Rank Robustness of Composite Indices

Many common multidimensional indices take the form of a 'composite index' or a weighted average of several dimension-specific achievements. Rankings arising from such an index are dependent upon an initial weighting vector, and any given judgment could, in principle, be reversed if an alternative weighting vector was employed. This paper examines a variable-weight robustness criterion for composite indicators that views a comparison as robust if the ranking is not reversed at any weight vector within a given set. We characterize the resulting robustness relations for various sets of weighting vectors and illustrate how they moderate the complete ordering generated by the composite indicator. We propose a measure by which the robustness of a given comparison may be gauged and illustrate its usefulness using data from the Human Development Index. In particular, we show how some country rankings are fully robust to changes in weights while others are quite fragile. We investigate the prevalence of the different levels of robustness in theory and practice and offer insight as to why certain datasets tend to have more robust comparisons.