Ratings and rankings: Voodoo or Science?

TL;DR

This paper examines how the importance of variables in composite indicators often diverges from nominal weights due to data heteroskedasticity and correlation, using correlation ratio to measure variable importance.

Contribution

It introduces the 'main effect' measure based on correlation ratio and analyzes the discrepancy between nominal weights and actual variable importance in composite indicators.

Findings

Main effects often differ significantly from nominal weights.

Data correlation structure limits the ability to achieve desired importance through weight adjustments.

Inversion of weight-to-effect mapping is generally not feasible in practice.

Abstract

Composite indicators aggregate a set of variables using weights which are understood to reflect the variables' importance in the index. In this paper we propose to measure the importance of a given variable within existing composite indicators via Karl Pearson's `correlation ratio'; we call this measure `main effect'. Because socio-economic variables are heteroskedastic 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 further discuss to what extent the mapping from nominal weights to main effects can be inverted. This analysis is applied to five 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 nonnegative numbers with unit sum.