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h_α: An index to quantify an individual's scientific leadership

10/03/2018
by   J. E. Hirsch, et al.
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The α person is the dominant person in a group. We define the α-author of a paper as the author of the paper with the highest h-index among all the coauthors, and an α-paper of a scientist as a paper authored or coauthored by the scientist where he/she is the α-author. For most but not all papers in the literature there is only one α-author. We define the h_α index of a scientist as the number of papers in the h-core of the scientist (i.e. the set of papers that contribute to the h-index of the scientist) where this scientist is the α-author. We also define the h'_α index of a scientist as the number of α-papers of this scientist that have ≥ h'_α citations. h_α and h'_α contain similar information, while h'_α is conceptually more appealing it is harder to obtain from existing databases, hence of less current practical interest. We propose that the h_α and/or h'_α indices, or other variants discussed in the paper, are useful complements to the h-index of a scientist to quantify his/her scientific achievement, that rectify an inherent drawback of the h-index, its inability to distinguish between authors with different coauthorships patterns. A high h index in conjunction with a high h_α/h ratio is a hallmark of scientific leadership.

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I introduction

The -index has gained wide acceptance as a bibliometric indicator of individual scientific achievement h1 ; h2 ; h3 ; h4 . Its positive features have been pointed out and analyzed. At the same time, many flaws and shortcomings of the -index have been identified and studied hc1 ; hc2 ; hc3 ; hc4 , and many other bibliometric indices have been proposed as alternatives to it hn1 ; hn2 ; hn3 ; hn4 ; hn5 ; hn6 ; hn7 ; hn8 . However, to date no other single bibliometric index has been identified that is clearly preferable to the -index. An alternative to replacing the -index with another better index is to supplement the -index with another bibliometric index that addresses at least some of its deficiencies hs1 ; hs2 ; hs3 ; hs4 ; hs5 . To supplement the index, that was originally proposed as ‘An index to quantify an individual’s scientific research output’hpaper , in this paper we aim to define an index to quantify an individual’s scientific leadership. We should point out that this issue has been independently addressed and a solution proposed to in earlier work by X. Hu, R. Rosseau and J. Chen hs6 .

Possibly the greatest shortcoming of the index is its inability to discriminate between authors that have very different coauthorship patterns. This question has been addressed in the literature, see e.g. hs6 ; hh1 ; hh2 ; hh3 ; hh4 ; hh5 ; hh6 . How does one compare a scientist that usually publishes with one or two coauthors with another scientist that has 10 or more coauthors in every paper? Most will agree that for equally accomplished scientists a larger -index is expected for the scientist with more coauthors, but how much larger? More importantly, consider two scientists with similar -indices where the first one is usually the leader in the multiauthored papers he/she publishes with mostly junior coauthors, while the second one is mostly a junior coauthor in his/her multiauthored papers. Most will agree that the first one is the more accomplished scientist, but the -indices will not reflect it.

In earlier work we have proposed the -index (h-bar index) to address these issues hbar . The -index only counts the papers that contribute to the -index of all its coauthors. Thus, it affects negatively authors that publish with a large number of coauthors and with more senior coauthors. This index has not gained wide acceptance, in part perhaps because it is difficult to obtain. Also, a shortcoming of the -index is that when a paper gains enough citations it will contribute to the -index of all its coauthors equally, independently of what the relative contribution of each coauthor to the paper was. More importantly, in many cases the -index may not be sufficiently different from the -index of an author to justify the substantial extra work needed to obtain it.

We have emphasized from the outset that the -index should be only one of many elements used in evaluating scientific achievement of an individual hpaper . Because it has become perhaps of outsized importance in the evaluation of scientists, we believe it is important to supplement it with a quantifiable assessment of the relative importance of the given scientist in the collaborative work that contributes to his/her -index. To do so, we propose the index in this paper. Its name refers to the fact that the person is the dominant person in a group alphawiki . The purpose of the index is to give a measure of those highly cited scientific contributions of a scientist for which the scientist is the dominant person in the collaboration resulting in a multiauthored paper, who we will call the -author. In other words, the index measures scientific leadership.

Identifying the -author in a collaboration is not a trivial matter, and may not even be a well-defined question. Is it the scientist that procured the funding, is it the most senior scientist, is it the one that provided the key idea that got the project started, or the one that did most of the work? In many cases these roles may be played by different coauthors, in other cases several coauthors play similarly important roles in these tasks. Nevertheless, we argue that in most situations it is possible to identify a key person as the person in a collaboration. For lack of a better criterion, we the -author of a paper to be the coauthor with highest index. Because a high -index is generally an indication of high scientific achievement, we argue that this is a reasonable criterion. To determine who is the author we use the indices at the present time, rather than the indices at the time the paper was published, which are not available in existing databases. Under the assumption that indices grow at similar rates, both choices would give similar results. Of course for single-author papers the author is the author.

