A theoretical model of the relationship between the hindex and other simple citation indicators
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DOI: 10.1007/s1119201723519
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 BertoliBarsotti, L. & Lando, T. Scientometrics (2017) 111: 1415. doi:10.1007/s1119201723519
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Abstract
Of the existing theoretical formulas for the hindex, those recently suggested by Burrell (J Informetr 7:774–783, 2013b) and by BertoliBarsotti and Lando (J Informetr 9(4):762–776, 2015) have proved very effective in estimating the actual value of the hindex Hirsch (Proc Natl Acad Sci USA 102:16569–16572, 2005), at least at the level of the individual scientist. These approaches lead (or may lead) to two slightly different formulas, being based, respectively, on a “standard” and a “shifted” version of the geometric distribution. In this paper, we review the genesis of these two formulas—which we shall call the “basic” and “improved” LambertW formula for the hindex—and compare their effectiveness with that of a number of instances taken from the wellknown Glänzel–Schubert class of models for the hindex (based, instead, on a Paretian model) by means of an empirical study. All the formulas considered in the comparison are “readytouse”, i.e., functions of simple citation indicators such as: the total number of publications; the total number of citations; the total number of cited paper; the number of citations of the most cited paper. The empirical study is based on citation data obtained from two different sets of journals belonging to two different scientific fields: more specifically, 231 journals from the area of “Statistics and Mathematical Methods” and 100 journals from the area of “Economics, Econometrics and Finance”, totaling almost 100,000 and 20,000 publications, respectively. The citation data refer to different publication/citation time windows, different types of “citable” documents, and alternative approaches to the analysis of the citation process (“prospective” and “retrospective”). We conclude that, especially in its improved version, the LambertW formula for the hindex provides a quite robust and effective readytouse rule that should be preferred to other known formulas if one’s goal is (simply) to derive a reliable estimate of the hindex.
Keywords
Journal ranking hindex for journals Journal impact factor Glänzel–Schubert formula Geometric distribution Lambert W functionMathematics Subject Classification
62P99JEL Classification
C46Introduction
as a function of C (Hirsch 2005),
as a function of T (Egghe and Rousseau 2006),
as a function of T_{1} (Burrell 2013a),
as a function of C and T (Glänzel 2006; Iglesias and Pecharroman 2007; Schubert and Glänzel 2007; Bletsas and Sahalos 2009; Egghe et al. 2009; Egghe and Rousseau 2012),
as a function of C, T_{1} and C_{1} BertoliBarsotti and Lando (2015);
In particular, in this paper we focus mainly on the problem of obtaining an explicit “universal” formula for estimating the actual value of the hindex. Recently, Burrell (2013b) and BertoliBarsotti and Lando (2015) introduced a model that has proved very effective in estimating the actual value of the hindex for individual scientists. More precisely, these approaches lead (or may lead) to two slightly different formulas, being based, respectively, on a “standard” and a “shifted” version of the geometric distribution. In the first part of section ‘Methods’ we present a (functional) equation, based on the geometric distribution, that constitutes a theoretical basis for both these approaches. Indeed, this equation allows us to derive a closedform estimator of the hindex, expressed as a function of (some of) the above citation metrics. We shall call this estimator, for reasons which will be apparent below, the LambertW formula for the hindex.
In the related scientific literature, authors often limit their analysis to the problem of estimating the unknown parameters of a suggested theoretical parametric model for the hindex, under the assumption of knowing the real values of the hindex. Instead, in this paper we consider the more practical (and in a certain sense, opposing) problem of determining the (unknown) hindex on the basis of a readytouse formula for it. Then, in our empirical analyses we will use the actual values of the hindex but only to evaluate, a posteriori, the performance of the proposed readytouse formulas and not to determine (maybe for interpretative reasons) unknown parameters of a theoretical parametric model. In this paper, we will concentrate on the case of the hindex for journals (Braun et al. 2006). One of the major differences between the cases of an individual scientist and a journal is that, in the latter, the hindex should be computed in a “timed” version, i.e. limited to suitable, usually relatively short, publication and citation time windows. In this regard, it should be noted that a familiar definition such as “a journal has index h if h of its publications each have at least h citations and the other publications each have no more than h citations” is somewhat inaccurate because it does not specify the time windows to be considered for the calculation of h. One of the aims of our study will also be to test the robustness of the formula empirically against different possible choices of (1) length of the time windows and (2) type of approach adopted for analyzing the citation process: “prospective” (diachronous) or “retrospective” (synchronous) (Glänzel 2004). We shall also focus on a comparison of effectiveness between the LambertW formula for the hindex and a popular class of alternative models, related to the socalled Glänzel–Schubert formula, that have already been proved to be highly correlated to the hindex.
In the second part of section ‘Methods’ we review the existing literature on the Glänzel–Schubert family of models (and related models) and discuss some problematic aspects linked to the presence of unknown parameters in their expressions. Then, in section ‘Two empirical studies’, we report the results of an empirical comparison between the LambertW formula for the hindex and these alternative models, using two different dataset of journals. For this task, we downloaded citation data from the Scopus database on about 100,000 and 20,000 publications, respectively, for the first and the second dataset. Based on the results of our research study, we conclude that the LambertW formula for the hindex provides an effective readytouse rule that should be preferred to other known formulas if one’s goal is (simply) to derive a reliable estimate of the hindex.
Methods
Models of the relationship between h and other simple metrics based on citation counts
A basic equation connecting h, T and C
 (a)Given h and T, we easily obtain an estimate \(P^{*}\) of the expectation P as follows:and$$P^{*} = \frac{{\left( {\frac{h}{T}} \right)^{1/h} }}{{1  \left( {\frac{h}{T}} \right)^{1/h} }},$$(5)
 (b)Given T and C, we obtain an estimate of h as follows. Equation (4) is equivalent to \(sa^{s} =  T\), where \(a = \frac{m}{1 + m}\) and \(s =  h\). Then, multiplying each side of the latter equation by log a, and substituting \(z = s\log a\), we obtain \(z e^{z} =  T\log a\), which leads immediately to the solutionwhere \(W\left( \cdot \right)\) represents the socalled LambertW function (Corless and Jeffrey 2015). Remember that the LambertW function is the function W(y) satisfying \(y = W\left( y \right) e^{W\left( y \right)}\), and can be currently computed using mathematical software, for example the Mathematica^{®} 10.0 software package (Wolfram Research, Inc. 2014; it is implemented in the Wolfram Language as “LambertW”), or also using the R statistical computing environment (R Development Core Team 2012).$$z = W\left( {  T\log a} \right),$$(6)Hencethat is, equivalently,$$ h\log \frac{m}{1 + m} = W\left( {  T\log \frac{m}{1 + m}} \right),$$(7)where we have adopted a new symbol for differentiating the “predicted” hindex, \(h_{W}^{\left( 0 \right)}\), from the actual value h of the hindex. Note that the GD approach has been previously suggested by Burrell (2007, 2013b, 2014) but without giving an explicit formula, in closed form, for the estimation of the hindex.$$h_{W}^{\left( 0 \right)} = \frac{{W\left( {T\log \left( {1 + m^{  1} } \right)} \right)}}{{\log \left( {1 + m^{  1} } \right)}},$$(8)
An equation connecting h, T_{1} and C
 (c)Given h and T_{1}, we obtainand$$Q^{*} = \left( {1  \left( {\frac{h}{{T_{1} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 {\left( {h  1} \right)}}} \right. \kern0pt} {\left( {h  1} \right)}}}} } \right)^{  1}$$(11)
 (d)Given T_{1} and C, and following a completely analogous sequence of steps as in the above point (b), we obtain the estimate of h$$h_{W}^{\left( 1 \right)} = \frac{  1}{{\log \left( {1  m_{1}^{  1} } \right)}} \cdot W\left( {\frac{{T_{1} }}{{1  m_{1}^{  1} }} \cdot \log \left( {1  m_{1}^{  1} } \right)} \right).$$(12)
A formula for the hindex, as a function of T_{1}, C and C_{1}
As is well known, citation distributions are highly skewed; hence the sample mean is distorted by extreme values. In particular, the presence of individual highlycited papers tends to overestimate C, and consequently \(h_{W}^{\left( 1 \right)}\), in comparison to the true hindex—that is clearly insensitive to a single very highly cited paper. In this sense, the use of a trimmed mean is simply a technique for reducing this possible bias.
