The posterior probability distributions of the ordinary and intraclass correlation coefficients.

1. The accurate form of the probability distribution of an estimated correlation coefficient in a simple, the true correlation being taken as a datum, has been given by Fisher in two well-known papers (1915, 1921). Pearson and others (1917) have studied the adaptation of the formula, which is rather complicated, for numerical work, but a transformation suggested by Fisher in his earlier paper and applied in his late one offers great advantages. The present paper deals with the inverse problem: given the estimate from a sample, what is the probability distribution of the true value, the latter being supposed initially unknown? Much of the mathematics of correlation depends on the hypothesis of the normal correlation surface. As this is a generalization of the normal law of error for one variable, it is subject to similar criticisms. It can be derived from the hypothesis that the two variables considered are subject to a number of independent component disturbances,m of which effect both in the same sense andn in opposite senses (Jeffreys 1935, pp. 213-17). If they are all equal in amount, then even ifm andn are fairly small they lead to a good approximation to the normal correlation surface with correlation (m - n )/(m + n ). We could regardm /(m + n ) as a frequency to be estimated from a sample, and such frequencies are taken, in the absence of special knowledge, as having their prior probabilities uniformly distributed. The idea of the normal correlation surface therefore suggests that we should take the prior probability of the correlation coefficient as uniformly distributed between - 1 and + 1.