Gender issues in fundamental physics: A bibliometric analysis

This paper originates from an observational opportunity: For the first time sociological issues in fundamental physics can be studied using the public InSpire database, which has accumulated bibliometric data about fundamental physics worldwide from around 1970 to now (InSpire, 2010). Fundamental physics is a subdiscipline of physics that deals with the fundamental aspects of the field and that presently focuses mostly on particle physics, cosmology, and astrophysics, from an experimental and theoretical point of view.

Such bibliometric data are being used to study various aspects of the field. Like other science, technology, engineering, and mathematics (STEM) fields, physics exhibits persistent gender differences, which I try to characterize and understand in the present paper. The bibliometric approach relies on large amounts of objective quantitative data about papers, authors, citations, and hires. Having a large amount of new data I will follow a data-driven approach. Enough statistics is sometimes needed to reveal effects, and to go beyond simple counting by devising dedicated analyses that target specific questions. I can do this, as I have the full database, not just access to some predefined metrics.

While a vast literature has studied gender differences in STEM, no previous studies have specifically focused on fundamental physics: The present study will fill this gap.¹ A main theme is understanding why women remain underrepresented in STEM fields, a worldwide phenomenon that has persisted for decades, despite interventions on its alleged social causes (Stoet & Geary, 2018). A limitation of bibliometric analyses is that authors start being scientifically active roughly at PhD level: In physics (as in other STEM fields) low female representation is already present at this entry level of bibliometric data. Earlier phases need to be explored with other tools.

An important clue is that a similar gender difference already appears in surveys of occupational plans and first choices of high-school students (Ceci, Ginther et al., 2014; Xie & Shauman, 2003). This is possibly mainly due to gender differences in interests (Ceci et al., 2014; Hyde, 2014; Lippa, 2010; Stoet & Geary, 2018; Su & Rounds, 2015; Su, Rounds, & Armstrong, 2009; Thelwall, Bailey et al., 2019). Gender differences in relative attitudes (girls with high mathematical ability tend to also have high verbal ability) also contribute to student choices (Ceci et al., 2014; Stoet & Geary, 2018; Wang, Eccles, & Kenney, 2013): Most of the gender gap in student intentions to study math disappears after taking into account their mathematics versus reading difference in PISA scores, while absolute results (boys outperform girls in mathematics, girls outperform boys in reading) are much less able to explain the gender gap (Breda & Napp, 2019).

Coming to the later phase that can be studied by experiments and by bibliometrics, initial small-scale experiments and anecdotal reports suggested biases against hypothetical female applicants (see, for example, Moss-Racusin, Dovidio et al. [2012] and Wennerås and Wold [1997]); see also Eaton, Saunders et al. [2019]. These findings have not been supported by more recent larger-scale experiments (see the review in Ceci and Williams [2011] and also Ceci and Williams [2015] and Williams and Ceci [2017]). Milkman, Akinola, and Chugh (2015) sent letters by fictional students seeking research opportunities to professors and measured their response rate. The result of this social experiment performed in the United States was that, looking at gender in isolation (rather than at “women and minorities”), female students received slightly more responses from public schools (the majority of the sample) with respect to men in the same racial group.

Experiments about hypothetical applicants often miss key elements of real applications, mostly based on scientific results reported in publications and evaluated by scientists who work in the same subfield. No significant biases have been found in examined real grant evaluations (Ceci et al., 2014; Ley & Hamilton, 2008; Marsh, Jayasinghe, & Bond, 2011; Mutz, Bornmann, & Daniel, 2012) and referee reports of journals (Borsuk, Aarssen et al., 2009; Ceci et al., 2014; Edwards, Schroeder, & Dugdale, 2018); the gender composition of applicants (Way, Larremore, & Clauset, 2016) and panels (Abramo, D’Angelo, & Rosati, 2015) has little effect. Real hires show a higher success rate among women (Ceci et al., 2014; Glass & Minnotte, 2010; National Research Council, 2009; Wolfinger, Mason, & Goulden, 2008), especially in those STEM fields where women are less represented (Ceci et al., 2014). Bibliometric attempts to recognize higher merit (Ceci et al., 2014) found that male faculty members write more papers (Abramo, D’Angelo, & Caprasecca, 2009a; Fox, 2005; Larivière, Ni et al., 2013; Levin & Stephan, 1998; Way, Larremore, & Clauset, 2016; Xie & Shauman, 1998) (see also Holman, Stuart-Fox et al. [2018] and Thelwall [2018]), predominate among first and last authors (prestigious in some fields) and in single-authored papers (Jagsi, Guancial et al., 2006; West, Jacquet et al., 2013). Such gender productivity gap persists after accounting for confounding factors such as seniority (Caplar, Tacchella, & Birrer, 2017; Ceci et al., 2014; Moldwin & Liemohn, 2018). Consistent with these results, Witteman, Hendricks et al. (2019) found that female grant applications in Canada are less successful when evaluations involve career-level elements. Some studies observed a small group of extremely productive, mostly male, “star authors” (Abramo, D’Angelo, & Caprasecca, 2009b; Abramo, Cicero, & D’Angelo, 2015; Bordons, Morillo et al., 2003). A smaller “leaky pipeline” rate of female authors is observed in STEM fields than in other fields with higher female representation (Ceci et al., 2014). Looking at PLOS medical journals, where each author declares his or her role (analysis, design, material, perform, write), Macaluso, Larivière et al. (2016) found that women were more involved in performing experiments, and men more involved in the other roles.

This paper is structured as follows.

In Section 2 I describe how I identify the gender of authors; how I obtain lists of hires; how I combine citations to define bibliometric indicators that can be used as reliable proxies for scientific merit, being significantly correlated to human evaluations such as scientific prizes; and how I deal with the confounding factor due to the gender evolution of the field.

