Ratings and rankings: voodoo or science?

TLDR: In this article, the authors measure the importance of a given variable within existing composite indicators via Karl Pearson's "correlation ratio"; they call this measure the main effect, and they discuss to what extent the mapping from nominal weights to main effects can be inverted.

Abstract: Summary. Composite indicators aggregate a set of variables by using weights which are understood to reflect the variables’ importance in the index. We propose to measure the importance of a given variable within existing composite indicators via Karl Pearson's ‘correlation ratio’; we call this measure the ‘main effect’. Because socio-economic variables are heteroscedastic and correlated, relative nominal weights are hardly ever found to match relative main effects; we propose to summarize their discrepancy with a divergence measure. We discuss to what extent the mapping from nominal weights to main effects can be inverted. This analysis is applied to six composite indicators, including the human development index and two popular league tables of university performance. It is found that in many cases the declared importance of single indicators and their main effect are very different, and that the data correlation structure often prevents developers from obtaining the stated importance, even when modifying the nominal weights in the set of non-negative numbers with unit sum.