Pedometron No. 21 7

Soil Bibliometrics The h index Budiman Minasny, Alex. McBratney & Alfred Hartemink Introduction Recently Hirsch (2005) proposed the h (or Hisrch) index as an assessment of the research performance of individual scientists. Hirsch defined it as:

“a scientist has index h if h of his/her Np papers have at least h citations each, and the other (Np − h) papers have no more than h citations each.”

Although the paper by Hirsch was just published in 2005, it is now being used in every discipline for ranking or assessing scientists’ performance. It was also used as one of the criterion for promotion in some departments. So we’d like to see the distribution of the h index for pedometrics and pedometricians. The algorithm to calculate h index is as follows: all papers are ranked based on their number of citations, from the most to the least cited (See Fig. 1). The paper number (rank) when the number of citations equals the paper number is the h index. Graphically, as in Fig. 1, the intersection of the 45 degree line with the curve of the number of citations versus the rank (paper number) gives the h index. The total number of citations is the area under the curve. The distribution of the number of citations (Fig. 1) usually can be modelled as a stretched exponential function (Hirsch, 2005; Laherrere and Sornette, 1998):

( ) 00

expcyN y Ny

β = −

(1)

where Nc(y) is the number of citations of the y-th paper (ordered from the most to least cited), N0 is maximum no. citations, and β, and y0 are empirical parameters with β ≤ 1. When plotted on a semi-log plot it gives a straight line, and in a log-log plot shows a curvature. The larger the value of β and y0, the larger h will be.

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Fig. 1. Total no. citations vs. paper number (rank), the intersect with a 1:1 line is the h index. The total no. of citations for Np papers can be found by:

( ),1

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c tot cN N y dy= ∫ (2)

In practicality, it is much easier to count the number of citations rather than doing the integral.

Data synthesis We selected 35 pedometricians randomly using the ISI web of science database, accessed in November 2005. The following parameters were recorded: the number of papers Np, total number of citations Nc,tot, year of the first paper published by the scientist, the average citations per paper (Nc,tot /Np), and the h index. The h index ranges form 1 to 32, with a median of 7. The largest one being our revered pedometrician, Richard Webster. The scientific age ranges from 1 to 41 years. Results and Discussion Figure 2 shows the relationship between h and total no. citations. h increases with the square root of no. citations, following the diffusion or sorptivity process:

Pedometron No. 21 8

,0.57 c toth N= . (3) We can see that h is closely related to the total no. citations. Thus it depends on how many citations you can earn in the pedometrics subject. Another factor that controls h is obviously the age of the researchers. Typically h increases linearly with time, assuming the researcher has a constant output of papers and the papers get cited. A linear relation with age is proposed by Hirsch (2005):

h = m t (4)

where t is the “scientific age” of the researcher, and m is the impact or productivity of the researcher. As a measure of t, it can be approximated by the number of years after the first published paper till the present. The year when the first paper is published usually occurs in the end of the PhD degree (approximately 25-30 years or age). For physics, Hirsch found that m = 1 characterised a successful scientists (meaning that after ten years the top 10 paper will be cited more than 10 times), and m = 2 is outstanding. The relationship assumes that the researcher has a constant output of p paper per year and each paper gets cited c times per year. This “standard” of course depends on the Fig. 3 shows the relationship between h and scientific age t indices for the 35 pedometricians. The h index for pedometrics is much smaller than other major science disciplines. The highest h index is 32, our revered pedometrician Richard Webster. We found the “average” productivity and impact curve for pedometrics: h = 0.7 t (5) This means that on average a pedometrician should get an increase of 0.7 in the h index. This is assuming that a pedometrician has a constant output of paper and the papers also get cited annually.

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h = 0.57 Nc0.5

Fig. 2. Relationship between total no. citations and h index for pedometricians.

h = 0.69 t

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Fig. 3. Relationship between scientific age and h index. The line of best fit represents the average impact curve for pedometrics.

Pedometron No. 21 9

Np = 2.3 t

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Fig. 4. Number of papers as a function of time. Pedometricians on average published 2 papers a year.

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Fig. 5. Average no. citations per paper as a function of time. Pedometrics paper on average get cited 0.75 time per year. From Figs. 4 & 5 we can deduce that on average pedometricians publish 2 papers per year and each paper get citied 0.75 times per year. Based on Eq. (6) you can set your benchmark of the h index. You can also used this as If you don’t have access to ISI, you can use Google scholar. This webpage from University of Århus Denmark gives you http://www.brics.dk/~mis/hnumber.html Ball, P. (2005), Index aims for fair ranking of scientists. Nature 436 : 900.

Banks, M.G. (2006), An extension of the Hirsch index: Indexing scientific topics and compounds. Scientometrics, 69 : 161-168. Hirsch, J. E. (2005), An index to quantify an individual’s scientific research output. Proceedings of the National Academy of Sciences of the United States of America, 102 : 16569–16572.