In Section 3 I present findings that exhibit interesting gender differences. New authors today appear with roughly 4:1 male:female proportion, with order one variations in different countries. I find that this entry difference in representation is negligibly affected by hiring, consistent with Ceci et al. (2014). As I use citations, in Section 3.1 I first verify that male (M) and female (F) authors cite in the same way. I achieve this by defining a gender asymmetry in citations sensitive only to a differential gender bias, not to gender differences in number or productivity of authors. In Section 3.2 I then compare reliable bibliometric indices based on citations finding that F authors are hired with indices that are, on average, not higher than those of M authors. Section 3.3 finds a productivity gap consistent with previous studies. This new difference is quantitatively studied in Section 3.4 finding that the M fraction progressively grows going from average to top authors, consistent with Abramo et al. (2009b, 2015), and Bordons et al. (2003). In Section 3.5 I consider self-references, finding no new gender differences. Data are made available as discussed in the “Data availability” section. As bibliometric data are influenced by a complicated background of social and historical accidents in the supplementary Section S1 I show that the results above persist after taking confounding variables into account. Statistical details are presented in the supplementary Section S2. Interpretations of the data are discussed in Section 4.

The public InSpire database (InSpire, 2010) maintained by CERN and other institutions offers a picture of fundamental physics worldwide from around 1970 to now. InSpire gives data on about 1.3 million scientific papers, 30 million references, 71,104 identified authors in over 7,000 institutes and over 6,000 collaborations. InSpire individually identified all authors (except occasional authors), solving the problem of name disambiguation. I expect that the database is negligibly affected by direct or indirect gender bias. Indeed, the database provides essentially full coverage of the scientific literature within “fundamental physics.” For decades this has been a self-contained, highly specialized subject (Sinatra, Deville et al., 2005), so that the “boundaries” of the database play a minor role. Papers in InSpire include all those published in some categories of the preprint bulletin arXiv and in some journals with topics considered relevant for fundamental physics. Human intervention is minor, such as adding extra papers considered relevant. Only a minor fraction of authors occasionally work in other fields or arrive from other fields. A possible gender difference in multidisciplinary attitudes is thereby expected to have negligible impact on the subsequent discussion. On the other hand, various authors work on multiple topics within the field, which cannot be sharply subdivided.

InSpire does not provide gender information: In Section 2.1 I describe the procedure to infer gender from full names and nationality of the authors. In Section 2.2 I describe how I obtain lists of hires in fundamental physics worldwide. Section 2.3 motivates the bibliometric index that I will use to indicate scientific merit.

I need to infer gender from names in an accurate and complete way.² Three main problems are encountered. First, the InSpire database provides only name initials for about 13% of the authors. These are mostly authors with little impact, as defined by any index. Second, some names such as Nicola are “ambiguous”: They correspond to different genders in different countries. Third, some authors have unusual names. The Mathematica machine learning function Classify (Wolfram Mathematica, n.d.) uses information about the first name only and leaves about 40% of authors with unclassified gender.

I tested two approaches to determine their strengths and to choose the best combination:

1. First, I run the online Ethnea (Torvik & Agarwal, 2016) tool, which uses the full name (first and family name) to infer both gender and ethnicity. Ethnea leaves 26% of the authors with unclassified gender.

2. Second, for each author I extract a “guessed” nationality from the earlier affiliations in his or her papers and use it to disambiguate “ambiguous” names. The obtained list of first names and nationalities is matched to a database of names and countries from the Worldwide Gender-Name Dictionary (WGND) (Raffo, 2016). This database contains 175,917 names with their associated countries. About 70% of authors have “unambiguous” names that are present in the WGND. Authors with “ambiguous” names present in the WGND are matched using the nationality inferred from their earliest affiliations. The size of this subset of authors is approximately 3% of the total, and the uncertainty induced by this procedure is below the per cent level. About 0.1% of the authors have “ambiguous” names and no nationality information: I match them to the most common gender corresponding to their name, defined as the one used in the largest number of countries. Twenty-three percent of the authors remain unclassified.

The results discussed in the following are affected in a minor way using the Ethnea or WGND classifications. By comparing them it can be seen that the Ethnea classification is less complete, leaving unclassified more authors with unusual names. On the other hand, the WGND classification leads to some authors with misidentified gender, typically arising due to a misidentification of their nationality. Different genders are found for 1.8% of all identified authors; the percentage rises to 5% among Chinese,³ Indian and Korean authors, and falls to 1% among European authors.

As a best choice, I adopt the Ethnea classification whenever available, and the WGND classification otherwise. Furthermore, I selected 1,000 of the top-cited authors in different time periods and systematically verified and correctly assigned their gender with no errors, using information available on the internet.

InSpire is integrated with HepNames, a database with biographical information about the various authors, including papers, affiliation history, experiments they participated in, PhD advisor, and graduate students. As an example, the internet page http://inspirehep.net/author/profile/A.Strumia.1 shows the profile of the present author. A user interface allows researchers to create and update HepNames records for themselves and for other authors, providing precise career information on a voluntary basis (to be validated by the InSpire team). Furthermore, large collaborations systematically provide complete author information upon submission of documents through a dedicated format.

From HepNames I obtained a database of about 10,000 first hires in fundamental physics worldwide, including dates and disambiguated institutions. Such InSpire hires might be biased if those F and M authors who need to self-report their data tend to do this differently. While InSpire is a widely used tool in the community, integrated with a job announcement system, funded by multiple official institutions and endorsed in Reviews of Particle Physics, occasional authors might not use it.