We propose two indices, which we call and indices. They contain similar information. They are not proposed to replace but rather to complement the index. As stated in the abstract, the index of a scientist is defined exactly the same way the -index is defined except that it refers only to -papers of a scientist, i.e. those papers where the scientist is the -author. It is possible that a paper could have two or more -authors but that generally will not be the case. So a scientist with -index has -index if that scientist has written papers that have citations each, and where all the coauthors of each of those papers have -indices lower or equal to .

The index as defined above is difficult to obtain from existing databases. For that reason we define the related index of a scientist as the number of papers in the -core of the scientist for which the scientist is the author. It may be the case that , in general . The reason to define is that it is easier to calculate from existing databases. One simply has to go through the list of papers in the -core of a scientist and eliminate those papers for which a coauthor has higher -index than the -index of the author under consideration.

It is clear from the definition that , and we will argue that the ratio gives useful information. Note also that the set of papers that contribute to a scientist’s index may be a subset of the set of papers contributing to the scientist’s index, in which case , or it may have some subset of it that belongs to the core and another subset that does not, in which case , and it may even be the case that the core and the core are disjoint sets, in which case ,

. However, the latter situation will probably be very rare except for very junior scientists.

Ii case study

Name pubs citations years field
A 25 8 0.32 1.39 59 2944 18 het
B 27 19 0.70 0.52 83 3649 52 het
C 32 8 0.25 1.10 127 4040 29 het
D 34 5 0.15 0.97 93 5377 35 cmt
E 34 22 0.65 0.97 133 3967 35 pt
F 36 16 0.65 1.89 104 4702 19 cmt
G 36 7 0.19 1.09 146 39,062 33 het
H 37 18 0.49 1.32 80 6285 28 het
I 39 5 0.13 1.63 130 5823 24 cmt
J 39 16 0.41 1.39 119 6582 28 bpt
K 40 2 0.05 1.48 273 6815 27 oap
L 43 12 0.28 1.39 104 5631 31 het
M 47 30 0.67 1.62 186 9943 29 het
N 50 27 0.54 2.17 268 12,536 23 cmt
AA 55 51 0.93 1.31 116 23,509 42 cmt
O 60 1 0.02 5.45 160 14,190 11 oap
P 60 14 0.23 3.16 224 11,068 19 oap
Table 1: Bibliometric data for 16 physicists at the physics department of a major research university in the United States mru (A through P), and one physicist at Princeton University (AA), listed in order of increasing -index. The data for the index introduced in this paper, and the ratio , are in boldface. ‘pubs’ is the number of papers published, ‘years’ is years from publication of first paper to the present, is the ratio . The research fields of these physicists are high energy theory (het), condensed matter theory (cmt), plasma theory (pt), biophysics theory (bpt) and observational astrophysics (oap). Note the very strong variations in the index and the ratio . For the explanation of the red coloring, see text.

We use the Web of Science for the bibliometric data, and in particular the database ResearcherId when possible rid . ResearcherId is a very useful feature because it provides name disambiguation. Table I shows publication and citation metrics for 13 theoretical physicists at the physics department of a major research university in the United States mru , henceforth called “MRU” (entries A through J and L through N), three observational astrophysicists at MRU (K, O, P), and one theoretical physicist at Princeton University (AA). The data are arranged in order of increasing -index and include all theorists in the department of physics at MRU with in the range . The table gives the seniority of the researcher by listing the number of years since publication of the first paper (‘years’), which is usually close to (typically 1-3 years before) the Ph.D. date.

The first thing to note from table I is that there is not a strong positive correlation of the index with seniority, or equivalently years from Ph.D. degree. This is of course not surprising, since different scientists produce research at different rates, and the quality and impact of the research differs widely.

Turning to , note the large differences in for physicists with similar indices, reflecting very different coauthorship patterns and degree of scientific leadership. Note also that the ratio in table I is not strongly correlated with ‘years’. In other words, more seniority does not necessarily lead to higher independence and scientific leadership, contrary to what might have been expected. This suggests that scientific leaders start leading early on in their career.

The ratio ( years from first published paper to the present) was defined in ref. hpaper , where it was pointed out that a high value of indicates ‘outstanding scientists’ independent of seniority. However that statement has to be tempered when taking into account . We can see from table I that values above 1.4 are sometimes associated with high values of (F, M, N) and sometimes with low values (I, K, O, P). In the latter case, particularly because high values of may result from many papers in large collaborations rather than from high individual achievement, what was stated in ref. hpaper quoted above obviously does not necessarily follow.

One of the motivations for the original introduction of the -index was that the alternative of considering total number of citations could easily lead to misleading results. Indeed, just looking at the “citations” column in table I would lead to the conclusion that physicist G is by far the most accomplished scientist on the list, with citations. In fact, this high number comes about because physicist G coauthored 9 review articles (“Review of Particle Physics”), each having several thousand citations and several hundred coauthors. The total number of citations of physicist G excluding those review articles is 3882, i.e. the total citation number is 10 times larger than the citations to non-review articles. Instead, these multi-authored review articles augment the index of this author by only, properly decreasing their importance. With the index, the effect of these review papers, which are not really representative of this author’s scientific accomplishments, is completely eliminated, since each of the review articles has several other authors with (much) higher index than this author.

