Some variations on the standard theoretical models for the h-index: A comparative analysis. C. Malesios 1

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1 *********Ths s the fnal draft for the paper submtted to the Journal of the Assocaton for Informaton Scence and Technology. For the offcal verson please refer to DOI: /as.23410********* Some varatons on the standard theoretcal models for the h-ndex: A comparatve analyss C. Malesos 1 1 Department of Rural Development, Democrtus Unversty of Thrace, 193 Pantazdou Str., Orestada, Greece, e-mal: malesos@agro.duth.gr Abstract There are varous mathematcal models proposed n the recent lterature for estmatng the h-ndex through bblometrc measures, such as number of artcles (P) and ctatons receved (C). These models have been prevously emprcally tested assumng a mathematcal model and predetermnng the models parameter values at some fxed constant. Here, by adoptng a statstcal modellng vew I nvestgate alternatve dstrbutons commonly used for ths type of pont data. I also show that the typcal assumptons for the parameters of the h-ndex mathematcal models n such representatons are not always realstc, wth more sutable specfcatons beng favorable. Predcton of the h- ndex s also demonstrated. Keywords: h-ndex, mathematcal model, Hrsch, ecology, forestry journals. Introducton Hrsch (2005) ntroduced an ndcator, the h-ndex, based on the dstrbuton of ctatons receved by a gven researcher s publcatons. By defnton: A scentst has ndex h f h of hs N p papers have at least h ctatons each, and the other (N p - h) papers have at most h ctatons each. Braun et al. (2005; 2006) recommended a Hrsch-based ndex to qualfy the mpact of scentfc journals. Specfcally, a journal has anh as journal h-ndex f the journal has publshed h papers, recevng each one at least h ctatons. The h-ndex s used more and more as an ndcator for the evaluaton of journals (Jokc, 2009;Malesos and Abas, 2012; Malesos and Arabatzs, 2012). There are also varous modfcatons of the h-ndex proposed n the lterature (see, e.g., Rousseau, 2007; Barendse, 2007;Molnar and Molnar 2008). Despte the ease of calculaton of the h-ndex knowng the dstrbuton of receved ctatons C, the attempts for a concse nterpretaton of the theoretcal propertes of the h-ndex has drven researchers to nvestgate the dependence of the ndex on the basc parameters of the ctaton dstrbuton (number of artcles P and ctatons receved C) and the ways that ths dependence arses through mathematcal functons. Hence, varous theoretcal models for the h-ndex based on P and C have been proposed n the lterature, wth three beng the man representatves (see Ye, 2009; 2011). These are the Hrsch model (Hrsch, 2005), the Egghe-Rousseau model (Egghe and Rousseau, 2006) and the Glänzel-Schubert model (Schubert and Glänzel, 2007), llustrated below (Table 1). For my nvestgaton I nclude, addtonally to the three standard models, a two-parameter specfcaton for the Hrsch model, suggested by Ye (2011). Ye (2011) and Franceschn and Masano (2011) pont out that these models are more sutable 1

