A noninformetric analysis of the relationship between citation age and journal productivity

A Noninformetric Analysis of the Relationship betweenCitation Age and Journal Productivity

L. EggheLUC, Universitaire Campus, B-3590 Diepenbeek, Belgium, and UIA, Universiteitsplein 1, B-2610 Wilrijk,Belgium. E-mail: [email protected]

A problem, raised by Wallace (JASIS, 37, 136–145, 1986),on the relation between the journal’s median citation ageand its number of articles is studied. Leaving open theproblem as such, we give a statistical explanation of thisrelationship, when replacing “median” by “mean” in Wal-lace’s problem. The cloud of points, found by Wallace, isexplained in this sense that the points are scattered overthe area in first quadrant, limited by a curve of the form

y5Ex2 1 1,

where E is a constant. This curve is obtained by usingthe Central Limit Theorem in statistics and, hence, hasno intrinsic informetric foundation. The article closeswith some reflections on explanations of regularities ininformetrics, based on statistical, probabilistic or infor-metric results, or on a combination thereof.

Introduction

Detecting and explaining regularities (functions, laws,distributions, formulae, etc.) are the most fundamental sub-jects of scientific research. Based on “raw” data one usuallyconstructs graphs of “clouds of points” show a certainregularity. If this regularity resembles a graph or a curve,one often tries to fit such a curve to this cloud of points via,for example, (nonlinear) regression. Many authors then stoptheir investigations assuming they have given an “explana-tion” of the observed facts. This is only partially true, or, Iwould prefer to say, this is partially wrong. The classical“joke” is fitting the first 10 or 20 points on the graph fory5 =x for x[N, the set of positive entire numbers. It is aneasy exercise to show that a linear fit works well with ahighly significant value of the correlation coefficient ofPearson. Authors stopping here would then pretend to haveshown thaty 5 =x (x 5 1, 2, . . ., 20, say) is a linearrelationship!

Is it wrong to apply such fitting techniques ? Not at all!They provide first basic information on the shape of a cloudof points, although several different fittings (with differentregularities) are possible (see the above example!). This is,however, only the first part in the scientific work. The nextpart is to start a kind of rationale, leading to, preferably, oneof the previously obtained regularities (functions, distribu-tions). A rationale is only possible if one knows where todepart from, i.e., what are the laws (regularities or othertools) that are (or can be) presupposed. In other words,given previously established results (or, at least, given someacceptable axioms), try to deduce (by reasoning) the (orsome) observed regularities. This is the same as in puremathematics, where one deduces from axioms (the notion of“acceptable” is not even existing here) results, which, thenagain, can be used to deduce further results such as regu-larities.

A very simple example of this methodology is given inEgghe and Rao (1992a), where one tries to explain agingcurves: the basic aging curve, the exponential decay, isgeneralized so that the initial fast increase of use (and thenfollowed by a slow decrease as in the case of an exponentialdecay) is also explained. In this reference one uses thelognormal distribution, which is fully explained in proba-bility theory (the argument is even repeated in Egghe &Rao, 1992a). In a similar way, growth can be studied (seeEgghe & Rao, 1992b).

Another “basic” example is the explanation of the clas-sical informetric laws as, for example, the ones of Lotka(1926), Bradford (1934), Zipf (1949), Mandelbrot (1954,1977), and so on. For some of these explanations we referthe reader to Egghe and Rousseau (1990), where severalexplanations, given by different authors, are reproduced(explanations given by Bookstein, 1977, 1990a, 1990b; DeSolla Price, 1976; Mandelbrot, 1954, 1977). These infor-metric laws all deal with the relationship between sourcesand items (items being produced by sources). Classicalexamples are: articles “produced” by journals or authors,citations “produced” by articles (here citations can be givenor can be received), and so on. In linguistics, one can talkabout words (type) and their use in texts (token). So these

Received July 6, 2000; revised September 8, 2000; accepted September8, 2000

© 2001 John Wiley & Sons, Inc. ● Published online 1 February 2001

JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY, 52(5):371–377, 2001

laws are also called type-token relations (see, e.g., Herdan,1960). In informetrics, they are (usually) called source-itemrelations, but essentially they have the same meaning orinterpretation.

A further step can be to use these above results andcombine them in a way as to explain further regularities.Examples (and reviews) of this are given in the last section.

But before we do this, the next section is devoted to anapparent informetric problem raised by Wallace (1986). Wal-lace writes literally (p. 137): “For a given subject literature, themedian citation ages of the journals contributing to that liter-ature will vary inversely with the productivity of thoses jour-nals, where productivity is measured in terms of the number ofarticles contributed by each journal.” If this is true, it would bea remarkable informetric result, useable in many other appli-cations. In any case, we are in need of an explanation. Thefound regularity is shown in the graph of Figure 1 (reproducedby permission of John Wiley & Sons, Inc.).