More generally, table I shows very little correlation between the -indices and indices. I argue that the index is essential information to take into account in the evaluation and comparison of these scientists.

For example, physicists O and P have the highest and highest -values but smaller indices (1 and 14) than physicists B, E, F, H, J, M, N and AA, sometimes substantially so. It would be wrong to just rely on the -index to conclude that O and P are the most accomplished of the list. The reason O’s and P’s indices are so high is because a large number of their papers are coauthored by between 10 and 40 authors, some of them with -index substantially larger than O’s and P’s, indicating that O and P are not the leaders in these collaborations. Because O and P are not the -authors in these papers, the papers don’t contribute to O’s and P’s indices, resulting in indices that are 60 times and 4 times smaller than their -indices respectively. Instead, the higher index and ratio of the other physicists in the comparison group reflect the fact that they are the leading authors in a substantially larger number of their highly cited papers, which suggests that they are more accomplished scientists.

The situation is the same for physicist K. With an , 27 years after the first paper, , and 6,815 total citations, one might have concluded that this is an outstanding scientist. However, physicist K’s is a mere 2, and the -ratio . Many of the papers of physicist K are written in collaboration with 20-40 coauthors, and both in those as well as other papers with fewer coauthors there are coauthors with indices higher than K’s, often substantially so. These data suggest that K is the scientific leader in only 2 out of the 40 papers in physicist K’s core.

Comparing high energy theorists B and L, it would be reasonable to conclude that physicist B, with an index of only 27 and of 19, is more accomplished than physicist L, with h=43 but only 12, contrary to what their relative indices suggest. Indeed, B has the rank of Distinguished Professor in the Department, while L has the lower rank of Professor.

Similarly, in a comparison between condensed matter theorists, one would reasonably conclude that physicist F, with h=36 and , is more accomplished than I, with but only . Physicist I writes many papers in large collaborations with high- scientists and it would be hard to believe that physicist I is the leader in these collaborations. His/her low index properly reflects this fact. In contrast, F is the scientific leader in a substantial fraction of his/her papers which are coauthored with his/her students and postdocs.

Comparing theorists D and E, assuming this is possible even though they are in different physics subfields, we learn that they have the same number of years since their first paper (35) and the same index (34), and D has somewhat more total citations than E (5377 versus 3967). One might have concluded from this information that D and E are similar, D somewhat more accomplished. However, their index differs by a factor of 4 (5 and 22), with E having the higher one. This indicates that the index of E results in large part from independent work where he/she is the leader, and that of D from collaborative work with more senior scientists where D is not likely to have played the leading role. Both D and E work in small collaborations involving at most a few coauthors.

There are 12 physicists on the list of table I that hold the rank of Professor in the department of physics at MRU, and 4 that hold the higher rank of Distinguished Professor dist . The latter ones are B, C, G, and M, colored red in the table. Could one have inferred this from the data given in table I? The answer is clearly no. To begin with, the Distinguished Professors are certainly not the ones with highest indices. Taking into account , the data in table I would suggests that if C and G are at the highest rank, E and F, that have comparable -indices to C and G but substantially higher indices, should certainly be at the highest rank, but they are not. E has also more seniority (35) than C and G (29 and 33). Similarly, while it seems clearly justified that M is a Distinguished Professor given his/her high and while L, O and P, with comparable indices but much lower -indices are not, it is surprising that , with a higher -index than M and almost as high an index, is not a Distinguished Professor. Comparing N with C and G, it seems incomprehensible that N, with higher than C and G, is at a lower academic rank than C and G. In this author’s opinion, that is informed by detailed knowledge of the scientific record of these physicists, these inconsistencies are not a reflection of shortcomings of the bibliometric indices and to quantify scientific achievement, but rather reflect the failure of the academic promotion process at this major research university to properly reward higher scientific achievement with a higher academic rank for the scientist and vice-versa.

Physicist AA in table I has a high -index, but not qualitatively different from that of others on the list, however has a remarkably high as well as ratio , the highest in the group by a large margin. We have found such high values of only among exceptionally accomplished scientists that have earned broad acclaim. Physicist AA is a Nobel laureate.

Iii more examples

In table II we list the bibliometric data of 10 mid-career active condensed matter theorists table2 . Their “age” (i.e. time since their first paper) ranges from 11 to 26 years, mostly clustered around 20 years, and their indices range from 16 to 32. No systematic rule was used in choosing these examples, other than keeping and ‘years’ within limited ranges, and choosing scientists where either themselves and/or their coauthors were known to the author of this paper, to facilitate the process of finding their -index. We have also computed for these scientists, which was considerably more time-consuming than computing .

As expected there is not a strong correlation between “age” and index in table II, in other words values vary widely, ranging from 0.75 to 1.82. None of these scientists works in large collaborations, their papers have typically one to a few coauthors. The average number of coauthors for papers in their core ranges from 1.3 to 3.5 as shown on table II, 2.7 is the overall average. Nevertheless their ’s are very different, hence so is their ratio .

