A theoretical model of the relationship between the h-index and other simple citation indicators
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DOI: 10.1007/s11192-017-2351-9
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- Bertoli-Barsotti, L. & Lando, T. Scientometrics (2017). doi:10.1007/s11192-017-2351-9
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Abstract
Of the existing theoretical formulas for the h-index, those recently suggested by Burrell (J Informetr 7:774–783, 2013b) and by Bertoli-Barsotti and Lando (J Informetr 9(4):762–776, 2015) have proved very effective in estimating the actual value of the h-index 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” Lambert-W formula for the h-index—and compare their effectiveness with that of a number of instances taken from the well-known Glänzel–Schubert class of models for the h-index (based, instead, on a Paretian model) by means of an empirical study. All the formulas considered in the comparison are “ready-to-use”, 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 Lambert-W formula for the h-index provides a quite robust and effective ready-to-use rule that should be preferred to other known formulas if one’s goal is (simply) to derive a reliable estimate of the h-index.
Keywords
Journal ranking h-index 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} Bertoli-Barsotti 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 h-index. Recently, Burrell (2013b) and Bertoli-Barsotti and Lando (2015) introduced a model that has proved very effective in estimating the actual value of the h-index 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 closed-form estimator of the h-index, expressed as a function of (some of) the above citation metrics. We shall call this estimator, for reasons which will be apparent below, the Lambert-W formula for the h-index.
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 h-index, under the assumption of knowing the real values of the h-index. Instead, in this paper we consider the more practical (and in a certain sense, opposing) problem of determining the (unknown) h-index on the basis of a ready-to-use formula for it. Then, in our empirical analyses we will use the actual values of the h-index but only to evaluate, a posteriori, the performance of the proposed ready-to-use 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 h-index 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 h-index 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 Lambert-W formula for the h-index and a popular class of alternative models, related to the so-called Glänzel–Schubert formula, that have already been proved to be highly correlated to the h-index.
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 Lambert-W formula for the h-index 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 Lambert-W formula for the h-index provides an effective ready-to-use rule that should be preferred to other known formulas if one’s goal is (simply) to derive a reliable estimate of the h-index.
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 so-called Lambert-W function (Corless and Jeffrey 2015). Remember that the Lambert-W 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” h-index, \(h_{W}^{\left( 0 \right)}\), from the actual value h of the h-index. 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 h-index.$$h_{W}^{\left( 0 \right)} = \frac{{W\left( {T\log \left( {1 + m^{ - 1} } \right)} \right)}}{{\log \left( {1 + m^{ - 1} } \right)}},$$(8)
An equation connecting h, T_{1} and C
- (c)Given h and T_{1}, we obtainand$$Q^{*} = \left( {1 - \left( {\frac{h}{{T_{1} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 {\left( {h - 1} \right)}}} \right. \kern-0pt} {\left( {h - 1} \right)}}}} } \right)^{ - 1}$$(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 h-index, 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 highly-cited papers tends to overestimate C, and consequently \(h_{W}^{\left( 1 \right)}\), in comparison to the true h-index—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 Lambert-W formulas for the h-index, 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 Bertoli-Barsotti and Lando (2015) for the estimation of the h-index for individual scientists.
Theoretical parametric models for the h-index related to the Glänzel–Schubert formula
- (a)Iglesias and Pecharroman (2007) derived the following one-parameter family of models of the h-index:where \(\eta > 0.5\) (the formula was reported by Iglesias and Pecharroman with parameter \({{\left( {1 - \eta } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \eta } \right)} \eta }} \right. \kern-0pt} \eta }\)). Glänzel (2008) estimated this model in an empirical comparative study of h-index 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 two-parameter 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 h-index 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 h-index as the numerical solution of the Eq. (17), that isfor a pre-specified 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 h-index considered in the present study, a closed-form 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 h-index, 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 one-parameter modelalso known as the Glänzel–Schubert model of the h-index. This model has been widely used (mainly for interpretative purposes—i.e. to provide a better understanding of the “mathematical properties” of the h-index) because several empirical studies suggest the existence of a strong correlation between h-index 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 ready-to-use formula for estimating the h-index 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 h-index. 