Asymmetric measures of association, closed data, and multivariate analysis

The association between constant-sum variables Xiand Xjexpressed as percentages can be calculated as a product-moment correlation between Xiand Xj/(100 - Xi) and a correlation between Xjand Xi/(100 - Xj). An asymmetric, square matrix may be formed from these coefficients, and multivariate analysis performed by two methods: singular value decomposition and canonical decomposition. Either analysis avoids problems in the interpretation of correlation coefficients determined from closed arrays, and provides information about dependencies among the variables beyond that obtained from the usual correlation coefficient between Xiand Xj. Two examples show the canonical decomposition to have the greater usefulness. ?? 1980 Plenum Publishing Corporation.