Let us start with physicist T, with and the smallest . This is the youngest of the group, with the highest and one of the largest total citation numbers. The impressive citation metrics (excluding ) come from collaborations with much more senior highly cited physicists such as D. Scalapino (h=97), S.C. Zhang (h=90), D. A. Huse (h=85), M.P.A. Fisher (h=73), S. Kivelson (h=62), F. Haldane (h=55), S. Kashru (h=50), S. Chakravarty (h=49). It would be difficult to believe that T is the leader in these collaborations. Even looking beyond the core, T has no single-author papers and only a handful of papers with few citations where T is the author. So in comparing T’s bibliometric record with that of other physicists it would be misleading to consider . In the absence of one would conclude from the bibliometric information that T is the most accomplished physicist in table II. Instead, knowing that , at least indicates that one has to have a closer look. One may assume that the reason for physicist T is because T is very junior, and in the future and will increase. This is suggested by the fact that T’s , mostly from recent papers. The future will tell.

At the other extreme we have physicist V, with comparable index to T, vs. , a substantially smaller but remarkably high and . V does have a few papers with scientists with higher including very senior scientists (N. Ashcroft (h=60), H. Kleinert (h=40), A. Sudbo (h=39)). However, V has a considerable number of highly cited single author papers (6 papers out of 25 in the core) and many highly cited papers with junior coauthors, resulting in the very high and alpha-ratio . V’s is also the highest among all the entries in table I other than AA, despite being more junior than 13 out of the 17 physicists on that list. These data suggest remarkable independence and scientific leadership for this relatively young physicist.

m pubs citations yrs coauth
Q 16 1 0.06 4 0.20 0.80 38 895 20 3.3
R 17 5 0.29 9 0.53 1 35 1032 17 2.2
S 17 8 0.47 10 0.56 0.94 51 1590 18 3.5
T 20 0 0.00 6 0.55 1.82 54 3468 11 2.9
U 22 3 0.14 4 0.19 1.05 40 3531 21 2.7
V 25 18 0.72 21 1.05 1.25 75 2096 20 1.3
W 27 6 0.22 12 0.46 1.04 109 2349 26 2.5
X 28 7 0.25 16 0.70 1.22 90 2590 23 3.5
Y 31 17 0.55 21 1.24 1.82 95 2616 17 2.7
Z 32 20 0.63 26 1.30 1.60 114 3059 20 2.1
Table 2: Bibliometric data for 10 condensed matter theorists of comparable age and indices. ‘coauth’ is the average number of coauthors for papers in the core of the author. ,

It is also apparent from table II that is not strongly correlated with age. Physicist Q, of the same age as V, has the second smallest , and the oldest physicist in the list, X, has a relatively small . The physicists with highest on this list are S, V, Y, Z, with and respectively and ages in the mid-range, respectively. It is however the case that it is rare to find physicists with smaller that have a large , like physicist S on this list. As indices become larger, larger values of become increasingly more common.

As expected, and give similar information. Do we learn anything new from ? Yes we do. Recall that also counts the -papers that are not in the -core. For example, comparing T and U, both have similar and . However, T has , , while U has , . This indicates that T has several papers not yet in the -core with appreciable number of citations, which are likely to enter the core in the near future and at that point increase T’s . In contrast, the fact that for U indicates that U does not have many papers with appreciable citations that are not in the core. This suggests that T’s is likely to be larger than U’s in the near future. Thus, while comparing the bibliometric data of T and U including but not may suggest that U is more accomplished, taking into account reverses this conclusion. A large difference between and , as seen in table II for authors T and X, indicates that the author is becoming increasingly independent and increasingly leading his/her research efforts.

We also list in table II the ratio , which gives the same information as but only for the papers where the author is the author. and give measures of the scientist that are independent of his/her seniority. We argue that gives a truer measure of the scientist than because it is less dependent on coauthorship patterns. For scientists where there is a large difference between and , such as Q, T, U, W, we suggest that this raises concern about how much the value of is a true reflection of the scientist.

We believe that the bibliometric information in table II, as well as in table I, clearly illustrates the importance of taking into account the proposed index and ratio , and if available, also and , to complement the bibliometric information given by and . Scientists Q and U look substantially weaker when the information is taken into account than in its absence. From the values of in table II one would conclude that physicists T, Y, Z, V are the most accomplished on the list, in decreasing order. Instead, according to , it is V, Z, Y, S, and according to it is Z, Y, V, X. Thus, S, T and X do not excel according to all these criteria, while V, Y and Z do, which is reassuring. The situation is rather different in table I, where several scientists that excel according to do not excel at all according to , nor presumably according to . Of course, detailed examination of all these authors’ publication records and other information could change these conclusions.

Iv technical details

Let us illustrate the procedure we use to obtain in more detail for the case of physicist AA in table I, where it is particularly easy because a large fraction of AA’s paper are single author or with very few coauthors. This will also allow us to suggest capabilities that could be incorporated in the existing bibliometrics databases to make the calculation of simpler. We will use the Web of Science.

