The tests for sampling differences and contingency

Abstract 1—The question discussed in these tests (Jeffreys 1935, 1936a) has the form: we have a sample or a pair of samples, in which the numbers of individuals with the pairs of properties ϕ. ψ, ϕ. ~ψ, ~ϕ. ψ, ~ϕ. ~ψ are respectively x, y, x', y': do the numbers afford evidence for or against the proposition that ϕ and ψ are associated? The conditions contemplated in the experiments, however, are somewhat different. In what I have called the sampling problem (1935, p. 203), large classes of ϕ's and ~ϕ's are already supposed separated, and we select from them at random arbitrarily assigned numbers of members x + y, x' + y'. In the contingency problem (1936a, pp. 426-29), the ϕ's and ~ϕ's are not already separated. We have only a single large class containing all four types, and in the sampling we choose arbitrarily a total number x + y + x' + y'. it would therefore be expected that the significance tests found in the two cases would differ slightly. In the sampling problem, the sampling errors of x and y are constrained to be equal and opposite, which is not so in the contingency problem. The tests actually found were different, and no surprise was felt; it is only recently that I have noticed that the difference is liable to be so large that in some cases, with the same x, y, x', y', an association could not be asserted in the sampling one. Again, if x' and y are very different, interchanging them makes a great difference to the result, so that in the sampling problem it apparently makes a large difference whether we sample first with regard to ϕ or ψ if one of them is a rare property. It turns out that the discrepancy is removed when we take into account an inequality that was overlooked. We denote the proposition that there is no association by q, that there is one by ~q. K is the ratio expressing the support for q given by the data θ, namely K = P (q | θh)/P(~q | θh)/P(q | h)/P(~q | h).