Tommaso Lando

A theoretical model of the relationship between the h-index and other simple citation indicators

Of the existing theoretical formulas for the h-index, those recently suggested by Burrell (J Informetr 7:774–783, 2013b) and by Bertoli-Barsotti and Lando (J Informetr 9(4):762–776, 2015) have proved very effective in estimating the actual value of the h-index Hirsch (Proc Natl Acad Sci USA 102:16569–16572, 2005), at least at the level of the individual scientist. These approaches lead (or may lead) to two slightly different formulas, being based, respectively, on a “standard” and a “shifted” version of the geometric distribution. In this paper, we review the genesis of these two formulas—which we shall call the “basic” and “improved” Lambert-W formula for the h-index—and compare their effectiveness with that of a number of instances taken from the well-known Glänzel–Schubert class of models for the h-index (based, instead, on a Paretian model) by means of an empirical study. All the formulas considered in the comparison are “ready-to-use”, i.e., functions of simple citation indicators such as: the total number of publications; the total number of citations; the total number of cited paper; the number of citations of the most cited paper. The empirical study is based on citation data obtained from two different sets of journals belonging to two different scientific fields: more specifically, 231 journals from the area of “Statistics and Mathematical Methods” and 100 journals from the area of “Economics, Econometrics and Finance”, totaling almost 100,000 and 20,000 publications, respectively. The citation data refer to different publication/citation time windows, different types of “citable” documents, and alternative approaches to the analysis of the citation process (“prospective” and “retrospective”). We conclude that, especially in its improved version, the Lambert-W formula for the h-index provides a quite robust and effective ready-to-use rule that should be preferred to other known formulas if one’s goal is (simply) to derive a reliable estimate of the h-index.

A New Bibliometric Index Based on the Shape of the Citation Distribution

In order to improve the h-index in terms of its accuracy and sensitivity to the form of the citation distribution, we propose the new bibliometric index . The basic idea is to define, for any author with a given number of citations, an “ideal” citation distribution which represents a benchmark in terms of number of papers and number of citations per publication, and to obtain an index which increases its value when the real citation distribution approaches its ideal form. The method is very general because the ideal distribution can be defined differently according to the main objective of the index. In this paper we propose to define it by a “squared-form” distribution: this is consistent with many popular bibliometric indices, which reach their maximum value when the distribution is basically a “square”. This approach generally rewards the more regular and reliable researchers, and it seems to be especially suitable for dealing with common situations such as applications for academic positions. To show the advantages of the -index some mathematical properties are proved and an application to real data is proposed.

The h-index as an almost-exact function of some basic statistics

As is known, the h-index, h, is an exact function of the citation pattern. At the same time, and more generally, it is recognized that h is “loosely” related to the values of some basic statistics, such as the number of publications and the number of citations. In the present study we introduce a formula that expresses the h-index as an almost-exact function of some (four) basic statistics. On the basis of an empirical study—in which we consider citation data obtained from two different lists of journals from two quite different scientific fields—we provide evidence that our ready-to-use formula is able to predict the h-index very accurately (at least for practical purposes). For comparative reasons, alternative estimators of the h-index have been considered and their performance evaluated by drawing on the same dataset. We conclude that, in addition to its own interest, as an effective proxy representation of the h-index, the formula introduced may provide new insights into “factors” determining the value of the h-index, and how they interact with each other.

On a formula for the h-index

The h-index is a celebrated indicator widely used to assess the quality of researchers and organizations. Empirical studies support the fact that the h-index is well correlated with other simple bibliometric indicators, such as the total number of publications N and the total number of citations C. In this paper we introduce a new formula h˜w=h˜w(N,C,cMAX), as a representative predictive formula that relates functionally h to these aggregate indicators, N, C and the highest citation count cMAX. The formula is based on the ‘specific’ assumption of geometrically distributed citations, but provides a good estimate of the h-index for the general case. To empirically evaluate the adequacy of the fit of the proposed formula h˜w, an empirical study with 131 datasets (13,347 papers; 288,972 citations) was carried out. The overall fit (defined as the capacity of h˜w to reproduce the true value of h, for each single scientist) was remarkably accurate. The predicted value was within one of the actual value h for more than 60% of the datasets. We found, in approximately three cases out of four, an absolute error less than or equal to 2, and an average absolute error of only 1.9, for the whole sample of datasets.

Measuring the citation impact of journals with generalized Lorenz curves

Abstract To improve comparisons of journals, which are typically based on single-value indicators, such as the journal impact factor (JIF), this paper proposes a functional approach. We discuss interpretatively three progressively finer dominance relations. The first one corresponds to a comparison between the quantile functions of the citation distributions. The second one consists in comparing the integrals of the quantile functions—namely, the generalized Lorenz curves (GLCs). The third one consists in comparing the integrals of the GLCs, where the integration is designed to emphasize the role of the “central body” of the articles of the journal. Although dominance relations are generally not complete orders, we demonstrate with an empirical analysis that it is possible to increase significantly the proportion of pairs of journals that are comparable by moving from the first to the second criterion, and then from the second to the third. Because, in practical applications, it may be convenient to reduce such a functional comparison to a scalar comparison between indicators, we follow an axiomatic approach to identify classes of indicators that are isotonic with the criteria introduced. We demonstrate that the established JIF may be usefully improved if it is corrected simply by multiplying it by one minus the Gini coefficient. The resulting index, defined as stabilized-JIF , has many attractive features and it is isotonic with all the dominance relations introduced.