The Correlation Coefficient as an Index of Relationship

Abstract

During the last few decades the technique of correlation has been in creasingly used in the social sciences to measure what is called the relation be tween variables. Thus it is usual to say that if two variables yield a correlation coefficient of +0.95 they are closely If the coeffi cient is near zero, we say they are un We use such language with more confidence if the coefficients in question are product-moment coefficients or ones considered equivalent to such coefficients. But the concept of meas uring relationship is quite general and is used to some extent in regard to a variety of different techniques. In view of the wide use of correlation and the considerable amount of mathematical discussion which it has evoked, it seems strange that there has been very little dis cussion of the meaning of the words rela tion and closeness of relationship as used in this connection. The present article is offered as a contribution to this problem. In what sense can we say that correlation measures relationship? It is hardly necessary to remark that correlation does not measure causal relation ship. For example, if it is shown that there is a high correlation between annual marriage rates and some index of general business condition, then the said correla tion does not prove that fluctuations in business cause changes in marriage rates, nor that marriage rates change the condi tion of business, nor that both variables depend on some third factor. These ques tions are to be decided from considerations quite apart from the mathematics of correla tion. Mathematics cannot measure causality. : At most it can tell us something about con ic omitan t variation. Since the term rela tionship may suggest the existence of caus al relationship, it would probably be bet ter to discard it altogether and use the less ambiguous term concomitant variation. \ATe shall not, however, endeavor to intro duce this somewhat awkward term but shall conti ue in this article to use the terms relation and relationship, bearing in mind the restrictions discussed in this para graph. Correlational analysis, then, is con cerned with concomitant variation. Before pursuing this subject further, it is worth noting that the use of correlation implies a type of problem quite strikingly different from that usual in the physical sciences. In the latter, relationship is simply consid ered to be present or absent. Generally speaking, the physical scientist refuses to recognize intermediate degrees of relation ship. If the points representing the paired valu s of the two variables fall along some reasonably simple mathematical curve, then he co siders that the variables are related. ! If the points are so scattered that such a curve cannot be drawn, he ordinarily sus pends Judgment. Not so in the social seieneesI Here the extremely simple type of relationship which may be expressed by a mathematical curve occurs but rarely. The social scien tist therefore introduces the new concept of closeness of relationship, that is, the degree to which the bivariate distribution in question approximates the perfect type of relationship expressible by some rela i tively simple mathematical function. To cha acterize this degree of approximation in a quantitative manner is the function of | correlation.