To summarize, we have: \(h_{W}^{\left( 0 \right)} = h_{W}^{\left( 0 \right)} \left( {C,T} \right)\) or also, equivalently, \(h_{W}^{\left( 0 \right)} = h_{W}^{\left( 0 \right)} \left( {T,m} \right)\), and \(\tilde{h}_{W}^{\left( 1 \right)} = \tilde{h}_{W}^{\left( 1 \right)} \left( {C,C_{1} ,T_{1} } \right)\) or also, equivalently, \(\tilde{h}_{W}^{\left( 1 \right)} = \tilde{h}_{W}^{\left( 1 \right)} \left( {T_{1} ,\tilde{m}_{1} } \right)\). We shall refer to these formulas as LambertW formulas for the hindex, respectively, in a “basic”, \(h_{W}^{\left( 0 \right)}\), and an “improved” version, \(\tilde{h}_{W}^{\left( 1 \right)}\). The formula \(\tilde{h}_{W}^{\left( 1 \right)}\) has been considered elsewhere BertoliBarsotti and Lando (2015) for the estimation of the hindex for individual scientists.
Theoretical parametric models for the hindex related to the Glänzel–Schubert formula
 (a)Iglesias and Pecharroman (2007) derived the following oneparameter family of models of the hindex:where \(\eta > 0.5\) (the formula was reported by Iglesias and Pecharroman with parameter \({{\left( {1  \eta } \right)} \mathord{\left/ {\vphantom {{\left( {1  \eta } \right)} \eta }} \right. \kern0pt} \eta }\)). Glänzel (2008) estimated this model in an empirical comparative study of hindex for journals. He found that the estimate of the power parameter depends on the length of the citation window considered. In particular, he found that the formula \(h_{\text{IP}} \left( {2/3} \right)\) (α = 2 in his notation, which corresponds to η = 2/3 in ours) is appropriate “for small windows comprising an initial period of about 3 years after publication”.$$h_{\text{IP}} \left( \eta \right) = \left( {\frac{2\eta  1}{\eta }} \right)^{\eta } m^{\eta } T^{1  \eta } ,$$(14)
 (b)
 (c)By starting from a continuous probability distribution—a Pareto distribution of the second kind,\(P\left( {II} \right)\left( {\sigma ,\theta } \right)\) (Johnson et al. 1994, p. 575; Arnold 1983, p. 44), also known as the Lomax distribution (Lomax 1954), where \(\sigma^{\theta } \left( {\sigma + x} \right)^{  \theta } ,\;\theta > 0,\;\sigma > 0\), represents the probability of observing a number greater than x, x > 0—and estimating its expectation \(\sigma \left( {\theta  1} \right)^{  1}\) (that exists if \(\theta > 1\)) by the sample mean m, Schubert and Glänzel (2007) (see also Glänzel 2006) derived a slightly more general twoparameter model:here defined as also reported by Bletsas and Sahalos (2009); see their Eq. (4)), as an approximate (and generalized) solution of the equation$$h_{\text{G}} \left( {\eta ,\gamma } \right) = \gamma m^{\eta } T^{1  \eta }$$(16)where \(\theta = \eta \left( {1  \eta } \right)^{  1}\). In words, model (16) states that “the hindex can be approximated by a power function of the sample size and the sample mean” (Schubert et al. 2009). It is important to note that the model \(h_{G} \left( {\eta ,\gamma } \right)\) is similar to but different from the above model \(h_{\text{IP}} \left( \eta \right)\), because in the former the proportionality constant is not merely a function of the power parameter η, while in the latter γ represents a free parameter. This gives rise to a more flexible model. Malesios (2015) estimated the parameters of model (16) in a study on 134 journals in the field of ecology and 54 journals in the field of forestry sciences. He obtained the best fit, respectively, with the estimates (0.64, 0.7) and (0.66, 0.78) for the pair (η, γ) (in our parameterization).$$Tm^{\theta } \left( {\theta  1} \right)^{\theta } \left( {\sigma + h} \right)^{  \theta } = h,$$(17)
 (d)The above Pareto distribution of the second kind \(P\left( {II} \right)\left( {\sigma ,\theta } \right)\) has also recently become known as the Tsallis distribution (Tsallis and de Albuquerque 2000). More specifically, with reparameterization \(\theta = \left( {q  1} \right)^{  1}\) and \(\sigma = \left( {q  1} \right)^{  1} \lambda^{  1} ,\;q > 1,\;\lambda > 0\), the probability of observing a number greater than x, x > 0, becomes equal to \(\left( {1 + \lambda \left( {q  1} \right)x} \right)^{{  \frac{1}{q  1}}}\) (see Bletsas and Sahalos 2009; Shalizi 2007). Bletsas and Sahalos (2009) suggest obtaining an estimate of the hindex as the numerical solution of the Eq. (17), that isfor a prespecified fixed value of the unknown parameter q. Let us call \(h_{\text{BS}} = h_{\text{BS}} \left( q \right)\) the (implicit) solution of Eq. (18). It is important to stress that, unlike all the other estimators of hindex considered in the present study, a closedform expression for h_{T} does not exist. Nevertheless, in an empirical application to a set of electrical engineering journals, Bletsas and Sahalos (2009) found a very good fit between measured and estimated values of the hindex, assuming Tsallis distribution with parameter q = 1.5 and q = 1.6. It is interesting to note that these values correspond, respectively, to η = 2/3 and η = 0.625, since \(\eta = q^{  1}\).$$T\left( {m\frac{2  q}{q  1}} \right)^{{\frac{1}{q  1}}} \left( {m\frac{2  q}{q  1} + h} \right)^{{\frac{1}{1  q}}} = h,$$(18)
 (e)For a special choice of the power parameter (η = 2/3 in the present parameterization) in model (16), Schubert and Glänzel (2007) derived the celebrated oneparameter modelalso known as the Glänzel–Schubert model of the hindex. This model has been widely used (mainly for interpretative purposes—i.e. to provide a better understanding of the “mathematical properties” of the hindex) because several empirical studies suggest the existence of a strong correlation between hindex and \(m^{2/3} T^{1/3}\). Its drawback (as with model (16)) is obviously that the value of the proportionality constant γ is unknown. Certainly, this parameter can be determined (ex post) empirically, but it is likely to vary from case to case (Prathap 2010a; Alguliev et al. 2014). Then, as a readytouse formula for estimating the hindex a priori, the Glänzel–Schubert model is in fact unusable. Sometimes researchers find an ex post least square estimate of the parameter γ, starting from known values of the hindex. In different contexts, and for different datasets, the estimate of the γ parameter has been found to vary appreciably, in that it turns out to range approximately from 0.7 to 0.95. Indeed, for example, Schubert and Glänzel (2007) found, for γ, the estimates 0.73 and 0.76, in a study on the hindex for journals, for two different sets of journals, while Csajbók et al. (2007) found an estimate of γ of 0.93 in a macrolevel analysis of the hindex for countries. Instead, other authors, among them Annibaldi et al. (2010), Bouabid et al. (2011) and Zhao et al. (2014), have found values of around 0.8. In quite different contexts (partnership ability and hindex for networks) Schubert (2012) and Schubert et al. (2009) have estimated the parameter γ of the model \(h_{\text{SG}} \left( \gamma \right)\), obtaining values within the range 0.6–0.96.$$h_{\text{SG}} \left( \gamma \right) = \gamma C^{2/3} T^{  1/3} = \gamma m^{2/3} T^{1/3} ,$$(19)
