V.—The General Expression for Boundary Conditions and the Limits of Correlation

Extract

The attempt to extract information from correlation coefficients concerning possible common factors among measured variables presents serious difficulties, since common factors are not determinate functions of the original variables. It is indeed possible, by the assumption of specific, group (common to some), and general (common to all) factors, to construct various hypothetical correlated variables which will produce correlations of a given magnitude. If it is required, for instance, to construct three variables to produce three given positive correlations, we may choose among various ways to produce the correlations by means of a triple factor (common to three variables) or dual factors (common to two), or both in co-operation. We are faced by a certain restriction, however, for whereas correlations of any magnitude can be produced by using a triple factor with or without dual factors, yet with dual factors unaided by a triple factor certain magnitudes are unattainable. The condition expressing this restriction is known as a “boundary condition.”