A Critique of Hirsch's Citation Index: A Combinatorial Fermi Problem

Abstract

1.1. Overview. In 2005, physicist J. E. Hirsch [Hi05] proposed theh-index to measure the quality of a researcher’s output. This metric is the largest integer n such that the person has n papers with at least n citations each, and all other papers have weakly less than n citations. Although the original focus of loc. cit. was on physicists, the h-index is now widely popular. For example, Google Scholar and the Web of Science highlight theh-index, among other metrics such as total citation count, in their profile summaries. An enticing point made in loc. cit. is that the h-index is an easy and useful supplement to a citation count (Ncitations), since the latter metric may be skewed by a small number of highly cited papers or textbooks. In Hirsch’s words: “I argue that two individuals with similar hs are comparable in terms of their overall scientific impact, even if their total number of papers or their total number of citations is very different. Conversely, comparing two individuals (of the same scientific age) with a similar number of total papers or of total citation count and very differenth values, the one with the higherh is likely to be the more accomplished scientist.” It seems to us that users might tend to eyeball differences of hs and citation counts among individuals during their assessments. Instead, one desires a quantitative baseline for what “comparable”, “very different” and “similar” actually mean. Now, while this would appear to be a matter for statisticians, we show how textbook combinatorics sheds some light on the relationship between the h-index and Ncitations. We present a simple model that raises specific concerns about potential misuses of theh-index.