Greenland's paradox, Fisher's example of the flowers and 2×2 tables

Abstract

A great amount has been written over the years in many different publications about the suitable test for independence in 2×2 tables: The conditional test (Fisher's exact test) or the unconditional test (Barnard's test). In this paper the authors carry out a review of what has been written and refute two of the principal arguments in favour of conditioning (in fact they are the only ones two that have not been refuted): Fisher's example of the flowers and the argument of causal models (which gives rise to Greenland's paradox). In agreement with Helland (1995) they show the importance of deciding which is the (real or conceptual) target population, and this, together with the existence or not of an ancillary statistic, is what allows a decision to be taken on the appropriateness or not of conditioning.