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?

This H0 has two important implications that are relevant to answering the question: First, the total number of survivors and nonsurvivors would be 3 and 3, respectively, regardless of which 3 patients were selected to receive the drug (and which other 3 received the placebo), so that the marginal totals of Table 1 would be fixed regardless of results of the randomization. Second, the number of survivors among the drug patients (which will be designated X hereafter), as well as the other 3 entries in the 2 × 2 table, would be determined by which patients were selected to receive the drug. For example, with the selection A, B, and C, the number of drug patients who would survive (X) would be 3, and the results of the study would be as displayed in Table 1. But with the selection of A, B, and D, then X would be 2, and the four cells of the 2 × 2 table would then have entries of [2 1] for row 1 and [1 2] for row 2 (note that since marginal totals of Table 1 are fixed, irrespective of which 3 patients are selected to receive the drug, knowledge of X determines entries in the other three cells of the 2 × 2 table; these entries are therefore redundant with X). With fixed marginal totals, the variable X is clearly relevant to tenability of the H0 relative to the hypothesis that [Page 356]the drug is effective. There were, in fact, 20 possible ways in which the 3 patients who were to receive the drug could have been selected; these 20 selections are listed in Table 3, together with values of X they would have determined under the H0.

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