number of stages of group testing that can be performed. We address situations in which only two stages of testing are used. We define efficiency to be the expected number of positives isolated per assay performed and assume gold-standard tests with unit sensitivity and specificity. Although practical tests never are golden, polymerase chain reaction (PCR) methods provide procedures for screening recombinant libraries that are strongly selective yet retain high sensitivity even when samples are pooled. Also, results for gold-standard tests serve as bounds on the performance of practical testing procedures. First we derive formulas for the efficiency of certain extensions of the popular rows-and-columns technique. Then we derive an upper bound on the efficiency of any two-stage strategy that lies well below the classical upper bound for situations with no constraint on the number of stages. This establishes that a restriction to only two stages necessitates performing many more assays than efficient multistage procedures need. Next, we specialize the bound to cases in which each item belonging only to pools that tested positive in stage 1 must be tested individually in stage 2. The specialized bound for such positive procedures is tight because we show that an appropriate multidimensional extension of the rows-and-columns technique achieves it. We also show that two-stage positive procedures in which the stage-1 groups are selected at random perform suboptimally, thereby establishing that efficient tests must be structured carefully.