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 h-index for journals, for two different sets of journals, while Csajbók et al. (2007) found an estimate of γ of 0.93 in a macro-level analysis of the h-index 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 h-index 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 ready-to-use formulaIn the framework of the analysis of the h-index for journals, ready-to-use formulas for estimating the h-index 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 so-called p-index 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 h-index 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 three-parameter 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 \(\eta = {\alpha \mathord{\left/ {\vphantom {\alpha {\left( {\alpha + 1} \right)}}} \right. \kern-0pt} {\left( {\alpha + 1} \right)}}\)). 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 h-index.$$h_{\text{BL}} \left( {\delta ,\gamma ,\eta } \right) = \delta + \gamma m^{\eta } T^{1 - \eta } .$$(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 two-parameter 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 h-index and the function \(m^{\eta } T^{1 - \eta }\) in a double logarithmic axis plot (log–log plot), one may define the following three-parameter 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 h-index 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 sub-area 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 h-index
Basic statistics for the S&MM list of journals and the approximations of the Hirsch h-index 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 | 1405-7425 | 42 | 152 | 24 | 6 | 3 | 3 | 3 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
2 | 1012-9367 | 276 | 360 | 111 | 14 | 6 | 5 | 6 | 4 | 4 | 5 | 5 | 6 | 4 | 5 | 6 |
3 | 0017-095X | 158 | 166 | 71 | 13 | 5 | 5 | 5 | 3 | 4 | 4 | 5 | 5 | 4 | 5 | 5 |
4 | 0315-3681 | 557 | 427 | 177 | 44 | 9 | 7 | 8 | 6 | 6 | 7 | 8 | 9 | 7 | 8 | 8 |
5 | 1081-1826 | 201 | 140 | 77 | 12 | 6 | 6 | 6 | 4 | 5 | 5 | 6 | 7 | 5 | 6 | 6 |
6 | 0957-3720 | 323 | 228 | 122 | 15 | 7 | 7 | 7 | 5 | 5 | 6 | 7 | 8 | 6 | 7 | 7 |
7 | 0002-9890 | 589 | 351 | 171 | 87 | 9 | 8 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 9 | 9 |
8 | 0361-0926 | 2033 | 1555 | 754 | 28 | 11 | 9 | 10 | 9 | 10 | 11 | 12 | 14 | 9 | 12 | 14 |
9 | 0117-1968 | 163 | 120 | 61 | 20 | 5 | 6 | 5 | 4 | 4 | 5 | 5 | 6 | 5 | 5 | 5 |
10 | 1210-0552 | 405 | 205 | 119 | 31 | 9 | 8 | 8 | 6 | 6 | 7 | 8 | 9 | 7 | 8 | 8 |
11 | 1056-2176 | 290 | 222 | 101 | 22 | 7 | 6 | 7 | 5 | 5 | 6 | 7 | 7 | 6 | 6 | 6 |
12 | 0165-4896 | 583 | 320 | 198 | 16 | 10 | 8 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 9 | 9 |
13 | 0315-5986 | 166 | 83 | 48 | 24 | 6 | 6 | 6 | 4 | 5 | 6 | 6 | 7 | 6 | 6 | 5 |
14 | 0736-2994 | 577 | 283 | 176 | 19 | 9 | 9 | 9 | 7 | 7 | 8 | 10 | 11 | 8 | 9 | 9 |
15 | 0399-0559 | 153 | 86 | 47 | 32 | 5 | 6 | 5 | 4 | 5 | 5 | 6 | 6 | 5 | 5 | 5 |
16 | 1303-5010 | 658 | 334 | 154 | 56 | 11 | 9 | 10 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 10 |
17 | 0927-7099 | 463 | 296 | 162 | 16 | 8 | 7 | 8 | 6 | 6 | 7 | 8 | 9 | 7 | 8 | 8 |
18 | 1351-1610 | 313 | 150 | 92 | 23 | 8 | 8 | 8 | 5 | 6 | 7 | 8 | 9 | 7 | 7 | 7 |
19 | 1292-8100 | 191 | 78 | 52 | 22 | 7 | 7 | 7 | 5 | 5 | 6 | 7 | 8 | 6 | 6 | 6 |
20 | 0361-0918 | 1036 | 635 | 369 | 45 | 9 | 9 | 9 | 8 | 8 | 10 | 11 | 12 | 9 | 10 | 11 |
21 | 0269-9648 | 263 | 172 | 84 | 16 | 7 | 7 | 7 | 5 | 5 | 6 | 7 | 7 | 6 | 6 | 6 |
22 | 1532-6349 | 308 | 141 | 93 | 15 | 7 | 8 | 8 | 6 | 6 | 7 | 8 | 9 | 7 | 7 | 7 |
23 | 0217-5959 | 522 | 261 | 155 | 33 | 9 | 8 | 9 | 6 | 7 | 8 | 9 | 10 | 8 | 9 | 9 |
24 | 1018-5895 | 424 | 189 | 115 | 25 | 9 | 8 | 9 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
25 | 0266-4763 | 2164 | 901 | 518 | 323 | 13 | 12 | 12 | 11 | 12 | 14 | 16 | 17 | 13 | 15 | 16 |
26 | 1471-678X | 336 | 138 | 92 | 23 | 8 | 8 | 8 | 6 | 7 | 7 | 8 | 9 | 8 | 8 | 8 |
27 | 0304-4068 | 737 | 433 | 265 | 25 | 9 | 9 | 8 | 7 | 8 | 9 | 10 | 11 | 8 | 9 | 10 |
28 | 0020-7276 | 480 | 265 | 158 | 13 | 8 | 8 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
29 | 0023-5954 | 813 | 337 | 208 | 36 | 11 | 10 | 11 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 11 |
30 | 1220-1766 | 526 | 193 | 137 | 31 | 10 | 10 | 9 | 7 | 8 | 9 | 10 | 11 | 9 | 10 | 9 |
31 | 1226-3192 | 457 | 271 | 137 | 20 | 10 | 8 | 8 | 6 | 6 | 7 | 8 | 9 | 7 | 8 | 8 |
32 | 1618-2510 | 305 | 172 | 90 | 31 | 8 | 7 | 7 | 5 | 6 | 7 | 7 | 8 | 7 | 7 | 7 |
33 | 1083-589X | 739 | 353 | 209 | 20 | 10 | 9 | 10 | 7 | 8 | 9 | 10 | 12 | 9 | 10 | 10 |