 (f)In the absence of a specific value of the proportionality constant γ, researchers sometimes decide to set γ equal to a fixed arbitrary value γ_{0}, obtaining a readytouse formulaIn the framework of the analysis of the hindex for journals, readytouse formulas for estimating the hindex with the formula \(h_{\text{SG}} \left( {\gamma_{0} } \right)\) have been adopted, for example, by Bletsas and Sahalos (2009), with the choice \(\gamma_{0} = 0.75\). Instead, for example, Ye (2009, 2010) and Elango et al. (2013) adopted the rule to set \(\gamma_{0} = 0.9\) for journals and \(\gamma_{0} = 1\) for other sources. Abbas (2012) and Vinkler (2013) also adopted the choice \(\gamma_{0} = 1\). It is worth noting that the latter value leads to the formula \(h_{\text{SG}} \left( 1 \right)\), which coincides with the socalled pindex defined by Prathap (2010b). Finally, note that \(h_{\text{SG}} \left( {4^{  1/3} } \right) = h_{\text{IP}} \left( {2/3} \right)\).$$h_{\text{SG}} \left( {\gamma_{0} } \right) = \gamma_{0} m^{2/3} T^{1/3} .$$(20)
 (g)As noted above, empirical analyses suggest a “strong linear correlation” between the hindex and the function \(m^{\eta } T^{1  \eta }\) (Schubert and Glänzel 2007; Glänzel 2007; Schreiber et al. 2012; Malesios 2015). Strictly speaking, this only means that when h is plotted against \(m^{\eta } T^{1  \eta }\), the data fall fairly close to a straight line. In other terms, h is approximately equal to \(\delta + \gamma m^{\eta } T^{1  \eta }\), for suitable choices of the parameters δ and γ. Indeed, the following threeparameter model has been considered in literature (see Bador and Lafouge 2010)In a comparative analysis of two samples of 50 journals (taken from the ‘‘Pharmacology and Pharmacy’’ and ‘‘Psychiatry’’ sections of the Journal Citation Reports 2006), Bador and Lafouge (2010) obtained the LS estimates of the parameters δ and γ for different fixed values of the power parameter η (values of “α close to 2”, in their parameterization, where \(\eta = {\alpha \mathord{\left/ {\vphantom {\alpha {\left( {\alpha + 1} \right)}}} \right. \kern0pt} {\left( {\alpha + 1} \right)}}\)). Their best estimates of the proportionality constant γ ranged from 0.7 to 0.8, with an intercept point always very close to 1. Based on these results, \(h_{\text{BS}} \left( {\eta ,\gamma } \right)\) and a fortiori\(h_{\text{SG}} \left( \gamma \right)\), underestimate the hindex.$$h_{\text{BL}} \left( {\delta ,\gamma ,\eta } \right) = \delta + \gamma m^{\eta } T^{1  \eta } .$$(21)
 (h)For the particular choice of the power parameter η = 2/3 in the above model \(h_{\text{BL}} \left( {\delta ,\gamma ,\eta } \right)\), we obtain the twoparameter modelThis model directly generalizes the above Glänzel–Schubert model \(h_{\text{SG}} \left( \gamma \right)\) by introducing a free intercept parameter, δ. Tahira et al. (2013) tested this model in a scientometric analysis of engineering in Malaysian universities. They found the estimates δ = −0.28 and γ = 0.97.$$h_{\text{TAB}} \left( {\delta ,\gamma } \right) = \delta + \gamma \cdot m^{2/3} T^{1/3} .$$(22)
 (i)Finally, by assuming a linear dependence between the hindex and the function \(m^{\eta } T^{1  \eta }\) in a double logarithmic axis plot (log–log plot), one may define the following threeparameter model (see Radicchi and Castellano 2013)Indeed, after taking logs, this corresponds to a regression relationship between log h and the linear model \(\xi + \varphi \cdot \log \left( {m^{\eta } T^{1  \eta } } \right)\), where \({\varrho } = e^{\xi }\). Needless to say, model \(h_{\text{RC}}\) is similar to but essentially different from the above models (a)–(h). Radicchi and Castellano (2013) analyzed the scientific profile of more than 30,000 researchers. They found a good linear correlation, in a log–log plot, between the true hindex and the values given by the model \(h_{\text{RC}} \left( {{\varrho },\varphi ,\eta } \right)\). Using this relationship, they obtained, in particular, the least square estimate of the parameter η: \(\hat{\eta } = 0.41\). It is quite puzzling to observe that the solution reached by Radicchi and Castellano is out of the parameter space of all the above models (η > 0.5).$$h_{RC} \left( {{\varrho },\varphi ,\eta } \right) = {\varrho }\left( {m^{\eta } T^{1  \eta } } \right)^{\varphi } .$$(23)
Two empirical studies
A first dataset of journals
Journal selection
 (a)
we considered all and only the journals (568 journals) belonging to the subarea S&MM;
 (b)
to facilitate possible comparisons between databases, the journals selected were subsequently restricted to only those (253) journals indexed by all three databases: WoS, Scopus and GS;
 (c)
we excluded 15 journals with incomplete issues within the period under investigation, 2010–2014;
 (d)
finally, in order to preserve the homogeneity of the sample, we excluded 6 journals with a “too large” number of published papers (more than 2000) and 1 journal that publishes only online.
Scopus “Subject Areas” of the 231 journals within the S&MM list
Subject area  Count  % 

Mathematics  239  38.3 
Decision sciences  79  12.7 
Computer science  63  10.1 
Social sciences  51  8.2 
Engineering  45  7.2 
Economics, econometrics and finance  37  5.9 
Medicine  23  3.7 
Business, management and accounting  17  2.7 
Environmental science  13  2.1 
Others  57  9.1 
Estimating the hindex
Basic statistics for the S&MM list of journals and the approximations of the Hirsch hindex calculated by means of different formulas (rounded values)
#  ISSN code  C  T  T_{1}  \(C_{\text{1}}\)  h  \(h_{W}^{\left( 0 \right)}\)  \(\tilde{h}_{W}^{\left( 1 \right)}\)  h_{SG} (.63)  h_{SG} (.7)  h_{SG} (.8)  h_{SG} (.9)  h_{SG} (1)  \(h_{\text{BS}} \;\left( {1.2} \right)\)  \(h_{\text{BS}} \;\left( {1.4} \right)\)  \(h_{\text{BS}} \;\left( {1.6} \right)\) 

1  14057425  42  152  24  6  3  3  3  1  2  2  2  2  2  2  2 
2  10129367  276  360  111  14  6  5  6  4  4  5  5  6  4  5  6 
3  0017095X  158  166  71  13  5  5  5  3  4  4  5  5  4  5  5 
4  03153681  557  427  177  44  9  7  8  6  6  7  8  9  7  8  8 
5  10811826  201  140  77  12  6  6  6  4  5  5  6  7  5  6  6 
6  09573720  323  228  122  15  7  7  7  5  5  6  7  8  6  7  7 
7  00029890  589  351  171  87  9  8  8  6  7  8  9  10  8  9  9 
8  03610926  2033  1555  754  28  11  9  10  9  10  11  12  14  9  12  14 
9  01171968  163  120  61  20  5  6  5  4  4  5  5  6  5  5  5 
10  12100552  405  205  119  31  9  8  8  6  6  7  8  9  7  8  8 
11  10562176  290  222  101  22  7  6  7  5  5  6  7  7  6  6  6 
12  01654896  583  320  198  16  10  8  8  6  7  8  9  10  8  9  9 
13  03155986  166  83  48  24  6  6  6  4  5  6  6  7  6  6  5 
14  07362994  577  283  176  19  9  9  9  7  7  8  10  11  8  9  9 
