Criteria for symmetric measures of association for nominal data

✍ Scribed by Jan Vegelius; Svante Janson

• Publisher: Springer Netherlands
• Year: 1982
• Tongue: English
• Weight: 337 KB
• Volume: 16
• Category: Article
• ISSN: 0033-5177
• DOI: 10.1007/bf00144075

✦ Synopsis

Most articles on association measures for nominal data suggest that such a measure should be interpretable as the reduction of the uncertainty of a dependent variable Y that is achieved by knowing the value of an explanatory variable X (see Goodman and Kruskal, 1954;Costner, 1965; S~trndal, 1974). In other words, the measure of association between the dependent variable Y and the independent variable X should be defined as

where U(Y) symbolizes the uncertainty in Y without knowledge of X, and U(Y/X) symbolizes the uncertainty in Y when X is known. For such a definition it is fundamental that Y is considered as dependent and X explanatory. Formally it is obvious that r(X,Y) may differ from r(Y, X).

Depending on the choice of U, various asymmetric coefficients have been proposed.

In many cases, however, it is very unnatural to choose one of the variables as dependent on the other. It may then be more reasonable to have symmetric association measures, i.e., measures for which X and Y are as much associated as are Y and X. In cases where certain multivariate analyses, such as hierarchical cluster analysis (Johnson, 1967) or truncated component analysis (Gorsuch, 1974) are to be used, it is necessary for the measure to be symmetric.

In this article we propose some criteria for a symmetric measure of association for nominal data. We also test some symmetric measures to determine whether or not they satisfy the criteria.