I therefore complement InSpire hires by computing unbiased “pseudohires,” defined as follows. For each paper, there is a list of disambiguated affiliations of each author. I consider an author as pyr-hired when he or she starts writing papers with the same affiliation for at least p years. Using this definition I obtain a database of about 40,000/19,000 5/10yr-first hires from 1960 to 2013/2008 (64,000/23,000, including multiple hires for the same author). However, in this way it is not possible to obtain a precise hiring date for the subset of authors hired by the same institution to which they were previously affiliated.

Therefore, InSpire hires will be used when a precise hiring date is more important than increased statistics, and pseudohires in the opposite situation, when full coverage is more important than precise timing. In each case, the other sample will be used as a control sample.

I here motivate the use of appropriate bibliometric indicators as a proxy for what is commonly considered as scientific merit.

Various authors have studied what citation counts do measure.

At theoretical level, two main models have been proposed. According to the normative interpretation, scientists primarily cite to give credit. A bibliometric index then provides a valid proxy of scientific merit, especially when highly correlated with scientific prizes or other human evaluations of scientific merit. According to the social-constructivist interpretation, citations are instead primarily a social persuasion tool; the concept of scientific merit itself is questioned: as reviewed in Bornmann and Daniel (2008) “scientific knowledge is socially constructed through the manipulation of political and financial resources and the use of rhetorical devices.” According to this point of view, citation counts could be correlated to prizes simply because both reflect social status. Some prizes in physics require established scientific results; others are awarded following rules that leave more space for sociological distortions.

Observational works supported the normative interpretation at high aggregation level (Bornmann & Daniel, 2008; Tahamtan & Bornmann, 2019); personal factors important at individual level average out when considering many authors. My data provides extra evidence in this direction: For example, authors of top-cited papers tend to be younger, rather than powerful senior scientists.

Citation counts are surely influenced by some confounding factors (Bornmann & Daniel, 2008): the citation intensity depends on time and field (my indicator will compensate for this), on language (essentially all physics literature is in English), on accessibility (physics literature has been freely available on the preprint bulletin arXiv since 1995), and on collaboration size.

Collaboration size is a big issue in fundamental physics, due to the presence of very large (up to 3,000 authors) and very productive (up to 6,000 papers) collaborations, mostly in high-energy experimental physics. Because of this main reason, traditional metrics (such as citation counts, h-index, paper counts) now fail to provide reasonable proxies for scientific merit in fundamental physics (Strumia & Torre, 2019). Signing more papers that one can read stretches the concept of authorship (Birnholtz, 2006). At a quantitative level, the problem is that the contribution of one big collaboration overwhelms the database (1.3 million papers) if 6,000 papers are counted as 3,000 × 6,000 = 1.8 million.

This situation can be corrected by “fractional counting” (Hooydonk, 1997; Leydesdorff & Park, 2016; Perianes-Rodriguez, Waltman, & van Eck, 2016): A fraction 1/N_aut of each paper (rather than the full paper) is equally attributed to its N_aut authors, as appropriate for an intensive quantity. All authors, including first and last authors, are treated on equal footing because authors are usually sorted alphabetically in fundamental physics, unlike in other fields.⁴ Thereby there is no way of telling who contributed what to multiauthored papers. When huge collaborations are involved there is no warranty that each author contributed to each paper. Despite this, the data show that the total fractionally counted bibliometric output of collaborations scales, on average, as their number of authors (Rossi, Strumia, & Torre, 2019), suggesting that large collaborations form when scientifically needed and that gift authorship does not play a large role.

Individual citations have the following meaning: An author who wrote N_icit fractionally counted papers of average impact in his or her field received N_icit individual citations. Table 7 of Strumia and Torre (2019) lists the 50 physicists who received the most individual citations, together with their scientific prizes. Physicists can read their names and consider whether N_icit is dominantly influenced by scientific achievements or by social constructivism. For practical purposes, an indicator provides an acceptable proxy of scientific merit if scientific merit positively affects the index more than confounding variables. A full or large correlation with scientific merit improves the sensitivity of the analysis, but some effects can be large enough that fine sensitivity is not needed to reveal them.

The use of bibliometric indices based on citation counts as a proxy for scientific merit comes with limitations and dangers. Some citations are given for negative reasons. On short timescales citations are more influenced by visibility, and some authors engage in boosting their citation counts in various ways: large collaborations, many references, self-references, citation networks, salami slicing into minimum publishable units, etc. Individual citations are not boosted by the first two strategies. As this paper is concerned with gender differences, it is reassuring that Section 3.5 will find no extra significant gender differences in self-referencing.

As “when a measure becomes a target, it ceases to be a good measure,” in the supplementary Section S1 I consider a metric based on citations more different from common targets, which is not enhanced by the latter three strategies. The CitationCoin ₡ is defined as the difference between the number of received and given individual citations (up to a correction factor that prevents systematically negative contributions from recent papers), such that it is not affected by self-citations or by networks of circular citations (Strumia & Torre, 2019). Authors who write too many poorly cited papers can even have a negative ₡ score.

Bibliometric indicators measure the average opinion of the community: While all opinions can be wrong, a better possibility could be to rely on the opinion of top authors. This is done by metrics based on the PageRank algorithm (such as those discussed in Chen, Xie et al. (2007), Ma, Guan, and Zhao (2008), Pinski and Narin (1976), Radicchi, Fortunato et al. (2009), Strumia and Torre (2019), and West, Jensen et al. (2014)). This is studied in the supplementary Section S1, where for completeness I also consider the widely used but naive bibliometric indicators based on paper counting and on the average number of citations per paper.