2 for aggregated levels of ctaton data (e.g. nsttutons) and less sutable for sngle researcher data. [See also Burrell (2013) for an exhaustve crtcsm on the approprateness of such type of models n estmatng the h-ndex of a sngle researcher]. model parameters range reference a h P a (1, ) Egghe-Rousseau model (Egghe and Rousseau, 2006) h C (3,5) Hrsch model (Hrsch, 2005) 1 1 / 1,c h cp C P (1, ); c (0, ) Glänzel-Schubert model (Schubert and Glänzel, 2007) C h 1/, (1, ); f Table 1. Theoretcal models for the h-ndex based on P and C. Hrsch model (Ye, 2011) Ye (2009) estmates journal and nsttutonal h-ndces based on the latter theoretcal models (except for the two-parameter Hrsch model) and examnes the ft of each one of those models to conclude that the Glänzel-Schubert model s best at estmatng the h-ndex. The analyss - utlzng journal and nsttutonal bblometrc data obtaned from the ISI Web of Scence (WoS) s based on determnstc mathematcal expressons and pre-determned values for the models parameters. Model ft and assessment s based on vsual nspecton of lne plots of both observed h-ndex values and predctons based on the 3 models. However, n the formulae for the h-ndex based on dfferent assumed theoretcal models for the ctaton dstrbuton, there should be no presumpton that the model parameter values are unversal and n practce the parameters should be estmated from each data set. In ths note, a statstcal modelng vew s adopted nstead of the determnstc mathematcal expressons utlzed prevously, specfcally I attempt to extend ths type of analyss by provdng parameter estmaton from a Bayesan modellng perspectve, n order to come up wth the most realstc estmatons of the parameters of nterest. The Bayesan approach was chosen for certan desrable propertes, such as the flexblty to ft varous models of hgh complexty or the ablty to nfluence the parameters of the ftted models by usng pror nformaton. Especally, the latter consttutes a desrable feature for the mathematcal models examned here snce the parameters are not completely unknown but restrcted to certan ntervals. The mathematcal models of Table 1 as orgnally proposed or presented by Ye (2009) generally mply a lnear relatonshp between the functons of C and/or P wth the journalh-ndex. For nstance, Schubert and Glänzel (2007) emprcally tests the 2

3 lnear regresson model Eh cp C / P between h and the product P C/ P. to fnd a strong lnear correlaton In addton to ths assumpton, I also ft the models usng alternatve dstrbuton specfcatons for the mean h-ndex Eh, correspondng to sutable dstrbutons for count data, such as the Posson and the negatve bnomal (NB), whch s usually utlzed as an overdspersed alternatve of the Posson. In ths way we have the advantage of estmatng the model s parameters from each specfc dataset nstead of usng fxed predetermned values as n Ye (2009). Addtonally, I assess model ft by formal model selecton crtera. The results confrm merely those of Ye (2009), n the sense that the results confrm model selecton, however the estmated parameters do not seem to concde wth those reported prevously. Data The proposed methodology s llustrated utlzng two dfferent datasets. The journal h-ndces, total number of publcatons P and assocated total ctatons C receved for journals n the felds of ecology and forestry scences were collected from WoS (Collecton date: March, 2013 and November 2011 for ecology and forestry journals respectvely). All values collected refer to the entre tme wndows of each one of the journals ncluded n the WoS database. Specfcally, a total of 134 journals were selected from the feld of ecology category, whereas 54 journals are ncluded n the forestry category. The ISI database was chosen manly due to that the WoS s a database generally deemed as vald and error-free by the scentometrcs communty. Methods Models I ft the four theoretcal bblometrc models presented n Table 1 as Bayesan regresson-type models, whch are of the followng form: f h, H ~ h g P C where s the random varable of the (theoretcal) h-ndex followng dstrbuton f (.e. one of the Gaussan, Posson and NB), = h denotes the lnk functon of the mean h-ndex, say Eh,to each one of the functons of Table 1, and fnally wth g we denote each one of the four theoretcal functons. Then, for each dstrbuton we have: 3