34 | 1048-5252 | 643 | 283 | 189 | 17 | 10 | 9 | 9 | 7 | 8 | 9 | 10 | 11 | 9 | 10 | 10 |
35 | 1004-3756 | 443 | 140 | 96 | 27 | 9 | 10 | 10 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
36 | 1009-6124 | 979 | 466 | 240 | 56 | 12 | 10 | 11 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 12 |
37 | 1120-9763 | 434 | 492 | 165 | 18 | 8 | 6 | 7 | 5 | 5 | 6 | 7 | 7 | 5 | 6 | 7 |
38 | 1369-1473 | 282 | 140 | 76 | 24 | 8 | 7 | 8 | 5 | 6 | 7 | 7 | 8 | 7 | 7 | 7 |
39 | 1230-1612 | 346 | 128 | 84 | 32 | 8 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
40 | 0026-1335 | 544 | 283 | 171 | 24 | 10 | 8 | 9 | 6 | 7 | 8 | 9 | 10 | 8 | 9 | 9 |
41 | 0218-348X | 476 | 167 | 129 | 30 | 9 | 10 | 9 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
42 | 0167-7152 | 3169 | 1546 | 945 | 40 | 16 | 12 | 13 | 12 | 13 | 15 | 17 | 19 | 13 | 16 | 18 |
43 | 0032-4663 | 154 | 103 | 58 | 13 | 6 | 6 | 6 | 4 | 4 | 5 | 6 | 6 | 5 | 5 | 5 |
44 | 0282-423X | 405 | 196 | 116 | 20 | 9 | 8 | 8 | 6 | 7 | 8 | 8 | 9 | 8 | 8 | 8 |
45 | 1748-670X | 1933 | 822 | 543 | 36 | 14 | 12 | 12 | 10 | 12 | 13 | 15 | 17 | 12 | 14 | 15 |
46 | 0094-9655 | 1649 | 695 | 425 | 55 | 14 | 12 | 12 | 10 | 11 | 13 | 14 | 16 | 12 | 14 | 15 |
47 | 0039-0402 | 365 | 129 | 86 | 34 | 9 | 9 | 9 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
48 | 0894-9840 | 615 | 331 | 184 | 29 | 9 | 9 | 9 | 7 | 7 | 8 | 9 | 10 | 8 | 9 | 9 |
49 | 0398-7620 | 679 | 303 | 170 | 66 | 10 | 9 | 10 | 7 | 8 | 9 | 10 | 12 | 9 | 10 | 10 |
50 | 0219-0257 | 336 | 159 | 102 | 31 | 7 | 8 | 7 | 6 | 6 | 7 | 8 | 9 | 7 | 8 | 7 |
51 | 0319-5724 | 511 | 206 | 129 | 36 | 10 | 9 | 9 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
52 | 0020-3157 | 772 | 285 | 189 | 60 | 11 | 11 | 10 | 8 | 9 | 10 | 12 | 13 | 10 | 11 | 11 |
53 | 0898-2112 | 597 | 228 | 149 | 26 | 11 | 10 | 10 | 7 | 8 | 9 | 10 | 12 | 9 | 10 | 10 |
54 | 1524-1904 | 669 | 301 | 155 | 42 | 12 | 9 | 11 | 7 | 8 | 9 | 10 | 11 | 9 | 10 | 10 |
55 | 0963-5483 | 719 | 272 | 179 | 24 | 11 | 10 | 11 | 8 | 9 | 10 | 11 | 12 | 10 | 11 | 11 |
56 | 1547-5816 | 770 | 290 | 201 | 37 | 11 | 10 | 10 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 11 |
57 | 0001-8678 | 821 | 269 | 201 | 37 | 11 | 11 | 11 | 9 | 10 | 11 | 12 | 14 | 11 | 12 | 11 |
58 | 0021-9002 | 1168 | 477 | 321 | 35 | 13 | 11 | 11 | 9 | 10 | 11 | 13 | 14 | 11 | 12 | 13 |
59 | 0257-0130 | 719 | 260 | 179 | 18 | 11 | 10 | 11 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 11 |
60 | 1026-0226 | 2306 | 1036 | 610 | 34 | 15 | 12 | 13 | 11 | 12 | 14 | 16 | 17 | 13 | 15 | 16 |
61 | 0378-3758 | 3899 | 1334 | 907 | 71 | 18 | 15 | 16 | 14 | 16 | 18 | 20 | 23 | 16 | 19 | 21 |
62 | 0377-7332 | 1353 | 597 | 348 | 38 | 15 | 11 | 12 | 9 | 10 | 12 | 13 | 15 | 11 | 13 | 13 |
63 | 1560-3547 | 735 | 249 | 182 | 25 | 11 | 11 | 11 | 8 | 9 | 10 | 12 | 13 | 10 | 11 | 11 |
64 | 0893-4983 | 793 | 297 | 200 | 36 | 12 | 11 | 11 | 8 | 9 | 10 | 12 | 13 | 10 | 11 | 11 |
65 | 1387-5841 | 645 | 305 | 178 | 26 | 10 | 9 | 10 | 7 | 8 | 9 | 10 | 11 | 9 | 10 | 10 |
66 | 0167-6377 | 1702 | 582 | 399 | 33 | 14 | 13 | 13 | 11 | 12 | 14 | 15 | 17 | 13 | 15 | 15 |
67 | 1747-7778 | 837 | 135 | 93 | 294 | 10 | 15 | 12 | 11 | 12 | 14 | 16 | 17 | 14 | 14 | 13 |
68 | 1054-3406 | 1098 | 429 | 277 | 40 | 13 | 11 | 12 | 9 | 10 | 11 | 13 | 14 | 11 | 12 | 13 |
69 | 1619-4500 | 493 | 125 | 89 | 38 | 12 | 11 | 11 | 8 | 9 | 10 | 11 | 12 | 10 | 10 | 10 |
70 | 0143-9782 | 761 | 258 | 179 | 31 | 12 | 11 | 11 | 8 | 9 | 10 | 12 | 13 | 11 | 11 | 11 |
71 | 1432-2994 | 512 | 207 | 146 | 29 | 9 | 9 | 9 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
72 | 0219-4937 | 304 | 178 | 102 | 21 | 7 | 7 | 7 | 5 | 6 | 6 | 7 | 8 | 6 | 7 | 7 |
73 | 0033-5177 | 1734 | 878 | 522 | 42 | 14 | 11 | 11 | 9 | 11 | 12 | 14 | 15 | 11 | 13 | 14 |
74 | 1748-006X | 779 | 238 | 184 | 31 | 11 | 11 | 11 | 9 | 10 | 11 | 12 | 14 | 11 | 12 | 11 |
75 | 1381-298X | 364 | 113 | 82 | 23 | 9 | 9 | 9 | 7 | 7 | 8 | 9 | 11 | 9 | 9 | 8 |
76 | 0277-6693 | 825 | 217 | 160 | 61 | 14 | 12 | 12 | 9 | 10 | 12 | 13 | 15 | 12 | 12 | 12 |
77 | 1435-246X | 735 | 263 | 175 | 43 | 11 | 11 | 11 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 11 |
78 | 1572-5286 | 587 | 158 | 114 | 25 | 12 | 11 | 11 | 8 | 9 | 10 | 12 | 13 | 11 | 11 | 10 |
79 | 1134-5764 | 458 | 246 | 128 | 59 | 8 | 8 | 8 | 6 | 7 | 8 | 9 | 9 | 8 | 8 | 8 |
80 | 0932-5026 | 829 | 396 | 210 | 26 | 11 | 10 | 11 | 8 | 8 | 10 | 11 | 12 | 9 | 10 | 11 |
81 | 0926-2601 | 769 | 286 | 196 | 78 | 10 | 10 | 10 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 11 |
82 | 0890-8575 | 333 | 119 | 74 | 47 | 8 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
83 | 0219-5259 | 803 | 254 | 179 | 32 | 12 | 11 | 11 | 9 | 10 | 11 | 12 | 14 | 11 | 12 | 11 |
84 | 0515-0361 | 447 | 150 | 89 | 37 | 11 | 10 | 10 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
85 | 0095-4616 | 626 | 192 | 135 | 46 | 11 | 11 | 11 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 10 |
86 | 0233-1934 | 1191 | 490 | 304 | 24 | 13 | 11 | 12 | 9 | 10 | 11 | 13 | 14 | 11 | 12 | 13 |
87 | 0167-5923 | 663 | 216 | 152 | 38 | 12 | 11 | 11 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 11 |