In practice, the differences in bibliometric indices among authors are so large that log-scale plots will be appropriate and refined metrics only make minor differences. Individual citations N_icit are used because this metric is simpler and closer to the commonly used number of citations N_cit, while allowing us to meaningfully deal with experimentalists, theorists, and astrophysicists by compensating for the vastly different typical number of co-authors N_aut of papers produced by these communities.

Among the 71,104 authors in fundamental physics listed in the InSpire database, 49,860 male and 9,205 female authors were identified. 16% of authors with identified gender are classified as female, and wrote 10% of the fractionally counted papers receiving 7% of the individual citations. These raw numbers, meant only to give a first rough idea of the field, are affected by a variety of historical accidents.

As documented in Strumia and Torre (2019), the field has expanded significantly: Due to increased publication intensity, about half of citations have been given after 2000, so that metrics based on citations favor recent authors (the metric N_icit automatically compensates for publication intensity). Furthermore, the F percentage grew with time, as shown by the raw data in the left panel of Figure 1.

The right panel shows that, within the countries that most contributed to fundamental physics, the female fractions range between 7% and 23%. It is interesting to explore if the female fraction is correlated with the Global Gender Gap Index (GGGI) of the countries (World Economic Forum, 2016), which measures the gap between women and men in education, politics, health, and economy, as this is a possible cause of the low female representation. The GGGI ranges between 0 and 1, with 1 indicating parity or a gap in favor of women (as the GGGI ignores imbalances to the advantage of women). The right panel of Figure 1 shows that the female fraction is not positively correlated with the GGGI, as similarly observed among students in STEM (Stoet & Geary, 2018).

Figure 2 shows that the female percentage is a factor of 2 higher in subfields dominated by large experimental collaborations than in theoretical fields.

Clearly, the field and its gender composition have evolved in the past 50 years. While describing such changes from a bibliometric point of view is an interesting subject, I try to focus on the general features that emerge from the complicated background of social factors. This will need to take into account possible confounding variables, by studying subperiods and subtopics or by trying to compensate for the above variations.

I want to investigate if citations are influenced by the gender of the cited authors, searching for a possible different tendency of the two genders to cite a given gender more often.

In principle, complete information could be extracted by comparing “how citations are” with “how citations would be” in the absence of gender discrimination. In practice, this strategy needs a theoretical model of citations, but such models are affected by questionable systematic issues. One can try controlling for the main factors (such as different numbers of M and F authors, different average seniorities, regional differences), but reality can contain more complicated effects, such as different scientific qualities. For example, Caplar, Tacchella, and Birrer (2017) claim (consistent with my later findings) that papers in astronomy written by F authors are less cited than papers written by M authors, even after trying to correct for some social factors. After considering attributing the remaining difference to gender bias, Caplar et al. (2017) conclude “of course we cannot claim that we have actually measured gender bias.”

I will follow a different strategy, which is often more useful in the presence of backgrounds that cannot be reliably modeled: I construct an asymmetry such that it is not affected by the backgrounds. The extracted information encoded in the asymmetry is reliable but partial, as I give up on the attempt to model the full citation process.

I extract A from the data by removing self-citations, which introduce a background of same-gender preference not due to an actual gender preference. This removal is done exactly, as I have a list of all references where all authors are identified with a unique code. The removal of self-citations reduces same-gender citations, introducing a small gender discrimination of order 1/$NGaut$ in the asymmetry: When considering many authors this bias is negligibly smaller than the statistical uncertainty of A, which scales as 1/$NGaut$⁠.⁷Figure 3 shows the time evolution of the gender asymmetry, found to be compatible with zero at all times.⁸ Restricting to papers after 2010 gives the results shown in Table 1.⁹ The uncertainty is shown as one standard deviation after the ± symbol. A hint of an asymmetry, A_other = (4.8 ± 1.2)%, is observed among about 10⁴ other papers (mostly unpublished) not included in the eight major arXiv categories relevant for fundamental physics. As a result, combining all single-author papers citing single-author papers gives an asymmetry A_published = (1.0 ± 0.5)% when restricting to published papers, or A_all = (1.9 ± 0.4)% when including all papers.

The definition of the gender asymmetry could be extended to multiauthored papers knowing how a hypothetical gender bias would depend on the relative amount of F and M authors. One simple possibility is just to generalize the definition of $NG→G′cit$ into ∑_citationsf_G$fG′$ where f_G(⁠$fG′$⁠) is the fraction of authors with gender G in each citing (cited) paper. All self-citations, now defined as whenever the cited and citing paper have at least one author in common, are now dropped. With the new $NG→G′cit$ I find the results in Table 2. Uncertainties (not shown) are there about five times smaller than in the single-author sample, if propagation of errors is naively applied to fractional counts.

Taking into account the definition of the asymmetry A and the relative number of F and M authors in my data, I conclude that A is so close to zero that a nonzero gender asymmetry in citations within its measured range would not significantly distort the bibliometric indices based on citations discussed in the following.

The lack of a gender asymmetry in citations means that there is no fracture along gender lines in the community about which research in fundamental physics is more relevant/used/visible. In Section 2.3 it was shown that appropriate bibliometric indices based on citations are useful proxies for scientific merit. I here use such indices to search for a possible gender difference in hiring. For each hired or pseudohired author I compute his or her bibliometric indices at the hiring moment, defined as in Section 2.2. From this I extract the mean bibliometric indices of hired F and M authors.

The left and right panels of Figure 4 show the mean number of fractionally counted papers and of individual citations N_icit, respectively, of authors at their hiring date as reported by InSpire. For the sake of clarity I use traditional color codes: blue for male and pink for female authors.