15  03990559  153  86  47  32  5  6  5  4  5  5  6  6  5  5  5 
16  13035010  658  334  154  56  11  9  10  7  8  9  10  11  9  9  10 
17  09277099  463  296  162  16  8  7  8  6  6  7  8  9  7  8  8 
18  13511610  313  150  92  23  8  8  8  5  6  7  8  9  7  7  7 
19  12928100  191  78  52  22  7  7  7  5  5  6  7  8  6  6  6 
20  03610918  1036  635  369  45  9  9  9  8  8  10  11  12  9  10  11 
21  02699648  263  172  84  16  7  7  7  5  5  6  7  7  6  6  6 
22  15326349  308  141  93  15  7  8  8  6  6  7  8  9  7  7  7 
23  02175959  522  261  155  33  9  8  9  6  7  8  9  10  8  9  9 
24  10185895  424  189  115  25  9  8  9  6  7  8  9  10  8  8  8 
25  02664763  2164  901  518  323  13  12  12  11  12  14  16  17  13  15  16 
26  1471678X  336  138  92  23  8  8  8  6  7  7  8  9  8  8  8 
27  03044068  737  433  265  25  9  9  8  7  8  9  10  11  8  9  10 
28  00207276  480  265  158  13  8  8  8  6  7  8  9  10  8  8  8 
29  00235954  813  337  208  36  11  10  11  8  9  10  11  13  10  11  11 
30  12201766  526  193  137  31  10  10  9  7  8  9  10  11  9  10  9 
31  12263192  457  271  137  20  10  8  8  6  6  7  8  9  7  8  8 
32  16182510  305  172  90  31  8  7  7  5  6  7  7  8  7  7  7 
33  1083589X  739  353  209  20  10  9  10  7  8  9  10  12  9  10  10 
34  10485252  643  283  189  17  10  9  9  7  8  9  10  11  9  10  10 
35  10043756  443  140  96  27  9  10  10  7  8  9  10  11  9  9  9 
36  10096124  979  466  240  56  12  10  11  8  9  10  11  13  10  11  12 
37  11209763  434  492  165  18  8  6  7  5  5  6  7  7  5  6  7 
38  13691473  282  140  76  24  8  7  8  5  6  7  7  8  7  7  7 
39  12301612  346  128  84  32  8  9  8  6  7  8  9  10  8  8  8 
40  00261335  544  283  171  24  10  8  9  6  7  8  9  10  8  9  9 
41  0218348X  476  167  129  30  9  10  9  7  8  9  10  11  9  9  9 
42  01677152  3169  1546  945  40  16  12  13  12  13  15  17  19  13  16  18 
43  00324663  154  103  58  13  6  6  6  4  4  5  6  6  5  5  5 
44  0282423X  405  196  116  20  9  8  8  6  7  8  8  9  8  8  8 
45  1748670X  1933  822  543  36  14  12  12  10  12  13  15  17  12  14  15 
46  00949655  1649  695  425  55  14  12  12  10  11  13  14  16  12  14  15 
47  00390402  365  129  86  34  9  9  9  6  7  8  9  10  8  8  8 
48  08949840  615  331  184  29  9  9  9  7  7  8  9  10  8  9  9 
49  03987620  679  303  170  66  10  9  10  7  8  9  10  12  9  10  10 
50  02190257  336  159  102  31  7  8  7  6  6  7  8  9  7  8  7 
51  03195724  511  206  129  36  10  9  9  7  8  9  10  11  9  9  9 
52  00203157  772  285  189  60  11  11  10  8  9  10  12  13  10  11  11 
53  08982112  597  228  149  26  11  10  10  7  8  9  10  12  9  10  10 
54  15241904  669  301  155  42  12  9  11  7  8  9  10  11  9  10  10 
55  09635483  719  272  179  24  11  10  11  8  9  10  11  12  10  11  11 
56  15475816  770  290  201  37  11  10  10  8  9  10  11  13  10  11  11 
57  00018678  821  269  201  37  11  11  11  9  10  11  12  14  11  12  11 
58  00219002  1168  477  321  35  13  11  11  9  10  11  13  14  11  12  13 
59  02570130  719  260  179  18  11  10  11  8  9  10  11  13  10  11  11 
60  10260226  2306  1036  610  34  15  12  13  11  12  14  16  17  13  15  16 
61  03783758  3899  1334  907  71  18  15  16  14  16  18  20  23  16  19  21 
62  03777332  1353  597  348  38  15  11  12  9  10  12  13  15  11  13  13 
63  15603547  735  249  182  25  11  11  11  8  9  10  12  13  10  11  11 
64  08934983  793  297  200  36  12  11  11  8  9  10  12  13  10  11  11 
65  13875841  645  305  178  26  10  9  10  7  8  9  10  11  9  10  10 
66  01676377  1702  582  399  33  14  13  13  11  12  14  15  17  13  15  15 
67  17477778  837  135  93  294  10  15  12  11  12  14  16  17  14  14  13 
68  10543406  1098  429  277  40  13  11  12  9  10  11  13  14  11  12  13 
69  16194500  493  125  89  38  12  11  11  8  9  10  11  12  10  10  10 
70  01439782  761  258  179  31  12  11  11  8  9  10  12  13  11  11  11 
71  14322994  512  207  146  29  9  9  9  7  8  9  10  11  9  9  9 
72  02194937  304  178  102  21  7  7  7  5  6  6  7  8  6  7  7 
73  00335177  1734  878  522  42  14  11  11  9  11  12  14  15  11  13  14 
74  1748006X  779  238  184  31  11  11  11  9  10  11  12  14  11  12  11 
75  1381298X  364  113  82  23  9  9  9  7  7  8  9  11  9  9  8 
76  02776693  825  217  160  61  14  12  12  9  10  12  13  15  12  12  12 
77  1435246X  735  263  175  43  11  11  11  8  9  10  11  13  10  11  11 
78  15725286  587  158  114  25  12  11  11  8  9  10  12  13  11  11  10 
79  11345764  458  246  128  59  8  8  8  6  7  8  9  9  8  8  8 
80  09325026  829  396  210  26  11  10  11  8  8  10  11  12  9  10  11 
81  09262601  769  286  196  78  10  10  10  8  9  10  11  13  10  11  11 
82  08908575  333  119  74  47  8  9  8  6  7  8  9  10  8  8  8 
83  02195259  803  254  179  32  12  11  11  9  10  11  12  14  11  12  11 
84  05150361  447  150  89  37  11  10  10  7  8  9  10  11  9  9  9 
85  00954616  626  192  135  46  11  11  11  8  9  10  11  13  10  11  10 
86  02331934  1191  490  304  24  13  11  12  9  10  11  13  14  11  12  13 
87  01675923  663  216  152  38  12  11  11  8  9  10  11  13  10  11  11 
88  14697688  2100  653  404  77  17  14  16  12  13  15  17  19  15  16  17 
89  10836489  1321  488  330  32  13  12  12  10  11  12  14  15  12  13  14 
90  13925113  747  202  138  52  13  12  12  9  10  11  13  14  11  12  11 
91  18638171  404  118  77  34  10  10  10  7  8  9  10  11  9  9  9 
92  13807870  379  170  103  39  9  8  8  6  7  8  9  9  8  8  8 
93  18624472  1866  652  438  32  15  13  14  11  12  14  16  17  13  15  16 
94  02198762  905  300  185  65  15  11  12  9  10  11  13  14  11  12  12 
95  02181274  5537  1370  1013  136  26  19  20  18  20  23  25  28  21  24  26 
96  07474938  649  149  113  54  12  12  12  9  10  11  13  14  12  12  11 
97  00207985  1280  417  268  28  16  12  13  10  11  13  14  16  12  14  14 
98  0047259X  3329  915  650  89  21  17  17  14  16  18  21  23  18  20  21 
99  03036898  868  256  188  31  12  12  12  9  10  11  13  14  12  12  12 
100  1471082X  405  134  88  35  9  9  9  7  7  9  10  11  9  9  9 
101  09246703  413  117  79  38  9  10  10  7  8  9  10  11  9  9  9 
102  03461238  337  128  79  28  9  8  9  6  7  8  9  10  8  8  8 
103  07488017  2076  534  380  31  19  15  16  13  14  16  18  20  16  17  18 
104  13894420  793  184  124  124  15  13  12  9  11  12  14  15  12  12  12 
105  01466216  737  215  155  30  12  11  12  9  10  11  12  14  11  11  11 
106  01605682  3870  853  663  90  21  19  19  16  18  21  23  26  20  22  23 
107  09600779  2712  570  443  118  20  18  18  15  16  19  21  23  19  20  20 
108  02460203  1019  266  206  33  14  13  13  10  11  13  14  16  13  13  13 
109  03067734  563  147  83  101  12  11  11  8  9  10  12  13  11  11  10 