88 | 1469-7688 | 2100 | 653 | 404 | 77 | 17 | 14 | 16 | 12 | 13 | 15 | 17 | 19 | 15 | 16 | 17 |
89 | 1083-6489 | 1321 | 488 | 330 | 32 | 13 | 12 | 12 | 10 | 11 | 12 | 14 | 15 | 12 | 13 | 14 |
90 | 1392-5113 | 747 | 202 | 138 | 52 | 13 | 12 | 12 | 9 | 10 | 11 | 13 | 14 | 11 | 12 | 11 |
91 | 1863-8171 | 404 | 118 | 77 | 34 | 10 | 10 | 10 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
92 | 1380-7870 | 379 | 170 | 103 | 39 | 9 | 8 | 8 | 6 | 7 | 8 | 9 | 9 | 8 | 8 | 8 |
93 | 1862-4472 | 1866 | 652 | 438 | 32 | 15 | 13 | 14 | 11 | 12 | 14 | 16 | 17 | 13 | 15 | 16 |
94 | 0219-8762 | 905 | 300 | 185 | 65 | 15 | 11 | 12 | 9 | 10 | 11 | 13 | 14 | 11 | 12 | 12 |
95 | 0218-1274 | 5537 | 1370 | 1013 | 136 | 26 | 19 | 20 | 18 | 20 | 23 | 25 | 28 | 21 | 24 | 26 |
96 | 0747-4938 | 649 | 149 | 113 | 54 | 12 | 12 | 12 | 9 | 10 | 11 | 13 | 14 | 12 | 12 | 11 |
97 | 0020-7985 | 1280 | 417 | 268 | 28 | 16 | 12 | 13 | 10 | 11 | 13 | 14 | 16 | 12 | 14 | 14 |
98 | 0047-259X | 3329 | 915 | 650 | 89 | 21 | 17 | 17 | 14 | 16 | 18 | 21 | 23 | 18 | 20 | 21 |
99 | 0303-6898 | 868 | 256 | 188 | 31 | 12 | 12 | 12 | 9 | 10 | 11 | 13 | 14 | 12 | 12 | 12 |
100 | 1471-082X | 405 | 134 | 88 | 35 | 9 | 9 | 9 | 7 | 7 | 9 | 10 | 11 | 9 | 9 | 9 |
101 | 0924-6703 | 413 | 117 | 79 | 38 | 9 | 10 | 10 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
102 | 0346-1238 | 337 | 128 | 79 | 28 | 9 | 8 | 9 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
103 | 0748-8017 | 2076 | 534 | 380 | 31 | 19 | 15 | 16 | 13 | 14 | 16 | 18 | 20 | 16 | 17 | 18 |
104 | 1389-4420 | 793 | 184 | 124 | 124 | 15 | 13 | 12 | 9 | 11 | 12 | 14 | 15 | 12 | 12 | 12 |
105 | 0146-6216 | 737 | 215 | 155 | 30 | 12 | 11 | 12 | 9 | 10 | 11 | 12 | 14 | 11 | 11 | 11 |
106 | 0160-5682 | 3870 | 853 | 663 | 90 | 21 | 19 | 19 | 16 | 18 | 21 | 23 | 26 | 20 | 22 | 23 |
107 | 0960-0779 | 2712 | 570 | 443 | 118 | 20 | 18 | 18 | 15 | 16 | 19 | 21 | 23 | 19 | 20 | 20 |
108 | 0246-0203 | 1019 | 266 | 206 | 33 | 14 | 13 | 13 | 10 | 11 | 13 | 14 | 16 | 13 | 13 | 13 |
109 | 0306-7734 | 563 | 147 | 83 | 101 | 12 | 11 | 11 | 8 | 9 | 10 | 12 | 13 | 11 | 11 | 10 |
110 | 1350-7265 | 1499 | 375 | 294 | 40 | 15 | 15 | 14 | 11 | 13 | 15 | 16 | 18 | 15 | 15 | 15 |
111 | 0021-9320 | 910 | 274 | 207 | 22 | 12 | 12 | 12 | 9 | 10 | 12 | 13 | 14 | 12 | 12 | 12 |
112 | 0218-4885 | 1036 | 297 | 202 | 81 | 13 | 13 | 13 | 10 | 11 | 12 | 14 | 15 | 12 | 13 | 13 |
113 | 1945-497X | 885 | 162 | 130 | 57 | 15 | 14 | 14 | 11 | 12 | 14 | 15 | 17 | 14 | 14 | 13 |
114 | 1352-8505 | 564 | 192 | 130 | 64 | 10 | 10 | 10 | 7 | 8 | 9 | 11 | 12 | 10 | 10 | 10 |
115 | 0003-1305 | 670 | 241 | 133 | 43 | 13 | 10 | 11 | 8 | 9 | 10 | 11 | 12 | 10 | 10 | 10 |
116 | 1076-2787 | 900 | 224 | 163 | 49 | 14 | 13 | 13 | 10 | 11 | 12 | 14 | 15 | 13 | 13 | 12 |
117 | 1862-5347 | 524 | 125 | 79 | 63 | 11 | 11 | 11 | 8 | 9 | 10 | 12 | 13 | 11 | 11 | 10 |
118 | 0022-4715 | 5302 | 1246 | 966 | 91 | 24 | 20 | 20 | 18 | 20 | 23 | 25 | 28 | 21 | 24 | 26 |
119 | 1133-0686 | 617 | 246 | 127 | 54 | 12 | 10 | 11 | 7 | 8 | 9 | 10 | 12 | 9 | 10 | 10 |
120 | 1539-1604 | 1075 | 286 | 194 | 183 | 13 | 13 | 12 | 10 | 11 | 13 | 14 | 16 | 13 | 13 | 13 |
121 | 1434-6028 | 7722 | 1849 | 1420 | 72 | 27 | 21 | 21 | 20 | 22 | 25 | 29 | 32 | 23 | 27 | 30 |
122 | 0304-4149 | 2652 | 791 | 577 | 44 | 15 | 15 | 15 | 13 | 15 | 17 | 19 | 21 | 16 | 18 | 19 |
123 | 0143-2087 | 1089 | 228 | 155 | 152 | 15 | 14 | 14 | 11 | 12 | 14 | 16 | 17 | 14 | 14 | 14 |
124 | 0323-3847 | 1221 | 327 | 230 | 129 | 15 | 13 | 13 | 10 | 12 | 13 | 15 | 17 | 13 | 14 | 14 |
125 | 0266-4666 | 1295 | 303 | 208 | 33 | 17 | 14 | 15 | 11 | 12 | 14 | 16 | 18 | 14 | 15 | 15 |
126 | 0925-5001 | 3452 | 849 | 611 | 61 | 22 | 18 | 19 | 15 | 17 | 19 | 22 | 24 | 19 | 21 | 22 |
127 | 1085-7117 | 682 | 183 | 129 | 49 | 13 | 12 | 12 | 9 | 10 | 11 | 12 | 14 | 11 | 11 | 11 |
128 | 0927-5398 | 1505 | 358 | 250 | 53 | 18 | 15 | 16 | 12 | 13 | 15 | 17 | 18 | 15 | 16 | 16 |
129 | 0899-8256 | 2942 | 696 | 512 | 76 | 20 | 17 | 18 | 15 | 16 | 19 | 21 | 23 | 18 | 20 | 21 |
130 | 0035-9254 | 1023 | 212 | 169 | 54 | 14 | 14 | 14 | 11 | 12 | 14 | 15 | 17 | 14 | 14 | 14 |
131 | 0893-9659 | 9519 | 1631 | 1295 | 95 | 35 | 26 | 27 | 24 | 27 | 31 | 34 | 38 | 29 | 33 | 35 |
132 | 0926-6003 | 2408 | 508 | 394 | 78 | 20 | 18 | 18 | 14 | 16 | 18 | 20 | 23 | 18 | 19 | 19 |
133 | 1368-4221 | 533 | 116 | 86 | 49 | 9 | 12 | 11 | 8 | 9 | 11 | 12 | 13 | 11 | 11 | 10 |
134 | 1386-1999 | 534 | 120 | 83 | 30 | 13 | 12 | 12 | 8 | 9 | 11 | 12 | 13 | 11 | 11 | 10 |
135 | 0254-5330 | 4505 | 1241 | 824 | 190 | 21 | 18 | 19 | 16 | 18 | 20 | 23 | 25 | 19 | 22 | 24 |
136 | 1180-4009 | 1611 | 325 | 236 | 52 | 18 | 16 | 17 | 13 | 14 | 16 | 18 | 20 | 16 | 17 | 16 |
137 | 0167-9473 | 7203 | 1541 | 1235 | 162 | 26 | 22 | 22 | 20 | 23 | 26 | 29 | 32 | 24 | 28 | 30 |
138 | 0013-1644 | 1350 | 262 | 214 | 78 | 16 | 16 | 15 | 12 | 13 | 15 | 17 | 19 | 16 | 16 | 15 |
139 | 1050-5164 | 2089 | 373 | 322 | 30 | 20 | 18 | 18 | 14 | 16 | 18 | 20 | 23 | 18 | 19 | 19 |
140 | 1544-6115 | 1073 | 260 | 199 | 56 | 15 | 14 | 13 | 10 | 11 | 13 | 15 | 16 | 13 | 14 | 13 |