It can be seen that hired F authors do not have, on average, bibliometric indicators above those of hired M authors. Rather, a tendency in the opposite direction seems present at all times, across the main subfields¹⁰ and most countries (statistical uncertainties become significant when restricting to some countries with not enough authors). This result persists after taking into account the possible confounding variables considered in the supplementary Section S1.1.

I next provide extra information.

Figure 5 shows the cumulative distribution of hired physicists as a function of their scientific age at hiring. It exhibits no significant gender difference. A gender difference could have been produced in various ways:

1. Some hiring committees might take into account career gaps due to maternity (about which no information is available): This would tend to increase the average scientific age of female hired scientists.

2. A gender discrimination in hiring would tend to reduce the average scientific age at hiring of scientists with the favored gender.

3. A gender difference in abandonment rates would tend to reduce the average scientific age at hiring of scientists with the higher abandonment rate.¹¹

A warning is necessary about the two next plots, which extend the analysis to authors who have not been hired. My analysis is restricted to InSpire authors listed in HepNames (described in Section 2.2), which misses many authors who leave the field after writing a few papers. This generates an extra systematic issue, which presumably tends to be gender neutral, such that gender ratios presumably are more reliable than absolute rates. Indeed, information for M and F authors presumably is similarly incomplete, as InSpire does not collect data about gender, especially of unknown authors.

Figure 6 shows the fraction of hired authors among those who started writing papers in given time periods. Significant gender differences are not seen. I used 10-year hiring because coverage is more important here than timing. Therefore the plot stops 10 years ago, and absolute numbers would be different using incomplete InSpire hiring. Furthermore, as warned above, extra un-hired authors not in InSpire would lower the hired fraction.

Figure 7 shows the abandonment rate per year as a function of scientific age. I considered departures during 2000–2015, counting as having left those authors who wrote no further papers up to 2018. Older authors started when the M fraction was higher and the abandonment rate was lower (as hinted by Figure 6): This confounder generates an apparently lower abandonment rate among M authors. I thereby compensate for gender history as described in Eq. 2. I find that the abandonment rate is maximal among older authors who retire, minimal among senior authors, and intermediate among junior authors (as warned above, the abandonment rate of very young authors who leave the field after writing just a few papers is underestimated). Abandonment rates show no significant gender difference, in agreement with the null result by Perley (2019) and in disagreement with Flaherty (2018) (these authors only considered astrophysics).

In conclusion, the gender gap in representation at the entrance level of research is negligibly affected by “leaky pipeline” effects consistent with Ceci et al. (2014) that finds large gender differences at PhD level in STEM, and mild differences in the subsequent progress; see also Allen-Hermanson (2017) and Miller and Wai (2015).

In this section I study scientific productivity as quantified through bibliometric indices. Of course, such indices say nothing about other activities of researchers that do not result in publications, such as teaching, mentoring, and outreach. Figures 1, 2, 8, and 11 show a possible gender gap in the fractionally counted number of papers: Male authors write, on average, 10% more papers. The gap is consistent with earlier findings in the literature (see, for example, Table 2 of Ceci et al. (2014), Abramo et al. (2009a, 2015). A slightly larger gap is found in the number of received individual citations.

Is such a gap due to the different average scientific age of M and F authors? To check this possibility, the left panel of Figure 8 shows the mean number of fractionally counted papers written by M and F authors as a function of their scientific age (time since their earliest paper). The gap persists. The right panel of Figure 8 similarly shows the mean number of received individual citations. In both cases it can be seen that junior M and F authors have similar productivity, and that a gap develops with their scientific age. A higher scientific age means going backwards in time to authors who started earlier, when the field was different and when the F percentage was smaller.

The averages in Figure 8 are shown separately for hired and not-hired authors, using 10-yr-hires in order to have more complete coverage. It can be seen that hiring is not the reason for the gap.

Furthermore, in Figure 8 only scientifically active authors are considered (those who wrote at least one paper after 2013), such that these results would not be affected by a gap in abandonment rates.

Figure 8 does not compensate for possible career gaps, as such gaps do not exhibit significant gender differences. This is shown in Figure 9, where for each author I computed the longest time gap between consecutive papers, using arXiv dates to have precise information about publication dates. The distribution of longest gaps among M and F authors shown in Figure 9 does not exhibit significant gender differences. A similar null result is found when restricting to hired authors. Stopping writing papers might, however, be the extremum of a tendency towards reduced productivity (possibly due to maternity issues). I therefore searched for consecutive years of reduced publication intensity: Some authors are more regular, other experience periods of relatively lower productivity, but again the distributions show no significant gender differences (see Figure 10). No significant gender differences are found looking at periods of relatively higher productivity.

A gender difference in abandonment rates or career gaps or periods of lower productivity would reduce the cumulative number of papers and of received citations of authors at career level. It is thereby interesting to test whether a gap persists in noncumulative productivity indices that avoid summing over author careers. The procedure is as follows: For each year the subset of scientifically active authors that produced papers is selected, and Figure 11 shows their average productivity, separately for M and F authors. I find that active F authors produce on average roughly 30% fewer papers than active M authors, and receive roughly half the number of citations.¹²

Furthermore, Figure 12 (left panel) analyzes the gap at the level of papers, finding a smaller F percentage among authors of top-cited papers, even when restricting to single-author papers (see also Jagsi et al., 2006; West et al., 2013). The right panel of Figure 12 shows that F authors tend to work in larger collaborations.

The supplementary Section S1.2 discusses other possible confounding variables, without finding anything that can remove the gender gap in productivity.

I now discuss some possible causes of such a gap.

In various countries F authors have earlier retirement ages. But gender differences show up before retirement in Figure 8. Furthermore, many physicists tend to remain scientifically active after retirement (although the productivity of most physicists tends to decline before retirement).