110  13507265  1499  375  294  40  15  15  14  11  13  15  16  18  15  15  15 
111  00219320  910  274  207  22  12  12  12  9  10  12  13  14  12  12  12 
112  02184885  1036  297  202  81  13  13  13  10  11  12  14  15  12  13  13 
113  1945497X  885  162  130  57  15  14  14  11  12  14  15  17  14  14  13 
114  13528505  564  192  130  64  10  10  10  7  8  9  11  12  10  10  10 
115  00031305  670  241  133  43  13  10  11  8  9  10  11  12  10  10  10 
116  10762787  900  224  163  49  14  13  13  10  11  12  14  15  13  13  12 
117  18625347  524  125  79  63  11  11  11  8  9  10  12  13  11  11  10 
118  00224715  5302  1246  966  91  24  20  20  18  20  23  25  28  21  24  26 
119  11330686  617  246  127  54  12  10  11  7  8  9  10  12  9  10  10 
120  15391604  1075  286  194  183  13  13  12  10  11  13  14  16  13  13  13 
121  14346028  7722  1849  1420  72  27  21  21  20  22  25  29  32  23  27  30 
122  03044149  2652  791  577  44  15  15  15  13  15  17  19  21  16  18  19 
123  01432087  1089  228  155  152  15  14  14  11  12  14  16  17  14  14  14 
124  03233847  1221  327  230  129  15  13  13  10  12  13  15  17  13  14  14 
125  02664666  1295  303  208  33  17  14  15  11  12  14  16  18  14  15  15 
126  09255001  3452  849  611  61  22  18  19  15  17  19  22  24  19  21  22 
127  10857117  682  183  129  49  13  12  12  9  10  11  12  14  11  11  11 
128  09275398  1505  358  250  53  18  15  16  12  13  15  17  18  15  16  16 
129  08998256  2942  696  512  76  20  17  18  15  16  19  21  23  18  20  21 
130  00359254  1023  212  169  54  14  14  14  11  12  14  15  17  14  14  14 
131  08939659  9519  1631  1295  95  35  26  27  24  27  31  34  38  29  33  35 
132  09266003  2408  508  394  78  20  18  18  14  16  18  20  23  18  19  19 
133  13684221  533  116  86  49  9  12  11  8  9  11  12  13  11  11  10 
134  13861999  534  120  83  30  13  12  12  8  9  11  12  13  11  11  10 
135  02545330  4505  1241  824  190  21  18  19  16  18  20  23  25  19  22  24 
136  11804009  1611  325  236  52  18  16  17  13  14  16  18  20  16  17  16 
137  01679473  7203  1541  1235  162  26  22  22  20  23  26  29  32  24  28  30 
138  00131644  1350  262  214  78  16  16  15  12  13  15  17  19  16  16  15 
139  10505164  2089  373  322  30  20  18  18  14  16  18  20  23  18  19  19 
140  15446115  1073  260  199  56  15  14  13  10  11  13  15  16  13  14  13 
141  10556788  1243  314  220  285  12  14  12  11  12  14  15  17  14  14  14 
142  10769986  655  148  110  60  11  12  12  9  10  11  13  14  12  12  11 
143  00255718  3127  595  488  60  22  20  20  16  18  20  23  25  20  22  22 
144  00361410  3275  618  514  85  21  20  20  16  18  21  23  26  21  22  22 
145  0740817X  1881  382  302  44  18  17  17  13  15  17  19  21  17  18  18 
146  01676687  2779  572  469  37  19  18  18  15  17  19  21  24  19  20  21 
147  0364765X  1237  227  180  61  17  16  16  12  13  15  17  19  15  16  15 
148  10170405  2048  426  308  190  19  17  17  14  15  17  19  21  17  18  18 
149  1369183X  2904  469  398  90  24  21  20  17  18  21  24  26  21  22  22 
150  15455963  3954  658  524  72  26  22  23  18  20  23  26  29  23  25  25 
151  10641246  1887  813  504  40  16  12  13  10  11  13  15  16  12  14  15 
152  00255564  2637  545  434  61  20  18  18  15  16  19  21  23  19  20  20 
153  00361399  2359  466  390  63  19  18  18  14  16  18  21  23  18  19  19 
154  00223239  4134  1005  685  112  24  18  20  16  18  21  23  26  20  22  23 
155  01979183  1062  195  144  131  15  15  15  11  13  14  16  18  15  15  14 
156  09492984  777  146  124  25  14  14  13  10  11  13  14  16  13  13  12 
157  01788051  1744  408  313  47  17  16  16  12  14  16  18  20  16  17  17 
158  14359871  1565  347  280  51  15  16  15  12  13  15  17  19  16  16  16 
159  00911798  2227  408  353  56  20  18  18  14  16  18  21  23  19  19  19 
160  08955646  742  123  103  43  13  14  14  10  12  13  15  16  13  13  12 
161  02668920  1994  281  226  98  22  20  20  15  17  19  22  24  20  20  19 
162  03630129  3796  661  534  112  25  21  22  18  20  22  25  28  22  24  24 
163  0144686X  1902  376  287  50  17  17  18  13  15  17  19  21  17  18  18 
164  10618600  1661  290  237  73  18  17  17  13  15  17  19  21  17  18  17 
165  10665277  3165  491  380  273  25  22  21  17  19  22  25  27  22  23  23 
166  00207721  5586  1031  815  180  25  23  23  20  22  25  28  31  24  27  28 
167  03038300  5093  1260  850  124  25  19  21  17  19  22  25  27  21  24  25 
168  0006341X  3854  717  565  75  24  21  21  17  19  22  25  27  22  24  24 
169  09601627  854  189  149  36  14  13  13  10  11  13  14  16  13  13  12 
170  03059049  886  209  157  56  12  13  13  10  11  12  14  16  13  13  12 
171  01678655  12,864  1417  1249  1129  40  35  33  31  34  39  44  49  38  42  43 
172  19328184  3207  648  414  74  24  19  22  16  18  20  23  25  20  22  22 
173  16139372  832  171  134  36  13  14  14  10  11  13  14  16  13  13  12 
174  14798409  461  115  74  46  11  11  11  8  9  10  11  12  10  10  9 
175  18748961  1560  275  206  73  19  17  18  13  14  17  19  21  17  17  17 
176  09603174  1891  408  284  109  19  16  17  13  14  16  19  21  17  18  17 
177  17425468  3572  1564  950  41  19  13  14  13  14  16  18  20  14  17  20 
178  0885064X  1081  185  149  96  14  16  15  12  13  15  17  18  15  15  14 
179  00071102  907  149  115  123  14  15  14  11  12  14  16  18  14  14  13 
180  01716468  1499  215  165  82  17  18  19  14  15  17  20  22  18  18  17 
181  19440391  484  201  81  28  11  9  11  7  7  8  9  11  9  9  9 
182  17262135  1007  115  112  66  16  17  16  13  14  17  19  21  17  16  14 
183  15448444  1703  242  210  56  17  19  19  14  16  18  21  23  19  19  18 
184  00324728  558  101  87  34  11  13  12  9  10  12  13  15  12  11  11 
185  00224065  752  113  88  34  14  15  15  11  12  14  15  17  14  13  12 
186  00393665  913  158  119  176  13  15  13  11  12  14  16  17  14  14  13 
187  01686577  536  93  80  53  12  13  12  9  10  12  13  15  12  11  10 
188  08869383  2339  365  286  128  22  20  20  16  17  20  22  25  20  21  20 
189  00189529  4175  469  387  94  29  27  28  21  23  27  30  33  27  28  27 
190  10541500  5630  936  774  80  27  24  24  20  23  26  29  32  25  28  29 
191  03044076  5332  723  609  165  30  26  26  21  24  27  31  34  27  29  29 
192  00063444  2406  392  314  85  22  20  20  15  17  20  22  25  20  21  20 
193  09641998  1287  234  177  50  17  16  16  12  13  15  17  19  16  16  15 
194  19326157  2740  524  373  102  22  19  20  15  17  19  22  24  19  21  21 
195  14681218  12,517  1271  1139  238  42  37  36  31  35  40  45  50  39  43  43 
196  00255610  3997  567  442  194  27  24  24  19  21  24  27  30  25  26  26 
197  14363240  3874  661  562  66  24  22  21  18  20  23  25  28  23  24  24 
