The asymptotic distribution of commonality components

Commonality components have been defined as a method of partitioning squared multiple correlations. In this paper, the asymptotic joint distribution of all 2^k – 1 squared multiple correlations is derived. The asymptotic joint distribution of linear combinations of squared multiple correlations is obtained as a corollary. In particular, the asymptotic joint distribution of commonality components are derived as a special case. Simultaneous and nonsimultaneous asymptotic confidence intervals for commonality components can be obtained from this distribution.This work was supported in part by the Spencer Foundation and the National Science Foundation.The authors are grateful to Bryna Siegel-Gorlick for her help in obtaining the data used in Example 4.3, and to the referees for their comments and suggestions.