A Note on the Practical Interpretation of 2 x 2 Tables

There have been intensive studies of the problem of whether the proportions into which any set is doubly dichotomized could reasonably be due to chance, and numerous tests and tables are available, although there is still disagreement about some fundamental assumptions underlying most test procedures. In practice, this problem should not commonly arise, since, outside genetics, there is almost always an a priori expectation of some inter-action, and, in such cases, a significance test is merely a method of demonstrating whether the size of the sample is sufficient to exhibit this interaction. In most cases the experimenter or observer is mainly interested in the extent of the interaction, and its confidence limits. There are three situations in which data are commonly assembled into 2 x 2 tables. (1) The classical situation of paired attributes. (2) Two populations, defined by some attribute, further dichotomized by the more or less arbitrary division of some continuously dis-tributed variate. (3) The dichotomizing of two continuously dis-tributed variates. CASE No. 1 In the case of paired attributes, it is necessary to distinguish between two distinct situations, which may be termed simple sampling and compound sampling. In simple sampling, a supposedly random sample is drawn from some population, and then classified into a 2 x 2 table by two pairs of exhaustive and exclusive attributes. In this case, taking the classification a b c d the most meaningful expression of any difference is that between the proportions a c b d c+d or anda+b d a+b c±dd Any difference in proportion may be most meaningfully expressed as either a ratio or a dif-ference depending on circumstances. Consider the case when a difference is relevant (for example, the cure rate between two remedies). If a, b, c, d represent NOT CURED CURED