A possible reason for the gender gap observed in various fields is children and maternity. See Ceci et al. (2014) for a recent summary of the literature, which is not univocal. Some studies find no or small effect (Cole & Zuckerman, 1987; Sax, Hagedorn et al., 2002; Stack, 2004; Xie & Shauman, 2003). Other studies find a negative impact on women (Fox, 1995; Ginther & Kahn, 2009), and on men and women equally (Hargens, McCann, & Reskin, 1978), while some studies found a positive impact on men (Ceci et al., 2014), possibly due to selection effects. Results vary depending on field (with physical sciences sometimes being an outlier, possibly a fluctuation) and are mostly focused on the situation in the United States and on the number of papers produced or worked hours. Ceci et al. (2014) conclude: “the presence of children cannot explain the overall gender productivity gaps.” While maternity would deserve a dedicated study, the InSpire data do not provide any personal information, so it is only possible to proceed indirectly. As has already been described, timing of publications does not show gender differences in periods of null or reduced productivity. Figure 8 indicates that the productivity gap opens at an age roughly consistent with maternity (but also consistent with the transition to scientific independence), and that it does not close at older ages. A similar situation is found when analyzing the salaries of physicists in the United States: no gender gap just after graduation; a 10% gap after 10–15 years according to Porter and Ivie (2019), who report large differences in personal life choices, in particular that women are four times more likely to have a career break.

As maternity laws are different in different countries, an alternative possible strategy is looking for national differences in the M/F gap, which seems stronger in Germany, the United Kingdom, and Italy; weaker in the United States and France, and null in Japan. However single-country statistics are poor and many other national differences can act as confounding factors.

In the previous section a productivity gap was found. I here characterize its statistical properties. Figure 13 shows the distributions in the number of individual citations N_icit received by female and male authors in fundamental physics, considering the whole InSpire database. The bell-shaped distributions spread through a few orders of magnitude in N_icit. The dotted curves show that each bell is well approximated, at least on its upper side, by a log-normal as a function of N_icit (namely, by a Gaussian as function of log N_icit, the variable used on the horizontal axis of Figure 13; log is the logarithm base 10). A log-normal, already observed in bibliometrics (Thelwall & Wilson, 2014), arises when many positive independent random variables contribute multiplicatively.¹³

The difference between the F and M distributions in Figure 13 is statistically significant. This is better shown by the black curve in Figure 13, that shows the ratio N_F/N_M of female versus male authors (left axis) as function of the number of received individual citations (horizontal axis).¹⁴ The N_F/N_M gender ratio is not constant: The M fraction progressively grows when going from average to top authors in terms of individual citations.

Again it is necessary to study whether such a difference can be a byproduct of confounding factors that affect my composite sample of data. This issue is discussed in supplementary Section S1.3: I do not find any confounder that washes away the trend. As discussed in Section 2.4 I compensate for one significant confounder: Male authors are presently on average more senior than female authors. This is relevant here because senior authors had more time to receive citations (and because younger authors contribute more to top-cited papers). This confounder does not remove the gender difference in N_icit, given that it is observed within sub­samples of authors with same scientific age (see supplementary Figure S11). Correcting for the age confounder is, however, needed to precisely quantify the difference. I proceed as described in Eq. 2. The left panel of Figure 13 shows age-corrected results, while the right panel shows raw data. Both averages and variances differ in raw distributions. The difference in variances persists among the upper sides of the age-corrected distributions, while no significant gender differences are seen on the lower sides of age-corrected distributions, more affected by social phenomena (lower sides would be mostly removed by restricting to hired authors).

Different ages can be accounted for by using a complementary strategy: I consider a new bibliometric index N_icit/Δ$tAp$ that approximatively compensates for the scientific age Δt_A = t_now − t_A of each author. In order to achieve the compensation I choose p ≈ 1.8, because N_icit averaged over authors approximatively scales as Δ$tA1.8$ within all main subfields. The result of such a correction is shown in the left panel of Figure 14: Both M and F distributions become narrower, having removed one source of their variability. I again find a higher male variance among the upper sides. The right panel of Figure 14 shows that applying both age corrections has negligible extra effect. The difference in variances on the upper side is seen in any case.

The M fraction is largest among top authors, as clear from the dots below the bells in the right panel of Figure 13, which show the individual authors. It is useful to focus on the subset of top authors. A physicist might read their names and conclude that no sociological confounder can wash away most of them. It is thereby interesting to show that the gender difference is statistically significant, even when restricting to top authors. This is done by considering the hypothesis of no gender difference: While being sociologically implausible it has precisely computable consequences that can be compared to data about top authors. Starting from raw data, the F author with most individual citations is in position F₁ = 69. Under the hypothesis of no gender difference one can mathematically compute that the probability of observing F₁ ≥ 69 is ℘₁ = m^F1−1 ≈ 3 10⁻⁶, where m = 1 − f ≈ 0.83 is the male fraction of the large sample (or m ≈ 0.87 restricting to theorists). Under the same assumption, the kth F author should be on average position 〈F_k〉 = k/f, with probability distribution f^km^Fk−k(⁠$Fk−1k−1$⁠) (Knapp, 2010). The observed positions are F₂ = 147 (the probability of being in this or lower position is ℘₂ ≈ 3 10⁻¹¹), F₃ = 191 (℘₃ ≈ 2 10⁻¹³), etc., roughly fitted by F_k ≈ 69k. Such low probabilities mean that gender differences are needed to account for data about top authors. Of course, it is already known that the time evolution of f = N_F/(N_M + N_F) is a significant confounder. I again find that such a confounder is not enough to remove the difference, as the female fraction f is larger than 1/69 at all relevant times. More precisely, after performing the age correction as in Eq. 2, the kth top female authors shift to F_k = {36, 82, 114, 126, …} ∼ 33k position. Considering the age-corrected metric N_icit/Δ$tA1.8$ the positions are F_k = {20, 57, 109, 180, …} ∼ 38k. After correcting for the age confounder, ℘-values remain small because the female fraction ∼1/33 or ∼1/38 found among top authors remains smaller than the fraction f of female authors in the full sample. An excess of male top authors is found even after correcting for the age confounder.¹⁵

