Scatterplots for Unordered Pairs

This department publishes articles of interest to statistical practitioners. Innovative applications of known methodology may be suitable, but sizable case studies should be submitted to other journals. Brief descriptions and illustrations of new developments that are potentially useful in statistical practice are appropriate. Acceptable articles should appeal to a substantial number of practitioners.

Intraclass correlation coefficients (ICC's) measure the strength of the similarity among measurements within a class (or family) relative to the total variability. Classically, this index of association has been most often treated as a ratio of variance components, namely the fraction that the family-to-family variability represents of the total variability. For the special case of twins, or any other set of unordered pairs, there has been no authoritative prescription as to which scatterplot one should use to graphically represent the ICC.

We propose that the set of all permutations within pairs of exchangeable measurements forms a legitimate basis for selecting a representative scatterplot. The recommended plot is the one whose interclass coefficient is closest to an ICC. For the product-moment correlation this implies that one selects a plot with r equal to the analysis of variance (ANOVA) estimator. We prove that Fisher's “symmetrical table” yields a coefficient that is smaller than all of the positive correlations in the permutation set.