
doi: 10.1086/225004
For selecting and interpreting appropriate measures of association, a "proportional-reduction-in-error" (P-R-E) criterion is useful. However, efforts to give a P-R-E interpretation to measures of ordinal association have not been successful, especially in delineating the "form" or "shape" of ordinal association. An effort is therefore made in this paper to introduce the notion of relevant forms of ordinal association, such as strict monotonic, monotonic, and nonmonotonic associations, and to suggest a few P-R-E measures that would assess such particular forms of association in ordinal data. When we talk about a measure of association, we mean an index designed to measure the degree to which two or more variables "go together" in some sense. In the loosest sense, any two variables may be said to be associated as long as the two are not statistically independent. In a stricter sense, however, two variables are said to be associated only if they show some patterned covariation. The pattern of our interest may be linear or curvilinear, monotonic or nonmonotonic, to name a few. Although lack of independence can be indicated without any reference to a specific form of relationship, the degree of association can be indicated only with reference to a specific pattern of relationship. Costner (1965) pointed out the importance of making the implied form of association explicit when he advocated the adoption of the "proportional-reduction-in-error" (P-R-E) criterion in selecting appropriate measures of association for sociological research. The P-R-E interpretation, as he cogently argued, will give us not only an exact probabilistic statemenlt about the degree of ani association but also a clear indication of the reference pattern of association under consideration. For instance, it is clear from the P-R-E interpretation of Pearsonian r that it measures the linear relationship, and that the value of r2 will reach unity (maximum) if and only if one variable is a linear function of the other (see Kruskal 1958; Hays 1963).
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