The fact that gender differences are maximal among those top authors who receive up to 3,000 more individual citations than average authors indicates that differences do not predominantly result from sociological factors that could give factors of 2 differences at individual level (such as harder working vs. career gaps, more research vs. more teaching, specialization on physics vs. wider interests, etc).

As the variation of N_F/N_M survives to confounding variables, I try to better understand this effect by investigating its quantitative shape.

Is this statistically strong preference an artefact of the complexity of the full data sample? To answer, the analysis is repeated within the independent subsamples of supplementary Figure S11: Plotted on a log-log scale they independently tend to show a parabolic (rather than linear) trend in N_F/N_M. The dotted curves in Figure S11 show how well each subsample can be fitted in terms of R_σ ≈ 2 and p ≈ 2. Furthermore, the probabilities ℘_i that test the hypothesis of no gender difference restricting to top authors are small within the subsamples of Figure S11 (where the top authors are plotted as points), consistent with Abramo et al. (2009b) and Bordons et al. (2003).

A similar difference in upper variances is also found when looking at different bibliometric indicators; see the supplementary Section S1.3.

Gender differences in self-references are an interesting topic on its own, as as well as a possible confounding factor to previous analyses. I verified that my previous results persist when dropping self-references, and I next justify why there is no need to drop self-references.

Bibliometric studies have found that men cite their own papers more than women: Cameron, White, and Gray (2016) focused on six ecology journals; Ghiasi, Larivière, and Sugimoto (2016) on the Web of Science database; King, Bergstrom et al. (2017) on the JSTOR database; and Hossenfelder (2018) on single-authored papers in arXiv. In agreement with such studies, restricting to solo papers, I find a ∼20–30% higher fraction of self-references among the papers written by M authors (7.3% versus 5.9%).

However, King et al. (2017) mention the possibility that this gender difference in self-references is just a reflection of the fact that male authors tend to write more solo papers, and thereby have more scientific reasons for self-references. King et al. (2017) could not check if this is a significant confounding factor, because authors are not disambiguated in their database. As authors are disambiguated in the InSpire database, I can perform this check finding that this confounding factor removes the gender difference in self-references: A similar fraction of self-references is found when comparing male and female authors who wrote the same number of papers N_pap. In other terms, the fraction of self-references can be a misleading indicator because the average number of self-references grows with the number of solo papers following a scale law $Ncitself$ ∝ $Npapp$ with power p > 1. My data suggest p ≈ 1.3.

As solo papers are a relatively small subsample that might be not representative of the full database, I extend the analysis to multiauthored papers. In order to do so, it is necessary to distinguish self-references from self-citations. I count a citation as self-citation whenever the citing paper has at least one author in common with the cited paper. I count it as a self-reference only for those authors who wrote both the cited and the citing paper. I clarify this with an example. One paper by authors A and B cites a paper by authors B and C: B gives a self-reference; both B and C receive a self-citation (B directly and C indirectly).

As usual, fractional counting is used. For each author I compute the total number of received individual citations N_icit, of received individual self-citations $Nicitreceived$⁠, of given individual self-references $Nicitgiven$⁠, and of fractionally counted papers N_pap (equal to the number of given individual references).

I consider the mean fraction of given individual self-references $Nicitgiven$/N_icit (left panels of Figure 16) and the mean fraction of received individual self-citations $Nicitreceived$/N_icit (right panels of Figure 16). The upper row shows some gender difference: Male authors tend to give themselves a higher fraction of self-references (13.2% instead than 12.0%); female authors tend to receive a higher fraction of indirect self-citations (23.5% instead t hen 19.8%). Such differences again disappear in the lower row of Figure 16, where the self fractions are computed as a function of the number N_pap of fractionally counted papers written by each author. Similarly to what we found in the solo sample, male authors tend to cite themselves and their collaborators more just because male authors tend to have more past papers. Similar results are found restricting within the main subfields.

I performed a bibliometric analysis of gender issues in fundamental physics worldwide from approximately 1970 to now. Bibliometrics gives quantitative data on activities of researchers that result in publications and tells nothing about other possible activities, such as teaching, mentoring, and outreach. Nevertheless, research is an interesting area in which individual talent can be expressed, as confirmed by the large differences between authors found in data. Concerning gender I find the following results:

1. First, the well-known initial gender difference in representation is seen: There are roughly four males for each female among new authors that appear at PhD-level. The initial female fraction is not positively correlated with the Global Gender Gap Index of the countries,¹⁷ and negligibly evolves in the subsequent career stages (Figure 7).

2. When citing works by others, authors exhibit no or small gender difference: Male and female authors have the same average opinion about which research in fundamental physics deserves to be cited (Figure 3).¹⁸ Furthermore, M and F authors give to their own papers a similar fraction of self-references (Figure 16), taking into account that M authors tend to write more papers, especially solo papers (Figure 12).

3. Female authors do not have, at hiring moments, higher average bibliometric indicators based on individual citations or fractionally counted papers than male authors (Figures 4 and 5).