Physicist AA is F.D.M. Haldane, he has authored 116 papers, 35 of which are single author, a remarkably high number compared to typical condensed matter theorists. Even more remarkably, 25 single-author papers are in his h-core, and 8 of his 10 most highly cited papers are single author. This alone demonstrates his remarkable independence and scientific leadership. In collaborative work he is almost always the author, resulting in his very high ratio.

To find his index we go through the list of his publications in order of decreasing citations. The first non-single-author paper is paper 7, coauthored with S. Raghu, with 768 citations. Clicking on the paper title, then on Raghu’s name, then on ”Create Citation Report” for Raghu, we learn that Raghu’s index is 21, smaller than Haldane’s 55, hence this paper contributes to Haldane’s index. Continuing down the list of Haldane’s publications we find the next collaborative paper is paper 9 with H. Li, also contributing to Haldane’s since Li’s index is smaller than 55. The next collaborative paper is number 13, with I. Affleck, whose index is 73, larger than Haldane’s 55, therefore this paper does not contribute to Haldane’s . Continuing this process we find many papers with coauthors of index lower than Haldane’s (e.g. Rezayi, Arovas, Auerbach, Bernevig, Bhatt) that contribute to Haldane’s , and papers with coauthors P.W. Anderson (h=108), P. Littlewood (h=61) and L. Balents (h=61) that have 55 citations (i.e. are in Haldane’s h-core) and do not contribute to Haldane’s because the coauthors have larger than 55. We continue this rather tedious procedure until reaching paper 56 in the publication list that has fewer than 55 citations, at this point we stop and have found a total of 51 of the 55 papers in Haldane’s -core that are -papers for Haldane, hence his index is 51 and his is .

An alternative procedure would be to click on the “Analyze Results” link in Haldane’s publication list, then click on “Authors” on the column on the left, to obtain the list of all of Haldane’s coauthors, ordered by “Record Count”. Next we would need to check the citation records of all the coauthors to find their indices. However, since we don’t know from this page whether the coauthored papers are or are not in Haldane’s core this is not an efficient procedure to find . If the Web of Science were to provide the index of the coauthors on this page, and allow to order the coauthors in order of decreasing , it would be very simple to find the coauthors with index larger than Haldane’s, then look for the papers coauthored with these that are in Haldane’s core, thus greatly simplifying the calculation of .

To obtain for Haldane, we continue down his publication list beyond paper 55. The next three are papers and have citations, hence , and , incredibly close to , which is a very unusual situation. Here, the extra work beyond computing for computing was negligible. However, for the cases in table II it took considerably longer to obtain because we had to look at the citations of many papers not in the core of the scientist.

V summary and discussion

This paper was partially motivated by the increasingly wide use of the index to rank and compare scientists. The shortcoming of the -index to differentiate scientists with different coauthorship patterns already existed at the time the index was created, but we believe it may have been exacerbated by the index itself in the ensuing years. There is no “cost” to the index of a scientist to work in large collaborations that include highly accomplished scientists, on the contrary, there is potentially a large benefit in resulting in a higher -index compared to the scientist pursuing his/her own independent ideas in small collaborations or single-author papers. Thus this provides an incentive for young scientists to join large collaborations and/or collaborations with prestigious coauthors even when there is not a compelling scientific motivation for it, and we believe this may result in a non-optimal use of the scientists’ abilities.

More generally, we have observed that there are many examples of scientists with comparable indices but very different profile as far as scientific leadership is concerned, which we believe is a very important aspect of what is generally understood as “scientific accomplishment”. Scientific advances result both from the contributions of scientific leaders and scientific followers, but only the former are irreplaceable. We believe it is important to identify and incentivize such scientists with proper citation metrics. The index alone increasingly doesn’t do it in this age of -inflation.

To address these issues, we aimed in this paper to introduce a measure of the scientific production of a scientist that counts only those papers where the scientist is the leading author. Who that person is for a given paper is a non-trivial question, except for single author papers. In some scientific subdisciplines, it is usually the last author in the author list. In others, it is usually the first author. Yet in others, authors are always listed alphabetically so the order of authors carries no information. Is there a general criterion to identify this person? We proposed that the coauthor with the highest index is the most likely candidate and called that author the -author, and the paper an -paper of that author.

One could argue that it would be reasonable to use a more inclusive criterion to define what is an paper of an author, that would allow for more than one author not only when top indices are identical. For example, if the index of the author is within of the highest -index of a coauthor, it may be argued that it is likely this scientist also played a leading role, and count that paper as an paper for that author also. Particularly for young scientists that collaborate with peers of similar seniority without more senior coauthors we believe that this would be a reasonable procedure. One could call such an index , where gives the percentage range for inclusion, i.e. in the above example, and similarly for . For the Haldane example, , , , .

