Exact Tests of Significance in Contingency Tables

This table (for given a and A) is denoted by (al , a2 , a3) with al < a2 _< a3 . As an example I take the case A = 10, a = 10. There are just 14 different configurations consistent with these marginal totals, and these are listed in the table. In assessing the significance of a particular observed table, two separate problems arise. Quoting Fisher (1950), these are "(i) the calculation of *.. the objective frequencies of the observational series we have obtained, and of other alternative observational series. (ii) The choice of a criterion by which the assemblage of possibilities is to be divided up in making a test of significance. The first facet is purely mathematical; only one answer is possible. The second takes account of the needs and prior ideas of the experimenter: by it he specifies the particular question he chooses to ask". For a table such as (1), the first problem is straightforward, the frequency of occurrence being given by