(1) In a paper communicated to the Royal Society in 1903 I gave very briefly in a footnote the properties of the
correlation ratio
. These properties were discussed more at length in my memoir, “On the General Theory of Skew Correlation and Non-linear Regression,” published in 1905. The two papers dealt only with the
total correlation ratio
, or the relation between two variates without consideration of any other correlated variates. The introduction of the correlation ratio enabled the measure of the relationship between two variates to be expressed by a single number, measuring its total intensity, in cases where the regression line was of any form. The ratio passed into the usual correlation coefficient when the regression line became straight. This correlation ratio has been generally accepted by statisticians as a useful measure of relationship in cases of skew correlation and non-linear regression. Shortly after the appearance of the above memoirs I generalised this coefficient in a manner comparable with the generalisation of the coefficient of correlation, namely, by the definitions of the
multiple correlation ratio
and of the
partial correlation ratio
. These ratios correspond to the multiple correlation coefficient and the partial correlation coefficient in multiple linear regression. Their importance is very considerable, as they enable us to measure the intensity of association between two variates when other correlated variates are considered as constant without any assumption that the regression is linear, still less that the frequencies follow the normal (or Laplace-Gaussian) surface. I had not intended to discuss the results of the present paper before the probable errors had been provided, but the recent revival of interest in skew regression, and its fundamental importance in all higher statistical inquiry, justifies, at least, the publication of those formulæ which are fundamental to the subject. (2) I deal first with the problem of three variates, although the extension to any number is not hard to make.
On Correctly Adjusting the Squared Multiple Correlation Coefficient in Linear Regression: Effect Size Estimation and Significance Testing with Application to Substance Abuse Research
