The general sampling distribution of the multiple correlation coefficient

Abstract

Of the problems of the exact distribution of statistics in common use that of the multiple correlation coefficient is the last to have resisted solution. It will be seen that the solution introduces an extensive group of distributions, occurring naturally in the most diverse types of statistical investigation, and which in their mere mathematical structure supply an extension in a new direction of the entire system of distributions previously obtained, which in their generality underlie the analysis of variance. The individual distributions of this system were in each case obtained by the exact investigation of a particular problem. It was realised only gradually that many of these distri­butions, disguised by the different notations appropriate to different problems, were in reality equivalent, and could be made available in practice by a single system of tables. The remaining cases, with the notable exception of the correlation coefficient, then fall into place as particular limiting forms of a single general distribution. As the practical utility of these earlier solutions depends greatly on a recognition of their place in a single system, a very brief account of their mutual relationship may be given. The only statistic derived from samples of a continuous variate, of which the distribution was known before the present century, appears to be the arithmetic mean of a sample drawn from the normal distribution. In addition, however, two distributions which may be regarded as distributions of statistics had also been found, namely, Bernoulli’s binominal distribution, and Poisson’s series. Both of these distributions possess the property that the aggregate of the values of a sample is itself distributed in a distribution of the same type. In all three classical cases, therefore, the distribution of the statistic derived from a finite sample was known only by a mathematical simplification of this special type. In all other cases, approximations of unknown accuracy based on the use of the standard error and the assumption of normal distributions had perforce to be used.