4. Among authors identified by InSpire HepNames profiles (which misses authors who write very few papers), I do not find a gender difference in hired percentages (Figure 6), in abandonment rates (Figure 7), in longest breaks between papers (Figure 9), in periods of reduced activity relatively to their average (Figure 10).

The above results are in line with the literature, as summarized in Ceci et al. (2014): “the overall picture is one of gender neutrality”; “no evidence of women having harder time getting tenure”; postbachelor gender differences in attrition rates are significantly smaller in STEM than in life science, psychology, and social science. The literature finds a second gender difference, in productivity: “women on average publish fewer papers than men”; “there are no sex differences in citations per article” (Ceci et al., 2014). I find:

• 5. 

  A productivity gap both in the fractionally counted number of publications and in their citational impact (Figure 8), which does not appear to be predominantly concentrated in specific countries, topics, periods, bibliometric indicators, journals (supplementary Figure S1), etc. The gap is also found without integrating over careers (see Figures 11 and 12).

• 6. 

  A gradually increasing male fraction when going from average to top authors in terms of individual citations (or other indices, such as fractionally counted publications). The quantitative shape of this trend appears predominantly due to a higher variance on the upper side of the M distributions (see Figure 15 and Eqs. 5 and 6), rather than due to a difference in averages.

I verified that my results still hold when ignoring hyperauthored papers (possibly affected by guest authorship) or restricting to single-authored papers.

While many social phenomena could produce different averages, producing different variances would need something that specifically disadvantages research by top female authors. Just to take one example of a social nature, a gender gap in research productivity could arise if better female authors receive more honours and leadership positions that drive them away from research. However, data also show an excess of young authors among those who produced top-cited papers: The excess is observed among both M and F authors. This suggests extending my considerations from possible sociological issues to possible biological issues.

It is interesting to point out that the gender differences in representation and productivity observed in bibliometric data can be explained at face value (one does not need to assume that confounders make things different from what they seem), relying on the combination of two effects documented in the scientific literature: differences in interests (Diekman, Johnston, & Clark, 2010; Lippa, 2010; Su, Rounds, & Armstrong, 2009; Su & Rounds, 2015; Thelwall, Bailey et al., 2019) and in variability (Deary, Irwing et al., 2007; Halpern, Benbow et al., 2007; Hyde, 2014; Stevens & Haidt, 2017; Wang et al., 2013).

Greater male interest in things and greater female interest in people is observed consistently across cultures and time and is large (d ≈ 1, i.e., distributions differ by about one standard deviation): Such a difference in interests predominantly accounts for the initial difference in representation.¹⁹ Difference in variability accounts for the difference in productivity. This is consistent with O’Dea, Lagisz et al. (2018), which confirms the difference in variabilities looking at grades, and observes that this difference alone cannot reproduce the representation gap.

The amount of higher male variability suggested by bibliometric data in fundamental physics is at the 10% level, roughly consistent with independent observations of presumably relevant traits. While such psychometrics observations predominantly probe the central, most populated, part of the distributions, I reasonably expect that physicists probe the upper tail²⁰ and that top-cited physicists reach the far-end upper tail. I estimate to reach about five standard deviations above the mean, given that this is the maximal deviation statistically expected from a pool of ∼10⁹ persons in a Gaussian approximation.

When dealing with complex systems, any simple interpretation can easily be incomplete, including a hypothetical gender discrimination. In any case, it is interesting that data can be explained without invoking such a hypothesis.

I conclude by addressing ethical and social values, given that a gender difference in variances is seen by some as offensive, like other ideas originally proposed by Darwin (Hill, 2017) (modestly keeping things in proportion in this comparison). The interpretation in terms of different variabilities implies that one should keep giving gender-neutral equal opportunities to everybody by considering each person based on his or her individual qualities, not as member of a demographic group (gender, nationality, or whatever). The refusal to consider population level differences in distributions when trying to understand gaps in representation can lead to discrimination allegedly aimed at establishing equal outcomes (see, for example, Strumia (2019) for a more extensive discussion of such issues).

I thank the referees for their comments; the InSpire team for clarifications; Guy Madison for discussions and suggestions; Riccardo Torre for (among many things) having implemented the WGND name-gender association; Sabine Hossenfelder for having independently replicated the results in Figure 8b and Section 3.1 using arXiv data (Hossenfelder, 2018); and more colleagues who prefer not to be mentioned.

The author has no competing interests.

No funding has been used for this research, performed outside working time.

This study is based on bibliometric data extracted from the InSpire database around mid-2018. While data are already public from the InSpire web site (InSpire, 2010), I selected the key data and converted them into user-friendly Mathematica format. The code used to extract data is available at GitHub (https://github.com/RiccardoTorre/InSpireTools) and the resulting data at Zenodo (https://doi.org/10.5281/zenodo.3482884). The data consist of a table with more than one million entries, one for each paper. Each entry contains the following information:

1. an integer number identifying the paper;

2. publication date;

3. publication date on arXiv when available;

4. number of authors;

5. number of references;

6. number of citations;

7. PageRank with self-citations;

8. PageRank without self-citations;

9. list of authors with each author identified by an integer number;

10. list of references, with each paper identified by the integer in item 1);

11. list of citations;

12. title;

13. arXiv main category when available;

14. 1 if published, 0 otherwise;

15. list of lists of affiliations of each author, with each institute identified by an integer number;

16. arXiv subcategory when available;

17. arXiv number when available;

18. journal, with each journal identified by an integer number;

19. collaboration(s), with each collaboration identified by an integer number;

20. number of individual citations.

The correspondence between integer numbers and papers, authors, institutes, journals, and collaborations is available from InSpire (for privacy reasons I avoid making my tables public).