kb/data/en.wikipedia.org/wiki/Group_testing-1.md

title	chunk	source	category	tags	date_saved	instance
Group testing	2/10	https://en.wikipedia.org/wiki/Group_testing	reference	science, encyclopedia	2026-05-05T09:50:23.496143+00:00	kb-cron

=== Variations and extensions === There are many ways to extend the problem of group testing. One of the most important is called noisy group testing, and deals with a big assumption of the original problem: that testing is error-free. A group-testing problem is called noisy when there is some chance that the result of a group test is erroneous (e.g. comes out positive when the test contained no defectives). The Bernoulli noise model assumes this probability is some constant,

    q
  

{\displaystyle q}

, but in general it can depend on the true number of defectives in the test and the number of items tested. For example, the effect of dilution can be modelled by saying a positive result is more likely when there are more defectives (or more defectives as a fraction of the number tested), present in the test. A noisy algorithm will always have a non-zero probability of making an error (that is, mislabeling an item). Group testing can be extended by considering scenarios in which there are more than two possible outcomes of a test. For example, a test may have the outcomes

    0
    ,
    1
  

{\displaystyle 0,1}

and

      2
      
        +
      
    
  

{\displaystyle 2^{+}}

, corresponding to there being no defectives, a single defective, or an unknown number of defectives larger than one. More generally, it is possible to consider the outcome-set of a test to be

      0
      ,
      1
      ,
      …
      ,
      
        k
        
          +
        
      
    
  

{\displaystyle {0,1,\ldots ,k^{+}}}

for some

    k
    ∈
    
      N
    
  

{\displaystyle k\in \mathbb {N} }

. Another extension is to consider geometric restrictions on which sets can be tested. The above lightbulb problem is an example of this kind of restriction: only bulbs that appear consecutively can be tested. Similarly, the items may be arranged in a circle, or in general, a net, where the tests are available paths on the graph. Another kind of geometric restriction would be on the maximum number of items that can be tested in a group, or the group sizes might have to be even and so on. In a similar way, it may be useful to consider the restriction that any given item can only appear in a certain number of tests. There are endless ways to continue remixing the basic formula of group testing. The following elaborations will give an idea of some of the more exotic variants. In the 'good–mediocre–bad' model, each item is one of 'good', 'mediocre' or 'bad', and the result of a test is the type of the 'worst' item in the group. In threshold group testing, the result of a test is positive if the number of defective items in the group is greater than some threshold value or proportion. Group testing with inhibitors is a variant with applications in molecular biology. Here, there is a third class of items called inhibitors, and the result of a test is positive if it contains at least one defective and no inhibitors.

== History and development ==

=== Invention and initial progress === The concept of group testing was first introduced by Robert Dorfman in 1943 in a short report published in the Notes section of Annals of Mathematical Statistics. Dorfman's report – as with all the early work on group testing – focused on the probabilistic problem, and aimed to use the novel idea of group testing to reduce the expected number of tests needed to weed out all syphilitic men in a given pool of soldiers. The method was simple: put the soldiers into groups of a given size, and use individual testing (testing items in groups of size one) on the positive groups to find which were infected. Dorfman tabulated the optimum group sizes for this strategy against the prevalence rate of defectiveness in the population. Stephen Samuels found a closed-form solution for the optimal group size as a function of the prevalence rate. After 1943, group testing remained largely untouched for a number of years. Then in 1957, Sterrett produced an improvement on Dorfman's procedure. This newer process starts by again performing individual testing on the positive groups, but stopping as soon as a defective is identified. Then, the remaining items in the group are tested together, since it is very likely that none of them are defective. The first thorough treatment of group testing was given by Sobel and Groll in their formative 1959 paper on the subject. They described five new procedures – in addition to generalisations for when the prevalence rate is unknown – and for the optimal one, they provided an explicit formula for the expected number of tests it would use. The paper also made the connection between group testing and information theory for the first time, as well as discussing several generalisations of the group-testing problem and providing some new applications of the theory. The fundamental result by Peter Ungar in 1960 shows that if the prevalence rate

    p
    >
    
      p
      
        u
      
    
  

{\displaystyle p>p_{u}}

, where

      p
      
        u
      
    
    =
    (
    3
    −
    
      
        5
      
    
    )
    
      /
    
    2
    ≈
    0.38
  

{\displaystyle p_{u}=(3-{\sqrt {5}})/2\approx 0.38}

, then individual testing is the optimal group testing procedure with respect to the expected number of tests, and if

    p
    <
    
      p
      
        u
      
    
  

{\displaystyle p<p_{u}}

, then it is not optimal. However, it is important to note that despite 80 years' worth of research effort, the optimal procedure is yet unknown for

    p
    <
    
      p
      
        u
      
    
  

{\displaystyle p<p_{u}}

and a general population size

    n
    >
    2
  

{\displaystyle n>2}

.