Group Testing To Eliminate Efficiently All Defectives in a Binomial Sample

Abstract

In group‐testing, a set of x units is taken from a total starting set of N units, and the x units (1 ≦ x ≦ N) are tested simultaneously as a group with one of two possible outcomes: either all x units are good or at least one defective unit is present (we don't know how many or which ones). Under this type of testing, the problem is to find the best integer x for the first test and to find a rule for choosing the best subsequent test‐groups (which may depend on results already observed), in order to minimize the expected total number of group‐tests required to classify each of the N units as good or defective. It is assumed that the N units can be treated like independent binomial chance variables with a common, known probability p of any one being defective; the case of unknown p and several generalizations of the problem are also considered.

Optimality of group testing in the presence of misclassification

“…This strategy and its variations developed later, often referred to as group testing or pooled testing, have received substantial attention for efficient identification of an event or estimation of the probability that the event occurs; see Sobel & Groll (1959), Sobel & Elashoff (1975), Le (1981), Gastwirth & Hammick (1989), Chen & Swallow (1990), Farrington (1992), Gastwirth & Johnson (1994), Hughes-Oliver & Swallow (1994), Litvak et al (1994), Tu et al (1995), Barcellos et al (1997), Brookmeyer (1999), Hung & Swallow (1999), Hughes-Oliver & Rosenberger (2000) and Tebbs & Swallow (2003). An attractive feature of group testing is that retesting on individuals is not necessary if one is only interested in estimation of the probability of a positive test.…”

Asymptotic efficiency of two-stage disjunctive testing

“…Although consists of all unresolved and positive items, the number of positive items is small; e.g., it is known (see [5]) not to exceed . Therefore, to obtain sufficiently good upper bounds to and one can use (12) to write (22) (Here means that the set of active items of the column of the matrix belongs to the set of active items of the syndrome for ). Continuing, we consider a probabilistic method to prove the existence of matrices with sufficiently small values .…”

“…This approach has been used in many applications beginning with an efficient blood testing problem in [7]. Other applications include (following [13] and [10]) quality control in product testing [22], searching files in storage systems [12], efficient accessing of computer memories [12], sequential screening of experimental variables [16], efficient contention resolution algorithms for multiple-access communications, [3], [23], [17], and screening of clone libraries [2], [4]. The books and review papers [6], [1], [8], and [13] testing because of its importance in modern biological applications such as monoclonal antibody generation and cDNA library screening.…”

Group Testing and Batch Verification

“…Since there were only a few thousand cases of the disease in millions of draftees, large subsets would come back negative, saving many individual tests. Group testing later found many industrial applications, a line of research initiated by Sobel and Groll [40]. In the past 50 years or so, a large literature has grown around the problem, and many variants have been considered.…”