Independent Subsets of Correlation and Other Matrices

Abstract

It is known that the set of correlation coefficients formed from $k$ independent normal samples exhibits pairwise independence of its members (Geisser and Mantel (1962)). Here it is shown that many much larger subsets of the matrix are fully independent. The main result characterises such subsets in a simple way. Because the results are framed in abstract terms, they also apply to rank correlation coefficients and $\chi^2$ statistics.