4 Gaussan, 2 : h ( ): h log r1 q, : h Posson NB r q Wth 1,2,...,130 and 1,2,...,54 forestry feld, respectvely. q for the journals n the ecology and the The Bayesan approach n fttng the above models makes use of the avalable nformaton, whch ncludes pror nformaton n the form of pror dstrbutons assgned to the model s parameters. By combnng the data wth pror nformaton we obtan the posteror dstrbuton of model parameters utlzng Markov chan Monte Carlo (McMC) samplng (Gelman et al., 2003). Through ths approach the robustness of the models s ncreased n comparson to models where the parameters are fxed by obtanng posteror dstrbutons and credble ntervals for the parameters of nterest. Inference I perform Bayesan nference usng an McMC samplng scheme. Weaklynformatve prors (.e. a truncated Gaussan wth zero mean and very large varance) sutably constraned to non-negatve values n accordance to the range of ther values as shown n Table 1 are assgned to the models parameters, except parameter of the Hrsch model constraned to the nterval (3,5). The analyss was conducted usng the WnBUGS statstcal software (Lunn et al., 2000) for model fttng. Model selecton was carred through the use of the posteror mean devance D (see Spegelhalter et al., 2002). Models wth smaller D value are better supported by the data.all results of posteror dstrbutons for the models parameters have been obtaned by usng 5,000 teratons as ntal burn-n perod and an addtonal sample of 50,000 teratons. Results and Dscusson Goodness of ft statstcs for the varous models are gven n the followng Table (Table 2). As concerns ther ft, we observe that the Glänzel-Schubert model under a Gaussan dstrbuton provded the best ft to the two sets of data, as ndcated by the values of the ft statstcs ( D = and 302.2for the ecology and forestry journals, respectvely). Worst ft was observed for all theoretcal models under a Posson specfcaton. It s evdent that the large varablty n the data of ths type consttutes the assumpton of a smple model based on the Posson dstrbuton for the h-ndex very unrelable. Ths s due to the fact that the Posson dstrbuton has one free parameter and does not allow for the varance to be adjusted ndependently of the mean. The alternatve for dealng wth overdsperson to the data, negatve bnomal model seems to perform better, especally for the two-parameter Hrsch model, where the ft outperforms those of all the rest of the models, except the Glänzel-Schubert Gaussan model. The NB specfcaton has been shown to be more sutable also for the Egghe-Rousseau model ( D =1210 and 398.7). 4

5 Model Dstrbuton D (ecology) D (forestry) Egghe-Rousseau Gaussan Hrsch Gaussan Glänzel-Schubert Gaussan Hrsch Gaussan Egghe-Rousseau Posson Hrsch Posson Glänzel-Schubert Posson Hrsch Posson Egghe-Rousseau NB Hrsch NB Glänzel-Schubert NB Hrsch NB Table 2. Mean devance ( D ) for the ftted models (wth bold ndcatng the three best model fts to the data). As concerns the theoretcal models, we see that the specfcaton of Ye (2011) based upon the two parameters for the Hrsch model, at least for the Gaussan and NB dstrbutonal specfcatons, presents tself as a useful alternatve to the Glänzel- Schubert model, although based solely on the number of ctatons C. Fgures 1 & 2 present a vsual nspecton of the ft of the Glänzel-Schubert Gaussan and the twoparameter NB Hrsch models. Although both models exhbt good ft, the graphs are also revealng of the superorty of the Glänzel-Schubert Gaussan model, especally n case of predctons for the hgher h-ndex values. Fgure 1.Observed vs predcted journal h-ndces for the two-parameter NB Hrsch model (left) and the Glänzel-Schubert Gaussan model (rght) [ecology feld]. 5

6 Fgure 2. Observed vs predcted journal h-ndces for the two-parameter NB Hrsch model (left) and the Glänzel-Schubert Gaussan model (rght) [forestry feld]. Posteror estmates results (.e. posteror medans along wth the correspondng 95% credble ntervals) of parameters of nterest obtaned from the Bayesan models are summarzed n Table A1 n the Appendx. My proposal offers credble ntervals for the varous parameters of the theoretcal models, and compares these ntervals wth the already proposed values. For example, we see that my estmatons for the parameter c n the (Gaussan)Glänzel Schubert model are around 0.7, concdng thus wth the emprcal estmatons reported n Schubert and Glänzel (2007) nstead of the fxed value of 0.9 adopted by Ye (2009). There are also substantal varatons between the parameter estmates for the two felds of research, ndcatng thus that the model parameter values are not unversal. In summary, I have shown through an emprcal statstcal analyss that alternatve formulatons based on the three standard mathematcal models for the h-ndex may result nmproved model ft. Specfcally, varatons related to dstrbutonal assumptons for the theoretcal models and to the parameters assocated wth these models n certan nstances resulted n better predctons. The latter were demonstrated usng journal ctaton data from the felds of ecology and forestry. Bblography Burrell, Q.L. (2013). Formulae for the h-ndex: A lack of robustness n Lotkaannformetrcs? Journal of the Amercan Socety for Informaton Scence and Technology, 64(7), Barendse, W. (2007). The Strke rate ndex: A new ndex for journal qualty based on journal sze and the h-ndex of ctatons. Bomedcal Dgtal Lbrares, 4:3. Braun, T., Glänzel, W. and Schubert, A. (2005). A Hrsch-type ndex for journals. The Scentst, 19, 8. Braun, T., Glänzel, W. and Schubert, A. (2006). A Hrsch-type ndex for journals. Scentometrcs, 69(1), Egghe, L. and Rousseau, R. (2006). An nformetrc model for the Hrsch-ndex. Scentometrcs, 69(1),

