Limitations of the Application of Fourfold Table Analysis to Hospital Data.*,†

In the biologic laboratory we have a method of procedure for determining the effect of an agent or process that may be considered typical. It consists in dividing a group of animals into two cohorts, one considered the “experimental group,” the other the “control.” On the experimental group some variable is brought to play; the control is left alone. The results are set up as in table 1-a. If the results show that the ratio a:a þb is different from the ratio c:c þd, it is considered demonstrated that the process brought to bear on the experimental group has had a significant effect. A similar method is prevalent in statistical practice, which I venture to think has come into authority because of its apparent equivalence to the experimental procedure. In Biometrika it is referred to as the fourfold table and it is used as a paradigm of statistical analysis. The usual arrangement is that given in table 1-b. The entries, a, b, c and d are manipulated arithmetically to determine whether there is any correlation between A and B. A considerable number of indices have been elaborated to measure this correlation. Pearson has given the formula for calculating the product-moment correlation coefficient from a fourfold table on the assumption that the distribution of both variates is normal; Yule has an index of association for the fourfold table; there are the chi-square test and others. In essence, however, all these indices measure in different ways whether and how much, in comparison with the variation of random sampling, the ratio a:a þb differs from the ratio c:c þd. If the difference departs significantly from zero, there is said to be correlation, and the correlation is the greater the greater the difference. Now there is a distinction between the method as used in the laboratory and as applied in practical statistics. In the experimental situation, the groups, B and not B, are selected before the subgroupings, A and not A, are effected; that is, we start with a total group of unaffected animals. In the statistical application, the groupings, B and not B, are made after the subgroupings, A and not A, are already determined; that is, all the effects are already produced before the investigation starts. In the end, the tables of the results which are drawn up look alike for the two cases, but they have been arrived at differently. Correlative to this difference, a different interpretation may apply to the results, and this paper deals with a specific case of a kind that arises frequently in a medical clinic or a hospital. I take an example.