There will certainly be situations where our proposed criteria do not reflect reality. For example, it is often the case that experimentalists that make samples have very high indices. In an experimental paper where such a sample is used the scientist providing the sample would be the -author even if he/she made otherwise no contribution to the scientific project, hence certainly didn’t “lead” the project. Similarly, in a purely theoretical paper where an experimentalist supplied data hence is a coauthor, a theorist with lower index may well be the leading author in the project, yet not identified as such by our criteria. These limitations underscore the fact that it is important to consider many factors besides bibliometric indices in the evaluation of scientists.

We defined the index in the same way as the index is defined, namely the number of -papers written by a scientist that have citations. is the more consistent way to quantify the criterion we are after. Unfortunately, is very time-consuming to obtain from the existing databases. One has to go through a large fraction of the papers of an author and find out whether or not it is an paper for that author and whether or not it contributes to his/her index. For senior scientists with many publications, citations, and coauthors, this is a very time-consuming process. For that reason we defined the index, which counts only the -papers of the author in the author’s core, a subset of the papers in the -core. The core is usually at least a factor of 3 smaller than the total number of papers of a scientist, thus reducing the time necessary to compute the index versus the index by a substantial factor. and have similar information.

A drawback of relative to is that there could be cases where the index of an author is high principally because of multiauthored collaborations with scientists with even higher -indices, yet the author may have quite a few other papers not in his/her -core that contribute to a fairly high index. An example of this was scientist T in table II. In such cases, which we believe are not very common, could be much smaller than and give a somewhat distorted picture of the author’s scientific achievement and leadership.

In fact, neither nor are ideal definitions. Imagine that physicist O in table I, with and , coauthors a paper with physicist B, with and . More likely than not, B would be the leader in the collaboration, having shown leadership and/or independence 19 times before, versus 1 time for physicist O. However, according to our definitions, O would be the -author having the higher index, and the joint paper would potentially contribute to O’s but never to B’s (and the same for ). To avoid this, we could instead define papers to be those papers of an author where the author has the highest index rather than -index among the coauthors, with as defined earlier, and a better index as: a scientist has index if that scientist has written -papers with citations each. With these definitions, the paper coauthored by B and O would be an -paper of B and not of O, hence potentially contribute to the -index of B but never to that of O. Finally, for a self-consistent definition of we could define a paper to be an paper of the author that has the highest instead of the highest . In any event, the non-self-consistent and even more so the self-consistent -indices would be very time consuming to obtain and for that reason of no practical interest, certainly at present.

Despite these caveats, we argue that the index proposed in this paper and its variants provide an essential complement to the index. They can provide a clear distinction between scientists with similar indices but very different coauthorship patterns, in particular distinguish between scientists publishing with few coauthors and those in large collaborations, and distinguish between scientific leaders and followers. For two scientists with similar indices but very different indices, we argue that it is highly likely that the scientist with higher index is the more accomplished one. For two scientists with reverse ordering in and the comparison has to be done with care. If forced to choose between and to rank scientists, this author believes that in the absence of other information, should carry more weight. However, both and carry important information and should be used together. If available, the index has additional important information that should also be considered.

The index can be obtained with moderate work using the existing bibliometric databases, and we argue that in assessing and comparing the achievements of scientists using bibliometric data one should do a comparison using the index alone without also using the index. Furthermore, for cases where is very small, as in some of the examples seen, it is important to consider the additional information that is provided by even if it involves additional substantial effort.

To the extent that consideration of in the assessment of scientists gains acceptance, we believe it will provide additional incentive for young scientists to pursue innovative work following their own ideas, versus joining collaborations with more senior scientists and following their established ideas that may not always be correct. We believe that this incentive would be beneficial to the vitality and innovative quality of the scientific enterprise.

In summary, we propose that taking into account the index of a scientist and ratio in addition to his/her index and -ratio, as well as and if available, should result in better and fairer decisions regarding allocation of funding resources, career advancement of scientists, decisions on scientific awards and elections to prestigious scientific bodies. To the extent that bibliometric databases such as Web of Science, Scopus and Google Scholar, introduce tools to facilitate the calculation of and indices, and even and indices, as they have done for the index, we believe that this will have a positive effect on the advancement of science.

Acknowledgements.
The author is grateful to a colleague for thoughtful comments.

