Which fields drive the h-index?

Published

September 16, 2024

Title

Which fields drive the h-index?

Authors

Paolo Giudici, Luca Boscolo.

DOI

10.62684/FSOZ4761

Keywords

H-index, Poisson models, Scaling

Paolo Giudici^(a), Luca Boscolo^(b).

^(a) Department of Economics and Management Sciences, University of Pavia, Italy.

^(b) Top Italian Scientists founder.

Abstract

The measurement of the quality of academic research is often done by means of the h-index measure. Although widely accepted, the h-index has some issues and, in particular, it may depend on the scientific field in which a researcher operates. To date there is not a definitive answer as to whether this difference holds, and to what extent it varies. To fill the gap, we propose to operationally measure the difference in h-index across the sectors of a relatively homogeneous population of all scientists of a nation. To answer the heterogeneity issue we apply three different explainable machine learning models: linear regression, Poisson regression and tree models. Our results show that the latter two models better explain the data. They show that the only sectors for which a difference in h-index is significant are Physics, Biology and Clinical Sciences.

Introduction

The measurement of the quality of academic research is a rather controversial issue. in the 2000s [11] has proposed a measure that has the advantage of summarizing in a single summary statistics the information that is contained in the citation counts of each author. From that seminal paper, a large amount of research has been produces, focusing on in particular on the development of correction factors to the h index ([13], [3], [9]), [2], [12] that may take into account differences between sectors.

In this stream of research, [9] analysed the mathematical properties of the h index, and [3]proposed to employ a stochastic model for an author’s production/citation patterns. Following this mathematical formalisation, it becomes possible to analyse the h-index of individual researchers, whether or not in different fields, and compare them with each other.

Along a more empirical research line, [13] proposed to use a simple multiplicative correction to the h index to take into account the differences among researchers coming from different sectors, thus allowing a fair and sustainable comparison. They propose in particular a table with such normalizing factors, according to specific distributional assumptions of the citations. Their approach provides a simple way to explain and measure differences between different scientific fields. In a similar vein, [2] propose a rescaling procedure based on the Gini entropy and [12] propose a different rescaling, that takes into account the number of coautors: the fractional h-index.

We employ both streams of research as a starting point. More precisely, we follow [6], who, expanding the contribution of [9], propose a statistical approach that indicates that a Poisson distribution is a well suited approximation for the distribution of the h-index. In this paper we will show that a Poisson distribution is well suited to explain the drivers of the h-index. And we will employ this theoretical result to understand whether the h-index of a scientist depends on his/her filed of research, following the research line of [13], also followed by [15].

The paper is organized as follows: in Methodology section we review the proposal of [6] and formalise the model; in Application section we apply the new approach to a database of scientists homogeneous by nationality and, therefore, by scientific culture. Finally, Discussion section contains some concluding remarks.

Methodology

The paper of [11] has proposed a ”transparent, unbiased and very hard to rig measure” ([1]): the h index.

According to the definition, a scientist has index h if [math]\displaystyle{ h }[/math] of his or her n papers have at least [math]\displaystyle{ h }[/math] citations each and the other [math]\displaystyle{ (n-h) }[/math] papers have [math]\displaystyle{ ≤ h }[/math] citations each.

Following the work of Hirsch, many papers have discussed its application, especially in the bibliometric community. Some papers have focused on the statistical learning aspects behind the h index, and, among them, [9] who has stressed relevance of a ”statistical background” for the h index. Recently [?] has provided a complete statistical framework for the h index that holds for all sample sizes and respects the discrete nature of the citations data which are behind the h-index. We now recall their proposal as it dorms the basis of our analysis.

Let [math]\displaystyle{ X_1, . . . , X_n }[/math] be random variables which describe the number of citations of the n articles of a scientist. We assume that [math]\displaystyle{ X_1, . . . , X_n }[/math] are independent with a common citation distribution function [math]\displaystyle{ F }[/math]. Let us then assume that [math]\displaystyle{ F }[/math] is continuous, at least asymptotically, although the citation counts are integers. According to this assumption, the h index can be formally defined by the following:

[math]\displaystyle{ h = 1 - F(h) = \frac{h}{n} }[/math]

Then, following [6], assume that [math]\displaystyle{ F }[/math] is discrete. Given a set of n papers of a scientist to which a citations count vector [math]\displaystyle{ \underline{X} }[/math] is associated, consider the ordered sample of citations [math]\displaystyle{ {X_{(i)}} }[/math], that is [math]\displaystyle{ X_{(1)} ≥ X_{(2)} ≥ . . . ≥ X_{(n)} }[/math], from which obviously [math]\displaystyle{ X_{(1)} (X_{(n)}) }[/math] denotes the most (the least) cited paper. The h index can be defined as follows:

[math]\displaystyle{ h = max(t : X_{(t)} ≥ t) }[/math]

The distribution of the h index can then be shown to be:

[math]\displaystyle{ p(h) = [F(X_{j(h)}) − F(X_{j(h)+1})]^{(n+1−h)} }[/math]

The previous expression, albeit elegant, is non parametric, and is not so transparent in the estimation process.

To formulate a more explainable parametric specification, [6] suggested to follow the Loss Distribution Approach (LDA) employed in operational risk modelling (see [7] and [8]) where the losses are categorized in terms of ’frequency’ and ’severity’ (or impact). The frequency is the random number of loss events occurred during a specific time frame, while the severity is the mean impact of all such events in terms of monetary losses.

In the context of the h-index, the frequency is the (random) number of published papers along the career of a scientist and the impact is the (random) mean number of citations received in the same time frame by all such papers.

Let [math]\displaystyle{ Xi = (X_{i1}, X_{i2}, . . . , X_{ini}) }[/math] be a random vector containing the citations of the [math]\displaystyle{ n_i }[/math] papers published by the [math]\displaystyle{ i-th }[/math] scientist.

It follows that the total impact of a scientist [math]\displaystyle{ i }[/math] can be defined as the sum of a random number [math]\displaystyle{ n_i }[/math] of random citations:

[math]\displaystyle{ Ci = X_{i1} + X_{i2} + . . . + X_{ini} }[/math]

It can be shown that the above formula can be equivalently expressed as follows:

[math]\displaystyle{ C_i = n_i × m_i }[/math]

(1)

where [math]\displaystyle{ m_i = \frac{\sum_{j=1}^{n_i} X_{ij}}{n_i} }[/math] is the mean impact of a scientist.

Assuming that the scientists [math]\displaystyle{ i = 1, . . . , I }[/math] belong to a homogeneous community, conditionally on the production of each scientist (with number of papers equal to [math]\displaystyle{ n_i }[/math]), the citations of the papers [math]\displaystyle{ X_{ij} }[/math] , for [math]\displaystyle{ j = 1, . . . , n_i }[/math] are independent and identically distributed random variables, with common distribution [math]\displaystyle{ k(m_i) }[/math]:

[math]\displaystyle{ k(x_{i1}) = k(x_{i2}) = . . . = k(x_{{in}_i}) = k(m_i) }[/math]

Declarations

Conflict of Interest

The Authors declare that there is no conflict of interest.