198  01676911  7259  731  617  351  37  32  32  26  29  33  37  42  34  35  35 
199  03050548  13,373  1261  1135  156  45  39  39  33  37  42  47  52  42  45  45 
200  00401706  1141  235  153  79  16  15  16  11  12  14  16  18  14  15  14 
201  01650114  7962  1106  818  108  33  28  31  24  27  31  35  39  30  33  34 
202  08837252  2055  286  234  108  22  20  20  15  17  20  22  25  20  20  19 
203  02724332  6416  871  687  86  33  27  29  23  25  29  33  36  29  31  31 
204  02776715  10,506  1780  1314  623  35  27  28  25  28  32  36  40  30  34  37 
205  15684539  976  119  106  109  15  17  16  13  14  16  18  20  16  15  14 
206  00222496  1417  199  160  82  19  18  18  14  15  17  19  22  18  18  16 
207  00333123  1431  231  172  288  14  17  16  13  14  17  19  21  17  17  16 
208  09518320  9529  926  850  95  37  35  35  29  32  37  42  46  37  39  39 
209  03043800  13,918  1689  1511  412  36  34  33  31  34  39  44  49  38  42  44 
210  13845810  2334  238  198  137  24  24  24  18  20  23  26  28  23  23  21 
211  01697439  5880  726  645  187  30  28  27  23  25  29  33  36  29  31  31 
212  15386341  1341  264  132  147  17  16  18  12  13  15  17  19  16  16  15 
213  0030364X  5098  554  487  120  30  29  29  23  25  29  32  36  29  30  30 
214  00987921  1855  198  153  143  22  22  22  16  18  21  23  26  21  21  19 
215  14654644  2347  304  253  142  23  22  21  17  18  21  24  26  22  22  21 
216  01990039  1110  140  108  95  16  18  17  13  14  17  19  21  17  16  15 
217  10526234  4321  414  345  765  25  29  26  22  25  28  32  36  29  29  28 
218  07350015  1932  245  186  258  22  21  20  16  17  20  22  25  20  20  19 
219  01679236  10,594  923  797  458  42  38  38  31  35  40  45  50  40  42  42 
220  01621459  5231  663  519  156  31  27  28  22  24  28  31  35  28  29  29 
221  00491241  803  115  99  148  14  15  13  11  12  14  16  18  14  14  13 
222  03788733  2879  231  214  391  22  28  25  21  23  26  30  33  27  26  24 
223  1470160X  16,653  1636  1516  214  44  40  39  35  39  44  50  55  43  48  49 
224  00703370  3714  420  376  74  26  26  26  20  22  26  29  32  26  27  26 
225  09622802  1476  211  153  102  21  18  19  14  15  17  20  22  18  18  17 
226  00905364  5835  486  433  315  31  33  33  26  29  33  37  41  34  34  33 
227  00273171  1886  196  151  460  18  22  19  17  18  21  24  26  21  21  19 
228  08834237  1909  237  151  375  21  21  20  16  17  20  22  25  20  20  19 
229  15324435  14,005  1121  841  966  55  42  45  35  39  45  50  56  45  48  47 
230  13697412  3186  169  149  475  23  32  30  25  27  31  35  39  31  29  26 
231  10705511  1374  187  152  94  18  18  18  14  15  17  19  22  18  17  16 
A second dataset of journals
Journal selection
Basic statistics for the EE&F list of journals and the approximations of the Hirsch hindex calculated by means of different formulas (rounded values)
#  ISSN code  C  T  T_{1}  \(C_{\text{1}}\)  h  \(h_{W}^{\left( 0 \right)}\)  \(\tilde{h}_{W}^{\left( 1 \right)}\)  \(h_{\text{SG}} \;\left( {.63} \right)\)  \(h_{\text{SG}} \;\left( {.7} \right)\)  \(h_{\text{SG}} \;\left( {.8} \right)\)  \(h_{\text{SG}} \;\left( {.9} \right)\)  \(h_{\text{SG}} \;\left( 1 \right)\)  \(h_{\text{BS}} \;\left( {1.2} \right)\)  \(h_{\text{BS}} \left( {1.4} \right)\)  \(h_{\text{BS}} \;\left( {1.6} \right)\) 

1  00220515  697  69  63  61  15  16  15  12  13  15  17  19  15  14  12 
2  15314650  1161  127  117  58  18  19  18  14  15  18  20  22  18  17  15 
3  15571211  1773  193  173  119  21  21  20  16  18  20  23  25  21  20  19 
4  15406261  1529  190  178  54  17  19  19  15  16  18  21  23  19  19  17 
5  08953309  995  133  111  44  15  17  16  12  14  16  18  20  16  15  14 
6  15477185  1196  153  143  41  17  18  17  13  15  17  19  21  17  17  15 
7  00920703  1015  140  128  111  15  17  15  12  14  16  18  19  16  15  14 
8  0304405X  2413  412  372  48  20  19  19  15  17  19  22  24  20  20  20 
9  14680262  1014  187  171  35  14  15  14  11  12  14  16  18  14  14  14 
10  15232409  434  81  71  26  10  11  11  8  9  11  12  13  11  10  9 
11  1537534X  483  92  79  56  10  12  11  9  10  11  12  14  11  11  10 
12  14657368  1389  288  256  38  16  16  15  12  13  15  17  19  15  16  15 
13  15406520  1062  175  147  52  15  16  15  12  13  15  17  19  15  15  14 
14  14786990  795  155  140  38  13  14  13  10  11  13  14  16  13  13  12 
15  19457790  516  113  103  22  10  12  11  8  9  11  12  13  11  11  10 
16  00028282  3303  723  562  48  21  19  19  16  17  20  22  25  19  21  22 
17  19457715  422  91  78  38  9  11  10  8  9  10  11  13  10  10  9 
18  17416248  361  55  52  52  10  11  10  8  9  11  12  13  10  10  9 
19  14695758  272  65  46  26  10  9  9  7  7  8  9  10  8  8  7 
20  01654101  517  118  99  22  11  11  11  8  9  11  12  13  11  11  10 
21  09255273  4678  1036  888  92  22  20  19  17  19  22  25  28  21  24  25 
22  15424774  641  148  122  74  10  12  11  9  10  11  13  14  12  12  11 
23  15375277  1086  234  213  24  12  14  13  11  12  14  15  17  14  14  14 
24  09213449  1723  421  363  33  15  15  14  12  13  15  17  19  15  16  16 
25  1467937X  688  192  147  32  11  11  11  9  9  11  12  14  11  11  11 
26  1945774X  422  109  93  49  8  10  9  7  8  9  11  12  10  10  9 
27  18736181  2683  667  565  26  16  17  16  14  15  18  20  22  17  19  20 
28  15477193  948  213  188  56  13  14  12  10  11  13  15  16  13  13  13 
29  10864415  324  57  49  36  10  10  10  8  9  10  11  12  10  9  8 
30  17412900  234  54  42  34  8  9  8  6  7  8  9  10  8  8  7 
31  15309142  1065  292  241  27  13  13  12  10  11  13  14  16  13  13  13 
32  15309290  887  242  208  38  11  12  11  9  10  12  13  15  12  12  12 
33  00014826  837  217  178  48  12  12  12  9  10  12  13  15  12  12  12 
34  10909516  639  154  134  23  12  12  11  9  10  11  12  14  11  11  11 
35  15477215  239  60  54  14  8  9  8  6  7  8  9  10  8  8  7 
36  19411383  246  66  51  33  8  9  8  6  7  8  9  10  8  8  7 
37  09218009  2620  675  567  34  17  16  16  14  15  17  19  22  17  19  19 
38  00246301  248  58  44  33  9  9  8  6  7  8  9  10  8  8  7 
39  14682710  586  142  122  36  10  12  11  8  9  11  12  13  11  11  10 
40  14680297  760  210  179  29  10  12  11  9  10  11  13  14  11  12  11 
41  10662243  355  85  73  27  9  10  9  7  8  9  10  11  9  9  8 
42  1475679X  398  111  86  21  10  10  10  7  8  9  10  11  9  9  9 
43  0308597X  1557  475  399  35  12  13  12  11  12  14  15  17  14  15  15 
44  00221996  794  247  191  22  11  11  11  9  10  11  12  14  11  12  11 
45  10960449  673  183  142  25  11  12  11  9  9  11  12  14  11  11  11 
46  15736938  340  99  72  68  7  9  8  7  7  8  9  11  9  9  8 
47  2041417X  178  55  35  26  7  7  7  5  6  7  7  8  7  7  6 
48  03069192  951  291  224  35  14  12  12  9  10  12  13  15  12  12  12 
