On the Sampling Distribution of the Multiple Correlation Coefficient

Abstract

The problem of finding the distribution of the, multiple correlation coefficient in samples from a normal population with a non-zero multiple correlation coefficient was solved in 1928 by Fisher' by the application of geometrical methtds. In his derivation he used the facts that the population value p of the multiple correlation cQefficient is invariant under linear transformations.of the independent variates, and that the distribution of the multiple correlation coefficient is independent of all populationi parameters except p . In this paper it will be shown that the distribution of the multiple correlation coefficient can be derived directly from Wishart's2 generalized product moment distribution without making use of geometrical notions and the property bf the invariance of p under linear transformations of the independent variates. Furthermore, it will not be necessary to show that the distributioni will be independent of all population parameters except p . The population value of the multiple correlation coefficient between a variate x and a set of variates x, X x iS the ordinary correlation coefficient between x, and that linear function of the variates x2, x3, . Z.which will make this correlationi a maximum. It can be expressed as p2=J a Al where A is the determinant of the correlations among all of the