References

  • (1) Van Raan A.F.J. (2006) Comparison of the Hirsch-index with standard bibliometric indicators and with peer judgment for 147 chemistry research groups. Scientometrics 67: 491-502.
  • (2) Bornmann L., Daniel HD (2007) What do we know about the h index? Jour. of the Am. Soc. for Information Science and Technology 58: 1381-1385. and references therein.
  • (3) Alonso S., Cabrerizo F.J., Herrera-Viedma E., Herrera F. (2009) h-Index: A review focused in its variants, computation and standardization for different scientific fields. Jour. of Informetrics 3: 273-289 and references therein.
  • (4) Bornmann, L. (2014) h-Index research in scientometrics: A summary. J. of Informetrics 8: 478-485.
  • (5) Waltman L., Nees J. (2011) The inconsistency of the h-index J. Am. Soc. Inf. Sci. Tech.: 406-415.
  • (6) Gibb B.C. Lies, damned lies and h-indices (2012) Nature Chemistry 4: 513-514.
  • (7) Prathap G. (2012) The Inconsistency of the H-Index. J. of the Am. Soc. for Inf. Sci. and Tech. 63: 1480-81.
  • (8) Schreiber, M. (2018) A skeptical view on the Hirsch index and its predictive power. Phys. Scripta 93:10201.
  • (9) Bornmann L., Mutz R., Daniel, H.D. (2008) Are there better indices for evaluation purposes than the h index? a comparison of nine different variants of the h index using data from biomedicine. Jour. of the Am. Soc. for Information Science and Technology 59: 830-837.
  • (10) Van Eck N.J. , Waltman L. (2008) Generalizing the h- and g-indices. Jour. of Informetrics 2: 263-271.
  • (11) Rousseau R., Ye F. (2008) A proposal for a dynamic h-type index. Jour. of the Am. Soc. for Information Science and Technology 59: 1853-1855.
  • (12) Egghe L., Rousseau R. (2008) An h-index weighted by citation impact. Information Processing and Management 44: 770-780.
  • (13) Guns R. , Rousseau R. (2009) Real and rational variants of the h-index and the g-index. Jour. of Informetrics 3: 64-71.
  • (14) Yaminfirooz M, Gholinia, H (2015) Multiple h-index: a new scientometric indicator. Electronic Library 33: 547:556.
  • (15) Perry M., Reny P. J. (2016) How to Count Citations If You Must. AER 106: 2722-41.
  • (16) Mazurek, J (2018) A modification to Hirsch index allowing comparisons across different scientific fields. Current Science 114:2238-2239.
  • (17) Jin B., Liang L.M., Rousseau R., Egghe L. (2007) The R- and AR-indices: Complementing the h-index. Chinese Science Bulletin 52: 863-863.
  • (18) Zhang, C-T. (2009) The e-Index, Complementing the h-Index for Excess Citations. PLOS ONE 4: e5429.
  • (19) Bornmann, L., Daniel, H-D. (2010) The citation speed index: A useful bibliometric indicator to add to the h index. J. of Informetrics 4: 444-446.
  • (20) Dorta-Gonzalez, P., Dorta-Gonzalez M.I. (2011) Central indexes to the citation distribution: a complement to the h-index. Scientometrics 88: 729-745.
  • (21) Lando T., Bertoli-Barsotti L. (2014) New tools for complementing the h-index: an empirical study. Mathematical Methods in Economics (MME 2014): 566-571.
  • (22) Hirsch J.E. (2005) An index to quantify an individual’s scientific research output. PNAS 102: 16569-16572.
  • (23) Hu X., Rosseau R., Chen J. (2010) In those fields where multiple authorship is the rule, the h-index should be supplemented by role-based h-indices. J. of Information Science 36: 73-85.
  • (24) Egghe L. (2008) Mathematical theory of the h- and g-index in case of fractional counting of authorship. Journal of the American Society for Information Science and Technology 59: 1608-1616.
  • (25) Schreiber M. (2009) A case study of the modified Hirsch index accounting for multiple coauthors. Journal of the American Society for Information Science and Technology 60: 1274-1282.
  • (26) Liu X. Z., Fang, H (2012) Modifying h-index by allocating credit of multi-authored papers whose author names rank based on contribution. Jour. of Informetrics 6: 557-565.
  • (27) Ancheyta, J. (2015) A correction of h-index to account for the relative importance of authors in manuscripts. Int. J. of Oil, Gas and Coal Tech. 10: 221-232.
  • (28) Ausloos M. (2015) Assessing the true role of coauthors in the h-index measure of an author scientific impact. Physica A 422: 136-142.
  • (29) Crispo E. (2015) A new index to use in conjunction with the h-index to account for an author’s relative contribution to publications with high impact. J. of the Assoc. for Information Sci. and Tech. 66: 2381-2383.
  • (30) Hirsch, J.E. (2010) An index to quantify an individual s scientific research output that takes into account the effect of multiple coauthorship. Scientometrics 85: 741-754.
  • (31) See Alpha (Wikipedia).
  • (32) See http://www.researcherid.com/.
  • (33)

    See list of Universities classified as “R1: Doctoral Universities Highest Research Activity” in

    wikipedia.org/wiki/List of research universities in the United States. According to this website, “These universities have a very high level of both research activity and per capita in such research activity”.
  • (34) There are six other theoretical physicists in this department at the rank of Distinguished Professor, all have indices higher than 50.
  • (35) Scientists interested to know whether they are listed in table II can obtain this information from the author upon request.