49  15372707  422  139  86  73  9  9  9  7  8  9  10  11  9  9  9 
50  00130095  175  51  39  26  8  7  7  5  6  7  8  8  7  7  6 
51  1052150X  265  70  57  17  8  9  8  6  7  8  9  10  8  8  7 
52  15334465  179  56  28  25  8  7  8  5  6  7  7  8  7  7  6 
53  1526548X  634  182  142  61  11  11  10  8  9  10  12  13  11  11  11 
54  18735991  1725  540  426  22  13  14  13  11  12  14  16  18  14  15  16 
55  13895753  231  64  56  17  8  8  8  6  7  8  8  9  8  7  7 
56  15723089  268  86  71  24  7  8  8  6  7  8  8  9  8  8  7 
57  14681218  2068  716  522  35  14  13  13  11  13  15  16  18  14  16  17 
58  03043878  876  295  220  35  13  11  11  9  10  11  12  14  11  12  12 
59  00472727  959  331  246  74  11  11  11  9  10  11  13  14  11  12  12 
60  09695931  652  213  172  16  9  11  10  8  9  10  11  13  10  11  10 
61  15328007  270  102  78  23  7  8  7  6  6  7  8  9  7  7  7 
62  10754253  245  80  69  10  7  8  7  6  6  7  8  9  7  7  7 
63  13864181  192  68  47  24  7  7  7  5  6  7  7  8  7  7  6 
64  02651335  252  82  62  12  8  8  8  6  6  7  8  9  8  7  7 
65  15375307  214  79  61  11  7  7  7  5  6  7  8  8  7  7  6 
66  03014207  490  165  122  30  9  10  9  7  8  9  10  11  9  9  9 
67  10961224  200  61  57  22  7  8  7  5  6  7  8  9  7  7  6 
68  14676419  349  121  90  18  9  9  8  6  7  8  9  10  8  8  8 
69  1932443X  163  53  47  11  6  7  6  5  6  6  7  8  6  6  6 
70  17566916  433  167  125  19  9  9  9  7  7  8  9  10  8  9  9 
71  03043932  389  154  105  45  8  9  8  6  7  8  9  10  8  8  8 
72  15723097  265  107  78  14  7  8  7  5  6  7  8  9  7  7  7 
73  14645114  358  119  106  19  7  9  8  6  7  8  9  10  8  8  8 
74  19113846  437  156  110  31  10  9  9  7  7  9  10  11  9  9  9 
75  10960473  220  87  62  17  7  7  7  5  6  7  7  8  7  7  6 
76  10959068  325  126  99  13  8  8  8  6  7  8  8  9  8  8  8 
77  13899341  817  325  252  17  10  10  10  8  9  10  11  13  10  11  11 
78  02174561  402  148  123  13  8  9  8  6  7  8  9  10  8  9  8 
79  15488004  238  101  77  8  7  7  7  5  6  7  7  8  7  7  7 
80  03044076  1037  404  305  28  12  11  10  9  10  11  12  14  11  12  12 
81  00380121  218  74  49  38  7  8  7  5  6  7  8  9  7  7  6 
82  09287655  340  133  93  38  8  8  8  6  7  8  9  10  8  8  8 
83  1747762X  205  91  60  38  6  7  6  5  5  6  7  8  6  6  6 
84  15660141  273  110  87  16  7  8  7  6  6  7  8  9  7  7  7 
85  13928619  368  117  79  45  9  9  9  7  7  8  9  10  9  9  8 
86  15730913  719  261  198  18  11  10  10  8  9  10  11  13  10  11  11 
87  14751461  244  83  64  26  8  8  7  6  6  7  8  9  7  7  7 
88  10991255  372  163  113  15  8  8  8  6  7  8  9  9  8  8  8 
89  01762680  416  179  135  18  7  9  8  6  7  8  9  10  8  8  8 
90  10966099  242  113  78  25  6  7  7  5  6  6  7  8  7  7  6 
91  14321122  175  89  64  8  5  6  6  4  5  6  6  7  6  6  6 
92  09291199  553  244  172  28  8  9  9  7  8  9  10  11  9  9  9 
93  15730697  2627  934  717  29  13  14  13  12  14  16  18  19  15  17  18 
94  14670895  159  57  44  10  6  7  7  5  5  6  7  8  6  6  6 
95  03784266  1993  893  621  36  13  12  11  10  12  13  15  16  12  14  15 
96  18778585  167  64  50  15  6  7  6  5  5  6  7  8  6  6  6 
97  11791896  272  127  88  9  6  7  7  5  6  7  8  8  7  7  7 
98  03085147  231  88  60  14  8  8  8  5  6  7  8  8  7  7  7 
99  1043951X  449  194  145  19  8  9  8  6  7  8  9  10  8  9  9 
100  01687034  176  74  41  13  8  7  7  5  5  6  7  7  6  6  6 
Estimating the hindex
In the same way as above, for each journal we manually computed the actual value h of the hindex. Table 3 reports, for each journal, the hindex, h, and the other indicators also considered in Table 2, namely \(h_{W}^{\left( 0 \right)}\), \(\tilde{h}_{W}^{\left( 1 \right)}\), \(h_{\text{SG}} \left( {\gamma_{0} } \right)\), for \(\gamma_{0} = 0.63, 0.7, 0.8, 0.9, 1\), the numerical solution \(h_{\text{T}} \left( {q_{0} } \right)\) of Eq. (18), for different values of q_{0}, namely \(q_{0} = 1.2, 1.4, 1.6\), as well as the simple basic metrics C, T, T_{1} and C_{1}.
Discussion and conclusion
The hindex is, today, one of the tools most commonly used to rank journals (Braun et al. 2006; Vanclay 2007, 2008; Schubert and Glänzel 2007; Bornmann et al. 2009; Harzing and van der Wal 2009; Liu et al. 2009; Hodge and Lacasse 2010; Bornmann et al. 2012; Mingers et al. 2012; Xu et al. 2015). Indeed, its value is currently provided by all the three major citation databases, WoS, Scopus and GS. In an earlier study (BertoliBarsotti and Lando 2015) the LambertW formula for the hindex \(\tilde{h}_{W}^{\left( 1 \right)}\) was proved to be a good estimator of the hindex for authors. In this paper, we have extended the empirical study to the case of the hindex for journals. One of the major differences between the case of an individual scientist and that of a journal, for the computation of the hindex, is the role played by publication and citation time windows, and the approach adopted for the analysis and interpretation of the citation process (“prospective” vs “retrospective”; Glänzel 2004). As stressed by Braun et al. (2006): “The journal hindex would not be calculated for a “lifetime contribution”, as suggested by Hirsch for individual scientists, but for a definite period”. In fact, “Hirsch did not limit the period in which the citations were received” (BarIlan 2010). Unlike the case of individual scientists, and in view of a comparative assessment, calculations of a journal’s hindex must be timed (note that a notion of “timed hindex” has also been recently introduced by Schreiber (2015), for the case of individual scientists), i.e. it must be referred to standardized time periods of journal coverage, for example of 2, 3 or 5 years, as is usually done for the computation of the impact factor, in order to limit the typical sizedependency of the hindex—that is, its dependency on the total number of publications (an indicator is said to be sizedependent if it never decreases when new publications are added, Waltman 2016). A journal’s “impact factor” is essentially a timelimited version of the average number of citations by papers published in the journal in a given period of time. Several types of “impact factors” may be defined, depending on different time windows considered for publication and citation data and, possibly, different approaches to the analysis of the citation process, leading to synchronous or diachronous impact factors (Ingwersen et al. 2001; Ingwersen 2012). In its WoS form, the publication window is 2 years (defining the 2year Impact Factor, IF) or 5 years (defining the 5year Impact Factor, IF5), while Scopus adopts a 3year publication window for its IPP. In all these cases, the impact factor is computed in a synchronous mode, i.e. the citations used for the calculation are all received during the same fixed period—1 year, in these cases.
