Fisher Exact Probability Test

The Fisher exact probability test (also called the Fisher-Irwin test) is one of several tests that can be [Page 355]used to detect whether one dichotomous variable is related to another. The rationale of this test, as well as its principal advantages and limitations, can be presented in the context of the following hypothetical small randomized experiment designed to determine whether a dichotomous “treatment” variable (Drug vs. Placebo) is related to a dichotomous “outcome” variable (Survival vs. Death).

A physician believes that a new antiviral drug might be effective in the treatment of SARS (severe acute respiratory syndrome). Assume that the physician carries out a randomized double-blind drug efficacy study involving 6 SARS patients (designated A, B, C, D, E, and F), 3 of whom (say, A, B, and C) were randomly selected from this group and given the drug and the remaining 3 of whom (D, E, and F) were given a placebo. Four months later, the 3 patients who received the drug were still alive whereas the 3 patients who received the placebo were not.

Results of this study may be summarized in a 2 × 2 table (see Table 1, which has 2 rows and 2 columns, ignoring row and column totals). More generally, results of any randomized treatment efficacy study involving dichotomous treatment and outcome variables may be summarized using the notation shown in Table 2. Do the results in Table 1 support the belief that the new drug is effective (relative to the placebo)? Or, more generally, do results of a randomized study that can be summarized as in Table 2 support the belief that the two dichotomous variables are related—for example, that patient outcomes are related to the treatments to which they have been exposed?

The fact that, in the physician's study, all the drug patients survived and all the placebo patients died (Table 1) would seem consistent with the belief that the patient outcomes were related to treatments they received. But is there a nonnegligible probability that such a positive result could have occurred if the treatment had in fact been unrelated to the outcome? Consistent with absence of relation of outcomes to treatments, let us hypothesize (this will be called the null hypothesis, and designated H0 hereafter) that patients A, B, and C, who actually survived, would have survived whether they received the drug or the placebo, and that D, E, and F, who actually died, would have died whether they received the drug or the placebo. Now we ask, Would the positive results in Table 1 have been unlikely if this H0 had been true?

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