141 | 1055-6788 | 1243 | 314 | 220 | 285 | 12 | 14 | 12 | 11 | 12 | 14 | 15 | 17 | 14 | 14 | 14 |
142 | 1076-9986 | 655 | 148 | 110 | 60 | 11 | 12 | 12 | 9 | 10 | 11 | 13 | 14 | 12 | 12 | 11 |
143 | 0025-5718 | 3127 | 595 | 488 | 60 | 22 | 20 | 20 | 16 | 18 | 20 | 23 | 25 | 20 | 22 | 22 |
144 | 0036-1410 | 3275 | 618 | 514 | 85 | 21 | 20 | 20 | 16 | 18 | 21 | 23 | 26 | 21 | 22 | 22 |
145 | 0740-817X | 1881 | 382 | 302 | 44 | 18 | 17 | 17 | 13 | 15 | 17 | 19 | 21 | 17 | 18 | 18 |
146 | 0167-6687 | 2779 | 572 | 469 | 37 | 19 | 18 | 18 | 15 | 17 | 19 | 21 | 24 | 19 | 20 | 21 |
147 | 0364-765X | 1237 | 227 | 180 | 61 | 17 | 16 | 16 | 12 | 13 | 15 | 17 | 19 | 15 | 16 | 15 |
148 | 1017-0405 | 2048 | 426 | 308 | 190 | 19 | 17 | 17 | 14 | 15 | 17 | 19 | 21 | 17 | 18 | 18 |
149 | 1369-183X | 2904 | 469 | 398 | 90 | 24 | 21 | 20 | 17 | 18 | 21 | 24 | 26 | 21 | 22 | 22 |
150 | 1545-5963 | 3954 | 658 | 524 | 72 | 26 | 22 | 23 | 18 | 20 | 23 | 26 | 29 | 23 | 25 | 25 |
151 | 1064-1246 | 1887 | 813 | 504 | 40 | 16 | 12 | 13 | 10 | 11 | 13 | 15 | 16 | 12 | 14 | 15 |
152 | 0025-5564 | 2637 | 545 | 434 | 61 | 20 | 18 | 18 | 15 | 16 | 19 | 21 | 23 | 19 | 20 | 20 |
153 | 0036-1399 | 2359 | 466 | 390 | 63 | 19 | 18 | 18 | 14 | 16 | 18 | 21 | 23 | 18 | 19 | 19 |
154 | 0022-3239 | 4134 | 1005 | 685 | 112 | 24 | 18 | 20 | 16 | 18 | 21 | 23 | 26 | 20 | 22 | 23 |
155 | 0197-9183 | 1062 | 195 | 144 | 131 | 15 | 15 | 15 | 11 | 13 | 14 | 16 | 18 | 15 | 15 | 14 |
156 | 0949-2984 | 777 | 146 | 124 | 25 | 14 | 14 | 13 | 10 | 11 | 13 | 14 | 16 | 13 | 13 | 12 |
157 | 0178-8051 | 1744 | 408 | 313 | 47 | 17 | 16 | 16 | 12 | 14 | 16 | 18 | 20 | 16 | 17 | 17 |
158 | 1435-9871 | 1565 | 347 | 280 | 51 | 15 | 16 | 15 | 12 | 13 | 15 | 17 | 19 | 16 | 16 | 16 |
159 | 0091-1798 | 2227 | 408 | 353 | 56 | 20 | 18 | 18 | 14 | 16 | 18 | 21 | 23 | 19 | 19 | 19 |
160 | 0895-5646 | 742 | 123 | 103 | 43 | 13 | 14 | 14 | 10 | 12 | 13 | 15 | 16 | 13 | 13 | 12 |
161 | 0266-8920 | 1994 | 281 | 226 | 98 | 22 | 20 | 20 | 15 | 17 | 19 | 22 | 24 | 20 | 20 | 19 |
162 | 0363-0129 | 3796 | 661 | 534 | 112 | 25 | 21 | 22 | 18 | 20 | 22 | 25 | 28 | 22 | 24 | 24 |
163 | 0144-686X | 1902 | 376 | 287 | 50 | 17 | 17 | 18 | 13 | 15 | 17 | 19 | 21 | 17 | 18 | 18 |
164 | 1061-8600 | 1661 | 290 | 237 | 73 | 18 | 17 | 17 | 13 | 15 | 17 | 19 | 21 | 17 | 18 | 17 |
165 | 1066-5277 | 3165 | 491 | 380 | 273 | 25 | 22 | 21 | 17 | 19 | 22 | 25 | 27 | 22 | 23 | 23 |
166 | 0020-7721 | 5586 | 1031 | 815 | 180 | 25 | 23 | 23 | 20 | 22 | 25 | 28 | 31 | 24 | 27 | 28 |
167 | 0303-8300 | 5093 | 1260 | 850 | 124 | 25 | 19 | 21 | 17 | 19 | 22 | 25 | 27 | 21 | 24 | 25 |
168 | 0006-341X | 3854 | 717 | 565 | 75 | 24 | 21 | 21 | 17 | 19 | 22 | 25 | 27 | 22 | 24 | 24 |
169 | 0960-1627 | 854 | 189 | 149 | 36 | 14 | 13 | 13 | 10 | 11 | 13 | 14 | 16 | 13 | 13 | 12 |
170 | 0305-9049 | 886 | 209 | 157 | 56 | 12 | 13 | 13 | 10 | 11 | 12 | 14 | 16 | 13 | 13 | 12 |
171 | 0167-8655 | 12,864 | 1417 | 1249 | 1129 | 40 | 35 | 33 | 31 | 34 | 39 | 44 | 49 | 38 | 42 | 43 |
172 | 1932-8184 | 3207 | 648 | 414 | 74 | 24 | 19 | 22 | 16 | 18 | 20 | 23 | 25 | 20 | 22 | 22 |
173 | 1613-9372 | 832 | 171 | 134 | 36 | 13 | 14 | 14 | 10 | 11 | 13 | 14 | 16 | 13 | 13 | 12 |
174 | 1479-8409 | 461 | 115 | 74 | 46 | 11 | 11 | 11 | 8 | 9 | 10 | 11 | 12 | 10 | 10 | 9 |
175 | 1874-8961 | 1560 | 275 | 206 | 73 | 19 | 17 | 18 | 13 | 14 | 17 | 19 | 21 | 17 | 17 | 17 |
176 | 0960-3174 | 1891 | 408 | 284 | 109 | 19 | 16 | 17 | 13 | 14 | 16 | 19 | 21 | 17 | 18 | 17 |
177 | 1742-5468 | 3572 | 1564 | 950 | 41 | 19 | 13 | 14 | 13 | 14 | 16 | 18 | 20 | 14 | 17 | 20 |
178 | 0885-064X | 1081 | 185 | 149 | 96 | 14 | 16 | 15 | 12 | 13 | 15 | 17 | 18 | 15 | 15 | 14 |
179 | 0007-1102 | 907 | 149 | 115 | 123 | 14 | 15 | 14 | 11 | 12 | 14 | 16 | 18 | 14 | 14 | 13 |
180 | 0171-6468 | 1499 | 215 | 165 | 82 | 17 | 18 | 19 | 14 | 15 | 17 | 20 | 22 | 18 | 18 | 17 |
181 | 1944-0391 | 484 | 201 | 81 | 28 | 11 | 9 | 11 | 7 | 7 | 8 | 9 | 11 | 9 | 9 | 9 |
182 | 1726-2135 | 1007 | 115 | 112 | 66 | 16 | 17 | 16 | 13 | 14 | 17 | 19 | 21 | 17 | 16 | 14 |
183 | 1544-8444 | 1703 | 242 | 210 | 56 | 17 | 19 | 19 | 14 | 16 | 18 | 21 | 23 | 19 | 19 | 18 |
184 | 0032-4728 | 558 | 101 | 87 | 34 | 11 | 13 | 12 | 9 | 10 | 12 | 13 | 15 | 12 | 11 | 11 |
185 | 0022-4065 | 752 | 113 | 88 | 34 | 14 | 15 | 15 | 11 | 12 | 14 | 15 | 17 | 14 | 13 | 12 |
186 | 0039-3665 | 913 | 158 | 119 | 176 | 13 | 15 | 13 | 11 | 12 | 14 | 16 | 17 | 14 | 14 | 13 |
187 | 0168-6577 | 536 | 93 | 80 | 53 | 12 | 13 | 12 | 9 | 10 | 12 | 13 | 15 | 12 | 11 | 10 |
188 | 0886-9383 | 2339 | 365 | 286 | 128 | 22 | 20 | 20 | 16 | 17 | 20 | 22 | 25 | 20 | 21 | 20 |
189 | 0018-9529 | 4175 | 469 | 387 | 94 | 29 | 27 | 28 | 21 | 23 | 27 | 30 | 33 | 27 | 28 | 27 |
190 | 1054-1500 | 5630 | 936 | 774 | 80 | 27 | 24 | 24 | 20 | 23 | 26 | 29 | 32 | 25 | 28 | 29 |
191 | 0304-4076 | 5332 | 723 | 609 | 165 | 30 | 26 | 26 | 21 | 24 | 27 | 31 | 34 | 27 | 29 | 29 |
192 | 0006-3444 | 2406 | 392 | 314 | 85 | 22 | 20 | 20 | 15 | 17 | 20 | 22 | 25 | 20 | 21 | 20 |
193 | 0964-1998 | 1287 | 234 | 177 | 50 | 17 | 16 | 16 | 12 | 13 | 15 | 17 | 19 | 16 | 16 | 15 |