7 Franceschn, F. and Masano, D. (2011). On the analogy between the evoluton of thermodynamc and bblometrc systems: a breakthrough or just a bubble? Scentometrcs, 89, Gelman, A., Meng, X.L., Stern, H.S. and Rubn, D.B. (2003). Bayesan Data Analyss, (2nd ed), Chapman and Hall, London. Hrsch, J.E. (2005).A n ndex to quantfy an ndvdual s scentfc research output. Proceedngs of the Natonal Academy of Scences, USA, 102(46), Jokc, M. (2009). H-ndex as a new scentometrc ndcator. BochemaMedca, 19, 5 9. Lunn, D.J., Thomas, A., Best, N. and Spegelhalter, D. (2000). WnBUGS - A Bayesan modellng framework: Concepts, structure, and extensblty. Statstcs and Computng, 10, Malesos, C. and Abas, Z. (2012). An Examnaton of the Impact of Anmal and Dary scence Journals Based on Tradtonal and Newly Developed Bblometrc Indces. Journal of Anmal Scence, Malesos, C. and Arabatzs, G. (2012). An evaluaton of forestry journals usng bblometrc ndces. Annals of Forest Research, 55(2), Molnar, J-F. and Molnar, A. (2008). A new methodology for rankng scentfc nsttutons. Scentometrcs, 75(1), Rousseau, R. (2007). A case study: Evoluton of JASIS Hrsch ndex. Scence Focus. Shubert, A. and Glänzel, W. (2007). A systematc analyss of Hrsch-type ndces for journals. Journal of Informetrcs, 1(2), Spegelhalter, D.J., Best, N.G., Carln, B.P. and van der Lnde. A. (2002). Bayesan measures of model complexty and ft. Journal of the Royal Statstcal Socety, Seres B, 64, Ye, F.Y. (2009). An nvestgaton on mathematcal models of the h-ndex. Scentometrcs, 81(2), Ye, F.Y. (2011). A unfcaton of three models for the h-ndex. Journal of the Amercan Socety for Informaton Scence and Technology, 62(1),

8 APPENDIX Model Dstrbuton parameters (ecology) (forestry) a b c a b c Egghe-Rousseau Gaussan ( ) ( ) Hrsch Gaussan Hrsch Gaussan ( ) ( ) ( ) ( ) (0.83-4) ( ) ( ) Glänzel-Schubert Gaussan ( ) ( ) ( ) Egghe-Rousseau Posson ( ) ( ) Hrsch Posson ( ) ( ) Posson Hrsch ( ) ( ) ( ) ( ) Glänzel-Schubert Posson ( ) ( ) ( ) ( ) Egghe-Rousseau NB ( ) ( ) Hrsch NB ( ) ( ) NB Hrsch (1-1.17) ( ) ( ) ( ) Glänzel-Schubert NB ( ) ( ) ( ) ( ) Table A1. Posteror parameter estmates (medans) along wth the 95% credble ntervals of the theoretcal models for the h-ndex. 8