References

  • (1) Van Raan A.F.J. (2006) Comparison of the Hirsch-index with standard bibliometric indicators and with peer judgment for 147 chemistry research groups. Scientometrics 67: 491-502.
  • (2) Bornmann L., Daniel HD (2007) What do we know about the h index? Jour. of the Am. Soc. for Information Science and Technology 58: 1381-1385. and references therein.
  • (3) Alonso S., Cabrerizo F.J., Herrera-Viedma E., Herrera F. (2009) h-Index: A review focused in its variants, computation and standardization for different scientific fields. Jour. of Informetrics 3: 273-289 and references therein.
  • (4) Bornmann, L. (2014) h-Index research in scientometrics: A summary. J. of Informetrics 8: 478-485.
  • (5) Waltman L., Nees J. (2011) The inconsistency of the h-index J. Am. Soc. Inf. Sci. Tech.: 406-415.
  • (6) Gibb B.C. Lies, damned lies and h-indices (2012) Nature Chemistry 4: 513-514.
  • (7) Prathap G. (2012) The Inconsistency of the H-Index. J. of the Am. Soc. for Inf. Sci. and Tech. 63: 1480-81.
  • (8) Schreiber, M. (2018) A skeptical view on the Hirsch index and its predictive power. Phys. Scripta 93:10201.
  • (9) Bornmann L., Mutz R., Daniel, H.D. (2008) Are there better indices for evaluation purposes than the h index? a comparison of nine different variants of the h index using data from biomedicine. Jour. of the Am. Soc. for Information Science and Technology 59: 830-837.
  • (10) Van Eck N.J. , Waltman L. (2008) Generalizing the h- and g-indices. Jour. of Informetrics 2: 263-271.
  • (11) Rousseau R., Ye F. (2008) A proposal for a dynamic h-type index. Jour. of the Am. Soc. for Information Science and Technology 59: 1853-1855.
  • (12) Egghe L., Rousseau R. (2008) An h-index weighted by citation impact. Information Processing and Management 44: 770-780.
  • (13) Guns R. , Rousseau R. (2009) Real and rational variants of the h-index and the g-index. Jour. of Informetrics 3: 64-71.
  • (14) Yaminfirooz M, Gholinia, H (2015) Multiple h-index: a new scientometric indicator. Electronic Library 33: 547:556.
  • (15) Perry M., Reny P. J. (2016) How to Count Citations If You Must. AER 106: 2722-41.
  • (16) Mazurek, J (2018) A modification to Hirsch index allowing comparisons across different scientific fields. Current Science 114:2238-2239.
  • (17) Jin B., Liang L.M., Rousseau R., Egghe L. (2007) The R- and AR-indices: Complementing the h-index. Chinese Science Bulletin 52: 863-863.
  • (18) Zhang, C-T. (2009) The e-Index, Complementing the h-Index for Excess Citations. PLOS ONE 4: e5429.
  • (19) Bornmann, L., Daniel, H-D. (2010) The citation speed index: A useful bibliometric indicator to add to the h index. J. of Informetrics 4: 444-446.
  • (20) Dorta-Gonzalez, P., Dorta-Gonzalez M.I. (2011) Central indexes to the citation distribution: a complement to the h-index. Scientometrics 88: 729-745.
  • (21) Lando T., Bertoli-Barsotti L. (2014) New tools for complementing the h-index: an empirical study. Mathematical Methods in Economics (MME 2014): 566-571.
  • (22) Hirsch J.E. (2005) An index to quantify an individual’s scientific research output. PNAS 102: 16569-16572.
  • (23) Hu X., Rosseau R., Chen J. (2010) In those fields where multiple authorship is the rule, the h-index should be supplemented by role-based h-indices. J. of Information Science 36: 73-85.
  • (24) Egghe L. (2008) Mathematical theory of the h- and g-index in case of fractional counting of authorship. Journal of the American Society for Information Science and Technology 59: 1608-1616.
  • (25) Schreiber M. (2009) A case study of the modified Hirsch index accounting for multiple coauthors. Journal of the American Society for Information Science and Technology 60: 1274-1282.
  • (26) Liu X. Z., Fang, H (2012) Modifying h-index by allocating credit of multi-authored papers whose author names rank based on contribution. Jour. of Informetrics 6: 557-565.
  • (27) Ancheyta, J. (2015) A correction of h-index to account for the relative importance of authors in manuscripts. Int. J. of Oil, Gas and Coal Tech. 10: 221-232.
  • (28) Ausloos M. (2015) Assessing the true role of coauthors in the h-index measure of an author scientific impact. Physica A 422: 136-142.
  • (29) Crispo E. (2015) A new index to use in conjunction with the h-index to account for an author’s relative contribution to publications with high impact. J. of the Assoc. for Information Sci. and Tech. 66: 2381-2383.
  • (30) Hirsch, J.E. (2010) An index to quantify an individual s scientific research output that takes into account the effect of multiple coauthorship. Scientometrics 85: 741-754.
  • (31) See Alpha (Wikipedia).
  • (32) See http://www.researcherid.com/.
  • (33)

    See list of Universities classified as “R1: Doctoral Universities Highest Research Activity” in

    wikipedia.org/wiki/List of research universities in the United States. According to this website, “These universities have a very high level of both research activity and per capita in such research activity”.
  • (34) There are six other theoretical physicists in this department at the rank of Distinguished Professor, all have indices higher than 50.
  • (35) Scientists interested to know whether they are listed in table II can obtain this information from the author upon request.