194 | 1932-6157 | 2740 | 524 | 373 | 102 | 22 | 19 | 20 | 15 | 17 | 19 | 22 | 24 | 19 | 21 | 21 |
195 | 1468-1218 | 12,517 | 1271 | 1139 | 238 | 42 | 37 | 36 | 31 | 35 | 40 | 45 | 50 | 39 | 43 | 43 |
196 | 0025-5610 | 3997 | 567 | 442 | 194 | 27 | 24 | 24 | 19 | 21 | 24 | 27 | 30 | 25 | 26 | 26 |
197 | 1436-3240 | 3874 | 661 | 562 | 66 | 24 | 22 | 21 | 18 | 20 | 23 | 25 | 28 | 23 | 24 | 24 |
198 | 0167-6911 | 7259 | 731 | 617 | 351 | 37 | 32 | 32 | 26 | 29 | 33 | 37 | 42 | 34 | 35 | 35 |
199 | 0305-0548 | 13,373 | 1261 | 1135 | 156 | 45 | 39 | 39 | 33 | 37 | 42 | 47 | 52 | 42 | 45 | 45 |
200 | 0040-1706 | 1141 | 235 | 153 | 79 | 16 | 15 | 16 | 11 | 12 | 14 | 16 | 18 | 14 | 15 | 14 |
201 | 0165-0114 | 7962 | 1106 | 818 | 108 | 33 | 28 | 31 | 24 | 27 | 31 | 35 | 39 | 30 | 33 | 34 |
202 | 0883-7252 | 2055 | 286 | 234 | 108 | 22 | 20 | 20 | 15 | 17 | 20 | 22 | 25 | 20 | 20 | 19 |
203 | 0272-4332 | 6416 | 871 | 687 | 86 | 33 | 27 | 29 | 23 | 25 | 29 | 33 | 36 | 29 | 31 | 31 |
204 | 0277-6715 | 10,506 | 1780 | 1314 | 623 | 35 | 27 | 28 | 25 | 28 | 32 | 36 | 40 | 30 | 34 | 37 |
205 | 1568-4539 | 976 | 119 | 106 | 109 | 15 | 17 | 16 | 13 | 14 | 16 | 18 | 20 | 16 | 15 | 14 |
206 | 0022-2496 | 1417 | 199 | 160 | 82 | 19 | 18 | 18 | 14 | 15 | 17 | 19 | 22 | 18 | 18 | 16 |
207 | 0033-3123 | 1431 | 231 | 172 | 288 | 14 | 17 | 16 | 13 | 14 | 17 | 19 | 21 | 17 | 17 | 16 |
208 | 0951-8320 | 9529 | 926 | 850 | 95 | 37 | 35 | 35 | 29 | 32 | 37 | 42 | 46 | 37 | 39 | 39 |
209 | 0304-3800 | 13,918 | 1689 | 1511 | 412 | 36 | 34 | 33 | 31 | 34 | 39 | 44 | 49 | 38 | 42 | 44 |
210 | 1384-5810 | 2334 | 238 | 198 | 137 | 24 | 24 | 24 | 18 | 20 | 23 | 26 | 28 | 23 | 23 | 21 |
211 | 0169-7439 | 5880 | 726 | 645 | 187 | 30 | 28 | 27 | 23 | 25 | 29 | 33 | 36 | 29 | 31 | 31 |
212 | 1538-6341 | 1341 | 264 | 132 | 147 | 17 | 16 | 18 | 12 | 13 | 15 | 17 | 19 | 16 | 16 | 15 |
213 | 0030-364X | 5098 | 554 | 487 | 120 | 30 | 29 | 29 | 23 | 25 | 29 | 32 | 36 | 29 | 30 | 30 |
214 | 0098-7921 | 1855 | 198 | 153 | 143 | 22 | 22 | 22 | 16 | 18 | 21 | 23 | 26 | 21 | 21 | 19 |
215 | 1465-4644 | 2347 | 304 | 253 | 142 | 23 | 22 | 21 | 17 | 18 | 21 | 24 | 26 | 22 | 22 | 21 |
216 | 0199-0039 | 1110 | 140 | 108 | 95 | 16 | 18 | 17 | 13 | 14 | 17 | 19 | 21 | 17 | 16 | 15 |
217 | 1052-6234 | 4321 | 414 | 345 | 765 | 25 | 29 | 26 | 22 | 25 | 28 | 32 | 36 | 29 | 29 | 28 |
218 | 0735-0015 | 1932 | 245 | 186 | 258 | 22 | 21 | 20 | 16 | 17 | 20 | 22 | 25 | 20 | 20 | 19 |
219 | 0167-9236 | 10,594 | 923 | 797 | 458 | 42 | 38 | 38 | 31 | 35 | 40 | 45 | 50 | 40 | 42 | 42 |
220 | 0162-1459 | 5231 | 663 | 519 | 156 | 31 | 27 | 28 | 22 | 24 | 28 | 31 | 35 | 28 | 29 | 29 |
221 | 0049-1241 | 803 | 115 | 99 | 148 | 14 | 15 | 13 | 11 | 12 | 14 | 16 | 18 | 14 | 14 | 13 |
222 | 0378-8733 | 2879 | 231 | 214 | 391 | 22 | 28 | 25 | 21 | 23 | 26 | 30 | 33 | 27 | 26 | 24 |
223 | 1470-160X | 16,653 | 1636 | 1516 | 214 | 44 | 40 | 39 | 35 | 39 | 44 | 50 | 55 | 43 | 48 | 49 |
224 | 0070-3370 | 3714 | 420 | 376 | 74 | 26 | 26 | 26 | 20 | 22 | 26 | 29 | 32 | 26 | 27 | 26 |
225 | 0962-2802 | 1476 | 211 | 153 | 102 | 21 | 18 | 19 | 14 | 15 | 17 | 20 | 22 | 18 | 18 | 17 |
226 | 0090-5364 | 5835 | 486 | 433 | 315 | 31 | 33 | 33 | 26 | 29 | 33 | 37 | 41 | 34 | 34 | 33 |
227 | 0027-3171 | 1886 | 196 | 151 | 460 | 18 | 22 | 19 | 17 | 18 | 21 | 24 | 26 | 21 | 21 | 19 |
228 | 0883-4237 | 1909 | 237 | 151 | 375 | 21 | 21 | 20 | 16 | 17 | 20 | 22 | 25 | 20 | 20 | 19 |
229 | 1532-4435 | 14,005 | 1121 | 841 | 966 | 55 | 42 | 45 | 35 | 39 | 45 | 50 | 56 | 45 | 48 | 47 |
230 | 1369-7412 | 3186 | 169 | 149 | 475 | 23 | 32 | 30 | 25 | 27 | 31 | 35 | 39 | 31 | 29 | 26 |
231 | 1070-5511 | 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 h-index 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 | 0022-0515 | 697 | 69 | 63 | 61 | 15 | 16 | 15 | 12 | 13 | 15 | 17 | 19 | 15 | 14 | 12 |
2 | 1531-4650 | 1161 | 127 | 117 | 58 | 18 | 19 | 18 | 14 | 15 | 18 | 20 | 22 | 18 | 17 | 15 |
3 | 1557-1211 | 1773 | 193 | 173 | 119 | 21 | 21 | 20 | 16 | 18 | 20 | 23 | 25 | 21 | 20 | 19 |
4 | 1540-6261 | 1529 | 190 | 178 | 54 | 17 | 19 | 19 | 15 | 16 | 18 | 21 | 23 | 19 | 19 | 17 |
5 | 0895-3309 | 995 | 133 | 111 | 44 | 15 | 17 | 16 | 12 | 14 | 16 | 18 | 20 | 16 | 15 | 14 |
6 | 1547-7185 | 1196 | 153 | 143 | 41 | 17 | 18 | 17 | 13 | 15 | 17 | 19 | 21 | 17 | 17 | 15 |
7 | 0092-0703 | 1015 | 140 | 128 | 111 | 15 | 17 | 15 | 12 | 14 | 16 | 18 | 19 | 16 | 15 | 14 |
8 | 0304-405X | 2413 | 412 | 372 | 48 | 20 | 19 | 19 | 15 | 17 | 19 | 22 | 24 | 20 | 20 | 20 |
9 | 1468-0262 | 1014 | 187 | 171 | 35 | 14 | 15 | 14 | 11 | 12 | 14 | 16 | 18 | 14 | 14 | 14 |
10 | 1523-2409 | 434 | 81 | 71 | 26 | 10 | 11 | 11 | 8 | 9 | 11 | 12 | 13 | 11 | 10 | 9 |
11 | 1537-534X | 483 | 92 | 79 | 56 | 10 | 12 | 11 | 9 | 10 | 11 | 12 | 14 | 11 | 11 | 10 |
12 | 1465-7368 | 1389 | 288 | 256 | 38 | 16 | 16 | 15 | 12 | 13 | 15 | 17 | 19 | 15 | 16 | 15 |
13 | 1540-6520 | 1062 | 175 | 147 | 52 | 15 | 16 | 15 | 12 | 13 | 15 | 17 | 19 | 15 | 15 | 14 |
14 | 1478-6990 | 795 | 155 | 140 | 38 | 13 | 14 | 13 | 10 | 11 | 13 | 14 | 16 | 13 | 13 | 12 |
15 | 1945-7790 | 516 | 113 | 103 | 22 | 10 | 12 | 11 | 8 | 9 | 11 | 12 | 13 | 11 | 11 | 10 |
