Simultaneous Decomposition of the Variance by Sources and Subgroups New Insights Academic Article uri icon


  • Decomposing measures of variability (or inequality) is a topic of interest. There are two types of decompositions: 1) of a linear combination of random variables into contributions of the individual variables (sources) and contributions of associations between them, and 2) of a population into the contributions of its subpopulations – the “within” and “between” components of total variation (SST). Simultaneous treatment of the two types is called for, which takes into account the correlations between sources within subpopulations and the correlations between subpopulation means. We consider the ANOVA wherein the response variable is a linear combination of m sources. We give a breakdown of SST into SSB and SSW, and then these two components are further broken down into subcomponents due to the sources and the correlations between and within sources. The expected values of the subcomponents are derived and the sensitivity of these components to the correlations among the sources within groups and among source group means is conducted. We illustrate the technique via an example involving decomposition of the variation of total income into two sources: local sales and exports for 3 industries in Israel during two years. We find that the correlations contribute 20-25% to the total variability.

publication date

  • January 1, 2013