 1.
that of another popular theoretical model for the hindex that has been successfully applied elsewhere to the same type of application, i.e. the Glänzel–Schubert formula, \(h_{\text{SG}} \left( {\gamma_{0} } \right)\), for different values of the free parameter γ_{0}, and secondly,
 2.
that given by the numerical solution \(h_{\text{BS}} \left( {q_{0} } \right)\) of Eq. (18), for different values of the free parameter q_{0}.
 1.As expected, the Pearson correlation between the actual value h of the hindex and each of its estimates \(h_{W}^{\left( 0 \right)}\), \(\tilde{h}_{W}^{\left( 1 \right)}\) and \(h_{\text{SG}} \left( {\gamma_{0} } \right)\), was very high, for both S&MM and EE&F datasets. In particular, this confirms previous empirical results concerning the formula \(h_{\text{SG}}\) (see Schubert and Glänzel 2007; Glänzel 2007). Indeed, ρ always exceeded 0.97. More specifically, we found the following: for the S&MM dataset, \(\rho ( {h,h_{W}^{( 0 )} }) = 0.97\) and \(\rho ( {h,\tilde{h}_{W}^{( 1 )} } ) = \rho ( {h,h_{\text{SG}} } ) = 0.98\); for the EE&F dataset,\(\rho ( {h,h_{W}^{( 0 )} } ) = \rho ( {h,h_{\text{SG}} } ) = 0.97\) and \(\rho ( {h,\tilde{h}_{W}^{( 1 )} } ) = 0.98\). Nevertheless, as can be seen from Figs. 2 and 4, a high correlation does not specifically identify a “good” estimator for the hindex. Formula \(\tilde{h}_{W}^{( 1 )}\) yielded similar levels of correlation, but a much lower level of MARE, see Figs. 1 and 3 (be aware that the figures refer to nonrounded values of the estimates). Note that the correlation between the hindex and \(h_{\text{SG}} \left( {\gamma_{0} } \right)\) does not depend on the unknown value of \(\gamma_{0}\), while, at the same time, the MARE of \(h_{SG} \left( {\gamma_{0} } \right)\) depends heavily on the choice of \(\gamma_{0}\). As can be seen from Table 4, at its best (among the values of \(\gamma_{0}\) tested), the error of \(h_{SG} \left( {\gamma_{0} } \right)\) reached its minimum (in terms of both MARE and RMSRE), for \(\gamma_{0} = 0.9\), for the dataset S&MM, while for the EE&F dataset the error of \(h_{SG} \left( {\gamma_{0} } \right)\) is at its minimum for a slightly different value of γ_{0}, i.e. γ_{0} = 0.8. This confirms that, for fixed values of γ_{0}, the effectiveness of the formula may depend on the length of the citation window considered (Glänzel 2008) and, finally, that there is no “universal” optimal value for the constant γ_{0} in the formula \(h_{\text{SG}} \left( {\gamma_{0} } \right)\). Instead, for both datasets, the formula \(\tilde{h}_{W}^{\left( 1 \right)}\) gives similar, and even smaller, levels of error (in terms of both MARE and RMSRE).Table 4
Relative accuracy, computed in terms of MARE and RMSRE (in italic), of different estimators of the hindex. For each dataset, the smallest error is indicated by a boldface number
Journal dataset
MARE RMSRE \(h_{W}^{\left( 0 \right)}\)
MARE RMSRE \(\tilde{h}_{W}^{\left( 1 \right)}\)
MARE RMSRE \(h_{\text{SG}} \;\left( {.63} \right)\)
MARE RMSRE \(h_{\text{SG}} \;\left( {.7} \right)\)
MARE RMSRE \(h_{\text{SG}} \;\left( {.8} \right)\)
MARE RMSRE \(h_{\text{SG}} \;\left( {.9} \right)\)
MARE RMSRE \(h_{\text{SG}} \;\left( 1 \right)\)
MARE RMSRE \(h_{\text{BS}} \;\left( {1.2} \right)\)
MARE RMSRE \(h_{\text{BS}} \;\left( {1.4} \right)\)
MARE RMSRE \(h_{\text{BS}} \;\left( {1.6} \right)\)
S&MM
0.104
0.076
0.272
0.193
0.099
0.076
0.163
0.103
0.065
0.076
0.133
0.100
0.283
0.207
0.122
0.117
0.198
0.129
0.094
0.103
EE&F
0.092
0.050
0.217
0.127
0.058
0.130
0.251
0.058
0.072
0.092
0.120
0.079
0.229
0.149
0.088
0.158
0.275
0.093
0.108
0.124
 2.
The approach that consists of obtaining the numerical solution \(h_{\text{BS}} \left( {q_{0} } \right)\) of Eq. (18) was also considered. We tentatively tested this method for nine different values of the free parameter q between 1 and 2, i.e. q_{0} = 1.1, 1.2,…,1.9. As expected, the resulting estimates were more or less accurate depending on the set value of q_{0}. Of the nine values of q_{0} tested, the smallest estimation error was obtained for a q_{0} value equal to around 1.4 (MARE = 0.065; RMSRE = 0.094), for the S&MM dataset, and for a q_{0} value equal to around 1.2 (MARE = 0.058; RMSRE = 0.093) for the EE&F dataset (see Table 4). Ultimately, h_{T} was found to be the most accurate estimator (if one takes q_{0} = 1.4), of those included in Table 4, for the S&MM dataset and the third best (if one takes q_{0} = 1.2), for the EE&F dataset. Overall, the errors are not dramatically different in the range of q between 1.2 and 1.6, and then a value of q_{0} = 1.5, also tested by Bletsas and Sahalos (2009), may be a good compromise solution. The Pearson correlation between the actual value h of the hindex and its estimate \(h_{\text{BS}} \left( {q_{0} } \right)\) varies slightly according to the selected value of q_{0}, but it is still very high: in particular, for q_{0} = 1.5, we obtain \(\rho \left( {h,h_{\text{BS}} \left( {q_{0} } \right)} \right) = 0.98\) for the S&MM dataset and \(\rho \left( {h,h_{\text{BS}} \left( {q_{0} } \right)} \right) = 0.96\) for the EE&F dataset. Hence, overall, the method may lead to a very good fit, but it has two main drawbacks. First, the expression of \(h_{\text{BS}} \left( {q_{0} } \right)\) is not given by any explicit formula. Second, this method continues to suffer from the problem of the conventional choice of an unknown parameter, in that we do not know a priori the value of the parameter q that will yield the “smallest” estimation error.
In conclusion, basically, the same type of equation (see Eqs. 4, 10), describes the relationship between the hindex and other simple citation metrics. The LambertW formula for the hindex works well (also) for estimating the hindex for journals—especially in its improved version (13). As can be deduced from our empirical study, this still holds true for different scientific areas, for different time windows for publication and citation, for different types of “citable” documents, and for different approaches to the analysis of the citation process (“prospective” vs “retrospective”; Glänzel 2004). At the same time, the Glänzel–Schubert class of models seems to be much less robust and reliable as an estimator of the hindex, because its accuracy closely depends on a conventional choice of one or more unknown parameters. We may accordingly conclude that \(h_{W}^{\left( 0 \right)}\) and \(\tilde{h}_{W}^{\left( 1 \right)}\) are quite effective “universal” (in the sense that they are readytouse) informetric functions that work well for estimating the hindex, for a sufficiently wide range of values. Indeed, our empirical analysis, though preliminary, suggests that the fit is very good, at least for the datasets that we studied, and for values of its arguments that are not too large, namely, h < 40, T < 2000 and m < 20, which may be considered standard values for the cases of both and scientists journals within timespans of 2–5 years.
Acknowledgements
This paper has been financed by the Italian funds ex MURST 60% 2015 and the Italian Talented Young Researchers project. The research was also backed through the Czech Science Foundation (GACR) under project n. 1723411Y (to T.L.).
Funding information
Funder Name  Grant Number  Funding Note 

European Social Fund 
 
Czech Science Foundation 
 
MURSTex60%2015  
SGS Research Project 

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