16 | 0002-8282 | 3303 | 723 | 562 | 48 | 21 | 19 | 19 | 16 | 17 | 20 | 22 | 25 | 19 | 21 | 22 |
17 | 1945-7715 | 422 | 91 | 78 | 38 | 9 | 11 | 10 | 8 | 9 | 10 | 11 | 13 | 10 | 10 | 9 |
18 | 1741-6248 | 361 | 55 | 52 | 52 | 10 | 11 | 10 | 8 | 9 | 11 | 12 | 13 | 10 | 10 | 9 |
19 | 1469-5758 | 272 | 65 | 46 | 26 | 10 | 9 | 9 | 7 | 7 | 8 | 9 | 10 | 8 | 8 | 7 |
20 | 0165-4101 | 517 | 118 | 99 | 22 | 11 | 11 | 11 | 8 | 9 | 11 | 12 | 13 | 11 | 11 | 10 |
21 | 0925-5273 | 4678 | 1036 | 888 | 92 | 22 | 20 | 19 | 17 | 19 | 22 | 25 | 28 | 21 | 24 | 25 |
22 | 1542-4774 | 641 | 148 | 122 | 74 | 10 | 12 | 11 | 9 | 10 | 11 | 13 | 14 | 12 | 12 | 11 |
23 | 1537-5277 | 1086 | 234 | 213 | 24 | 12 | 14 | 13 | 11 | 12 | 14 | 15 | 17 | 14 | 14 | 14 |
24 | 0921-3449 | 1723 | 421 | 363 | 33 | 15 | 15 | 14 | 12 | 13 | 15 | 17 | 19 | 15 | 16 | 16 |
25 | 1467-937X | 688 | 192 | 147 | 32 | 11 | 11 | 11 | 9 | 9 | 11 | 12 | 14 | 11 | 11 | 11 |
26 | 1945-774X | 422 | 109 | 93 | 49 | 8 | 10 | 9 | 7 | 8 | 9 | 11 | 12 | 10 | 10 | 9 |
27 | 1873-6181 | 2683 | 667 | 565 | 26 | 16 | 17 | 16 | 14 | 15 | 18 | 20 | 22 | 17 | 19 | 20 |
28 | 1547-7193 | 948 | 213 | 188 | 56 | 13 | 14 | 12 | 10 | 11 | 13 | 15 | 16 | 13 | 13 | 13 |
29 | 1086-4415 | 324 | 57 | 49 | 36 | 10 | 10 | 10 | 8 | 9 | 10 | 11 | 12 | 10 | 9 | 8 |
30 | 1741-2900 | 234 | 54 | 42 | 34 | 8 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 7 |
31 | 1530-9142 | 1065 | 292 | 241 | 27 | 13 | 13 | 12 | 10 | 11 | 13 | 14 | 16 | 13 | 13 | 13 |
32 | 1530-9290 | 887 | 242 | 208 | 38 | 11 | 12 | 11 | 9 | 10 | 12 | 13 | 15 | 12 | 12 | 12 |
33 | 0001-4826 | 837 | 217 | 178 | 48 | 12 | 12 | 12 | 9 | 10 | 12 | 13 | 15 | 12 | 12 | 12 |
34 | 1090-9516 | 639 | 154 | 134 | 23 | 12 | 12 | 11 | 9 | 10 | 11 | 12 | 14 | 11 | 11 | 11 |
35 | 1547-7215 | 239 | 60 | 54 | 14 | 8 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 7 |
36 | 1941-1383 | 246 | 66 | 51 | 33 | 8 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 7 |
37 | 0921-8009 | 2620 | 675 | 567 | 34 | 17 | 16 | 16 | 14 | 15 | 17 | 19 | 22 | 17 | 19 | 19 |
38 | 0024-6301 | 248 | 58 | 44 | 33 | 9 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 7 |
39 | 1468-2710 | 586 | 142 | 122 | 36 | 10 | 12 | 11 | 8 | 9 | 11 | 12 | 13 | 11 | 11 | 10 |
40 | 1468-0297 | 760 | 210 | 179 | 29 | 10 | 12 | 11 | 9 | 10 | 11 | 13 | 14 | 11 | 12 | 11 |
41 | 1066-2243 | 355 | 85 | 73 | 27 | 9 | 10 | 9 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 8 |
42 | 1475-679X | 398 | 111 | 86 | 21 | 10 | 10 | 10 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
43 | 0308-597X | 1557 | 475 | 399 | 35 | 12 | 13 | 12 | 11 | 12 | 14 | 15 | 17 | 14 | 15 | 15 |
44 | 0022-1996 | 794 | 247 | 191 | 22 | 11 | 11 | 11 | 9 | 10 | 11 | 12 | 14 | 11 | 12 | 11 |
45 | 1096-0449 | 673 | 183 | 142 | 25 | 11 | 12 | 11 | 9 | 9 | 11 | 12 | 14 | 11 | 11 | 11 |
46 | 1573-6938 | 340 | 99 | 72 | 68 | 7 | 9 | 8 | 7 | 7 | 8 | 9 | 11 | 9 | 9 | 8 |
47 | 2041-417X | 178 | 55 | 35 | 26 | 7 | 7 | 7 | 5 | 6 | 7 | 7 | 8 | 7 | 7 | 6 |
48 | 0306-9192 | 951 | 291 | 224 | 35 | 14 | 12 | 12 | 9 | 10 | 12 | 13 | 15 | 12 | 12 | 12 |
49 | 1537-2707 | 422 | 139 | 86 | 73 | 9 | 9 | 9 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
50 | 0013-0095 | 175 | 51 | 39 | 26 | 8 | 7 | 7 | 5 | 6 | 7 | 8 | 8 | 7 | 7 | 6 |
51 | 1052-150X | 265 | 70 | 57 | 17 | 8 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 7 |
52 | 1533-4465 | 179 | 56 | 28 | 25 | 8 | 7 | 8 | 5 | 6 | 7 | 7 | 8 | 7 | 7 | 6 |
53 | 1526-548X | 634 | 182 | 142 | 61 | 11 | 11 | 10 | 8 | 9 | 10 | 12 | 13 | 11 | 11 | 11 |
54 | 1873-5991 | 1725 | 540 | 426 | 22 | 13 | 14 | 13 | 11 | 12 | 14 | 16 | 18 | 14 | 15 | 16 |
55 | 1389-5753 | 231 | 64 | 56 | 17 | 8 | 8 | 8 | 6 | 7 | 8 | 8 | 9 | 8 | 7 | 7 |
56 | 1572-3089 | 268 | 86 | 71 | 24 | 7 | 8 | 8 | 6 | 7 | 8 | 8 | 9 | 8 | 8 | 7 |
57 | 1468-1218 | 2068 | 716 | 522 | 35 | 14 | 13 | 13 | 11 | 13 | 15 | 16 | 18 | 14 | 16 | 17 |
58 | 0304-3878 | 876 | 295 | 220 | 35 | 13 | 11 | 11 | 9 | 10 | 11 | 12 | 14 | 11 | 12 | 12 |
59 | 0047-2727 | 959 | 331 | 246 | 74 | 11 | 11 | 11 | 9 | 10 | 11 | 13 | 14 | 11 | 12 | 12 |
60 | 0969-5931 | 652 | 213 | 172 | 16 | 9 | 11 | 10 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 10 |
61 | 1532-8007 | 270 | 102 | 78 | 23 | 7 | 8 | 7 | 6 | 6 | 7 | 8 | 9 | 7 | 7 | 7 |
62 | 1075-4253 | 245 | 80 | 69 | 10 | 7 | 8 | 7 | 6 | 6 | 7 | 8 | 9 | 7 | 7 | 7 |
63 | 1386-4181 | 192 | 68 | 47 | 24 | 7 | 7 | 7 | 5 | 6 | 7 | 7 | 8 | 7 | 7 | 6 |
64 | 0265-1335 | 252 | 82 | 62 | 12 | 8 | 8 | 8 | 6 | 6 | 7 | 8 | 9 | 8 | 7 | 7 |
65 | 1537-5307 | 214 | 79 | 61 | 11 | 7 | 7 | 7 | 5 | 6 | 7 | 8 | 8 | 7 | 7 | 6 |
66 | 0301-4207 | 490 | 165 | 122 | 30 | 9 | 10 | 9 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
67 | 1096-1224 | 200 | 61 | 57 | 22 | 7 | 8 | 7 | 5 | 6 | 7 | 8 | 9 | 7 | 7 | 6 |
68 | 1467-6419 | 349 | 121 | 90 | 18 | 9 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
69 | 1932-443X | 163 | 53 | 47 | 11 | 6 | 7 | 6 | 5 | 6 | 6 | 7 | 8 | 6 | 6 | 6 |
70 | 1756-6916 | 433 | 167 | 125 | 19 | 9 | 9 | 9 | 7 | 7 | 8 | 9 | 10 | 8 | 9 | 9 |
