Lucio Bertoli-Barsotti

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.

Improving a decomposition of the h-index

Dear Sir, In a recent note, Bartolucci (2012) introduced an interesting decomposition of the h-index (Hirsch, 2005). As noted by the author, the h-index is defined by Hirsch as follows: “A scientist has index h if h of his or her Np papers have at least h citations each and the other (Np h) papers have h citations each” (p. 16569). In his contribution, Bartolucci pointed out that the h-index may be factored as h = mr, where m N N p c tot = ⎡⎣ ⎤⎦ ( ) min , , , and where Nc,tot is the overall number of citations of the Np papers, and [z] represents the function integer part of the number z. The author claims that r may be interpreted as a measure of concentration of the citations of the first (that is, most cited) m papers. But, strictly speaking, this is not always the case, because in general the h-index depends on the concentration of the citations on a smaller number of papers, say m m ≤ . In other words, the upper bound for h may be lowered. This is due to the exchangeable role played by the notions of “papers” and “citations” in the computation of the h-index. I illustrate this using a symmetry argument. First of all, note that construction h does not depend on papers without citations. Then, let n, n Np, denote the number of the papers, say p1, p2, . . . , pn, with at least one citation each (cited papers). Let ci 1 be the number of citations of paper pi, i = 1, . . . , n, and let (x1, . . . , xn) be the ordered version of vector (c1, . . . , cn), with x1 x2 . . . xn. Additionally, let (y1, . . . , yk) be the vector with elements yj = number of papers with at least j citations j = 1, . . . , k, where k represents the maximum number of citations per paper. Note that, by construction, k = x1, n = y1, and N x y c tot i i n j j k , = = = = ∑ ∑ 1 1 . Then, it is easy to see that both

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.

Refusal to Answer Specific Questions in a Survey: A Case Study

It is well known that non ignorable item non response may occur when the cause of the non response is the value of the latent variable of interest. In these cases, a refusal by a respondent to answer specific questions in a survey should be treated sometimes as a non ignorable item non response. The Rasch-Rasch model (RRM) is a new two-dimensional item response theory model for addressing non ignorable non response. This article demonstrates the use of the RRM on data from an Italian survey focused on assessment of healthcare workers’ knowledge about sudden infant death syndrome (that is, a context in which non response is presumed to be more likely among individuals with a low level of competence). We compare the performance of the RRM with other models within the Rasch model family that assume the unidimensionality of the latent trait. We conclude that this assumption should be considered unreliable for the data at hand, whereas the RRM provides a better fit of the data.

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.

Identifying Guttman Structures in Incomplete Rasch Datasets

In applications of IRT, it often happens that many examinees omit a substantial proportion of item responses. This can occur for various reasons, though it may well be due to no more than the simple fact of design incompleteness. In such circumstances, literature not infrequently refers to various types of estimation problem, often in terms of generic “convergence problems” in the software used to estimate model parameters. With reference to the Partial Credit Model and the instance of data missing at random, this article demonstrates that as their number increases, so does that of anomalous datasets, intended as those not corresponding to a finite estimate of (the vector parameter that identifies) the model. Moreover, the necessary and sufficient conditions for the existence and uniqueness of the maximum likelihood estimation of the Partial Credit Model (and hence, in particular, the Rasch model) in the case of incomplete data are given – with reference to the model in its more general form, the number of response categories varying according to item. A taxonomy of possible cases of anomaly is then presented, together with an algorithm useful in diagnostics.

Reply to the comments of Prathap

In a recent article (Bertoli-Barsotti and Lando 2017), a new ‘‘ready-to-use’’ formula for predicting the value of the h-index as a function of simple citation indicators has been studied. To preliminarily summarize some facts about terminology, I would emphasize the distinction between the case of (1) a parametric formula/model for the h-index and (2) a ‘‘ready-to-use’’ formula for (estimating) the h-index. In a broad sense, I would refer to the latter terminology to denote the case in which the formula is no longer dependent on unknown constants (parameters) but is only a function of one or a few instances of basic statistics, i.e. simple citation indicators such that:

Estimating a Rasch Model via Fuzzy Empirical Probability Functions

The joint maximum likelihood estimation of the parameters of the Rasch model is hampered by several drawbacks, the most relevant of which are that: (1) the estimates are not available for item or person with perfect scores; (2) the item parameter estimates are severely biased, especially for short tests. To overcome both these problems, in this paper a new method is proposed, based on a fuzzy extension of the empirical probability function and the minimum Kullback–Leibler divergence estimation approach. The new method warrants the existence of finite estimates for both person and item parameters and results very effective in reducing the bias of joint maximum likelihood estimates.