Fisher-Irwin  (test)

We have two Bernoulli populations with respectively parameters p₁ and p₂. The Fisher-Irwin test is about the hypothesis according to which these two parameters have the same value. More specifically, it tests :

    * The null hypothesis  H₀:  p₁ = p₂ ,

    * Against the alternative hypothesis H₁: p₁  p₂.

This can be illustrated by the following example. We have two coins :

    * C₁, that generates "Heads" with probability  p₁.

    * C₂, that generates "Heads" with probability  p₂.

The null hypothesis H₀ states that both coins are in fact identical. To test this hypothesis :

     * C₁ is tossed n₁ times, thus generating x₁ "Heads",

     * C₂ is tossed n₂ times, thus generating x₂ "Heads",

The question is : "Are the observed values x₁ and x₂ inconsistent with the hypothesis that the two coins are identical ?". Note that it is not asked to estimate the probabilities p₁ and  p₂, but only to assess whether it is likely that these two probabilities are equal.

The cornerstone of the Fisher-Irwin test is the following result :

    * Let X and Y be two independent binomial random variables, with the same p but possibly different sizes m and n. Choose an integer k, and consider the distribution of X under the condition X + Y = k. This distribution is hypergeometric, and does not depend on p.

 This important property is demonstrated here.

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An example of use of the Fisher-Irwin test is in quality control. C₁ and C₂ are then machines that manufacture the same part, and that are supposed to be identical. The proportions p₁ and p₂ of defective parts made by these two machines should then be equal. But if one of the machines drifted off optimal tuning, these proportions will become different. To test whether both machines have the same (hopefully optimal) tuning, one control sample is collected on each of the machines, and all the parts in these control samples are then tested individually. The Fisher-Irwin test then answers the question "Are the numbers x₁ and x₂ of defective parts in the control samples consistent with the hypothesis that both machines are equally well tuned ?" 

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The Fisher-Irwin test is described in the following Tutorial :

THE FISHER-IRWIN TEST

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Related readings :

Interactive animation :