71 | 0304-3932 | 389 | 154 | 105 | 45 | 8 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
72 | 1572-3097 | 265 | 107 | 78 | 14 | 7 | 8 | 7 | 5 | 6 | 7 | 8 | 9 | 7 | 7 | 7 |
73 | 1464-5114 | 358 | 119 | 106 | 19 | 7 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
74 | 1911-3846 | 437 | 156 | 110 | 31 | 10 | 9 | 9 | 7 | 7 | 9 | 10 | 11 | 9 | 9 | 9 |
75 | 1096-0473 | 220 | 87 | 62 | 17 | 7 | 7 | 7 | 5 | 6 | 7 | 7 | 8 | 7 | 7 | 6 |
76 | 1095-9068 | 325 | 126 | 99 | 13 | 8 | 8 | 8 | 6 | 7 | 8 | 8 | 9 | 8 | 8 | 8 |
77 | 1389-9341 | 817 | 325 | 252 | 17 | 10 | 10 | 10 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 11 |
78 | 0217-4561 | 402 | 148 | 123 | 13 | 8 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 9 | 8 |
79 | 1548-8004 | 238 | 101 | 77 | 8 | 7 | 7 | 7 | 5 | 6 | 7 | 7 | 8 | 7 | 7 | 7 |
80 | 0304-4076 | 1037 | 404 | 305 | 28 | 12 | 11 | 10 | 9 | 10 | 11 | 12 | 14 | 11 | 12 | 12 |
81 | 0038-0121 | 218 | 74 | 49 | 38 | 7 | 8 | 7 | 5 | 6 | 7 | 8 | 9 | 7 | 7 | 6 |
82 | 0928-7655 | 340 | 133 | 93 | 38 | 8 | 8 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
83 | 1747-762X | 205 | 91 | 60 | 38 | 6 | 7 | 6 | 5 | 5 | 6 | 7 | 8 | 6 | 6 | 6 |
84 | 1566-0141 | 273 | 110 | 87 | 16 | 7 | 8 | 7 | 6 | 6 | 7 | 8 | 9 | 7 | 7 | 7 |
85 | 1392-8619 | 368 | 117 | 79 | 45 | 9 | 9 | 9 | 7 | 7 | 8 | 9 | 10 | 9 | 9 | 8 |
86 | 1573-0913 | 719 | 261 | 198 | 18 | 11 | 10 | 10 | 8 | 9 | 10 | 11 | 13 | 10 | 11 | 11 |
87 | 1475-1461 | 244 | 83 | 64 | 26 | 8 | 8 | 7 | 6 | 6 | 7 | 8 | 9 | 7 | 7 | 7 |
88 | 1099-1255 | 372 | 163 | 113 | 15 | 8 | 8 | 8 | 6 | 7 | 8 | 9 | 9 | 8 | 8 | 8 |
89 | 0176-2680 | 416 | 179 | 135 | 18 | 7 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 8 | 8 |
90 | 1096-6099 | 242 | 113 | 78 | 25 | 6 | 7 | 7 | 5 | 6 | 6 | 7 | 8 | 7 | 7 | 6 |
91 | 1432-1122 | 175 | 89 | 64 | 8 | 5 | 6 | 6 | 4 | 5 | 6 | 6 | 7 | 6 | 6 | 6 |
92 | 0929-1199 | 553 | 244 | 172 | 28 | 8 | 9 | 9 | 7 | 8 | 9 | 10 | 11 | 9 | 9 | 9 |
93 | 1573-0697 | 2627 | 934 | 717 | 29 | 13 | 14 | 13 | 12 | 14 | 16 | 18 | 19 | 15 | 17 | 18 |
94 | 1467-0895 | 159 | 57 | 44 | 10 | 6 | 7 | 7 | 5 | 5 | 6 | 7 | 8 | 6 | 6 | 6 |
95 | 0378-4266 | 1993 | 893 | 621 | 36 | 13 | 12 | 11 | 10 | 12 | 13 | 15 | 16 | 12 | 14 | 15 |
96 | 1877-8585 | 167 | 64 | 50 | 15 | 6 | 7 | 6 | 5 | 5 | 6 | 7 | 8 | 6 | 6 | 6 |
97 | 1179-1896 | 272 | 127 | 88 | 9 | 6 | 7 | 7 | 5 | 6 | 7 | 8 | 8 | 7 | 7 | 7 |
98 | 0308-5147 | 231 | 88 | 60 | 14 | 8 | 8 | 8 | 5 | 6 | 7 | 8 | 8 | 7 | 7 | 7 |
99 | 1043-951X | 449 | 194 | 145 | 19 | 8 | 9 | 8 | 6 | 7 | 8 | 9 | 10 | 8 | 9 | 9 |
100 | 0168-7034 | 176 | 74 | 41 | 13 | 8 | 7 | 7 | 5 | 5 | 6 | 7 | 7 | 6 | 6 | 6 |
Estimating the h-index
In the same way as above, for each journal we manually computed the actual value h of the h-index. Table 3 reports, for each journal, the h-index, 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 h-index 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 (Bertoli-Barsotti and Lando 2015) the Lambert-W formula for the h-index \(\tilde{h}_{W}^{\left( 1 \right)}\) was proved to be a good estimator of the h-index for authors. In this paper, we have extended the empirical study to the case of the h-index for journals. One of the major differences between the case of an individual scientist and that of a journal, for the computation of the h-index, 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 h-index would not be calculated for a “life-time 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” (Bar-Ilan 2010). Unlike the case of individual scientists, and in view of a comparative assessment, calculations of a journal’s h-index must be timed (note that a notion of “timed h-index” 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 size-dependency of the h-index—that is, its dependency on the total number of publications (an indicator is said to be size-dependent if it never decreases when new publications are added, Waltman 2016). A journal’s “impact factor” is essentially a time-limited 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 2-year Impact Factor, IF) or 5 years (defining the 5-year Impact Factor, IF5), while Scopus adopts a 3-year 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 h-index 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 h-index 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 h-index. 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 non-rounded values of the estimates). Note that the correlation between the h-index 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 h-index. 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 h-index 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 h-index and other simple citation metrics. The Lambert-W formula for the h-index works well (also) for estimating the h-index 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 h-index, 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 ready-to-use) informetric functions that work well for estimating the h-index, 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 time-spans 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. 17-23411Y (to T.L.).
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