A first glance indeed indicates a decreasing relationship,although it is also immediately clear that the cloud of pointsis filling up an area in the first quadrant delimited by adecreasing function. In other words, the cloud of pointssatisfiesy # f(x) (with x,y $ 0) rather thany 5 f(x), wheref is this decreasing function. This simple remark showsalready that (as is clear by visual inspection and as is alsoremarked in Wallace, 1986) that the smaller median citationages occur for highly as well as less productive journals.The hypothesis as such is hence already rejected but stillone wants to know why we have a relation of the formy# f(x), and we also want to know the form off. In Wallace(1986) an indication is given forf(x) 5 c/x (c 5 a constant)but any

f~x! 5c

xa ~a . 0!

FIG. 1. Journal productivity plotted against journal median citation age.

372 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY—March 2001

or another decreasing function is possible. Because we(more or less) rejected the hypothesis, we had the impres-sion that an informetric explanation ofy # f(x) was notpossible.

In the next section we indeed show that an explanationcan be given by only using the Central Limit Theorem inprobability theory, hence showing that the regularity inFigure 1 probably has no informetric basis. This is nocriticism on the work of Wallace (1986); on the contrary, asmany regularities as possible must be detected, but thentheir explanations need to be classified into informetric ornot (i.e., statistical, probabilistic, informetric or a combina-tion thereof). This will be done at the end of the article,where we examine a few explanations that exist in theliterature.

The Relation between Journal Median (Mean)Citation Age and the Number of Articles ina Journal

From an informetric point of view it is hard to believethat Wallace’s hypothesis is true. Although it is true that inany discipline the median (mean) citation ages might differfrom journal to journal, it would be a strange fact that theseages are related (let alone inversely related) with journals’productivity (as measured in terms of articles per journal).Only in special cases this might be true, for example, wherethe top journals in the field are the largest ones and wherethe articles are used so heavily so that they have the smallestmedian (mean) citation ages (the median (mean), measuredover all articles in the journal). An opposite example canalso be given, and this in any discipline: the case of reviewjournals. They are usually large (in terms of number ofarticles), but their median and mean citation ages are largetoo, usually.

To start a rationale on this matter we will limit ourselvesto the case of mean ciation ages and we will fix (as inWallace, 1986) the subject, represented by a set of journalsconsisting of articles, each with their (diachronous or syn-chronous) citation age distribution. Considering all meancitation ages of all these articles, in any journal in this set,we can calculate their average: the mean citation age of thefield. Here we have considered the field as consisting ofarticles: this approach is called the global one, and is dif-ferent from the approach where the field is consisting ofjournals and where we take the averages over the averagescalculated per journal (cf. Egghe and Rousseau, 1996a,1996b). We note that even a third approach is possiblewhere we consider the field consisting of citations (the“smallest” unit), but we will not go into this because we donot need it here.

Let us now consider the journals in this set, consisting ofA articles (A[N, fixed). Let us call this theA-subfield of theentire field. ThisA-subfield again can be considered asconsisting of articles (as we did above with the entire field).As above, this yields the mean citation amA of the A-subfield. The hypothesis of Wallace, rephrased in this ter-

minology, is thatmA is a decreasing function ofA. To showthat this hypothesis is not necessarily linked with Figure 1,and that the regularity in Figure 1 can be explained in anoninformetric way, we will assume thatmA is a constant(saym) of A, i.e., thatmA does not depend onA. We assumethis for the sake of simplicity but also (and more impor-tantly) because explaining Figure 1 with this assumption isthe most “spectacular” explanation because, in our argu-ments, we deny Wallace’s hypothesis, but we will never-theless be able to explain the regularity of Figure 1. In thesame way we assume that the variancess2

A are alsoA-independent:s2

A 5 s2, for all A [ N.For eachA [ N fixed and for each journal withA articles

(hence belonging to theA-subfield), we have that this jour-nal’s mean citation age is a number which is the mean of asample ofA articles. The Central Limit Theorem (CLT) thenyields that this mean belongs to a 100(12 a)% confidenceinterval (aroundm) of the form

m 6 Z~a!s

ÎA 2 1, (1)

where Z(a) is the abscis such that, on the graph of theGaussian distribution (i.e., the standard normal distribution)the tails, determined byZ(a) and2Z(a) have a total area ofa. More concretely, for example, fora 5 0.05, a 95%confidence interval is given by

m 6 1.96s

ÎA 2 1. (2)

In general, the values ofZ(a) can be read from the tableof the standard normal distribution, which is available inany book on statistics or probability theory.

Expression (1) contains the key to the explanation of thegraph in Figure 1. Indeed, for each fixedA[N (i.e., ordinatein Fig. 1) we have that the “sample” journals withA articleshave a mean citation age between the values given by (1),for 100(12 a)% sure. The lowerA, the larger this confi-dence interval. From (1), it follows that the deviation fromm to the right is equal to

m 5 Z~a!s

ÎA 2 1(3)

where m is the abscissa in Figure 1. BecauseA is theordinate, we will invert (3), yielding

A 5 SZ~a!s

m D 2

1 1

or, more simply

A 5Ea

m2 1 1, (4)

JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY—March 2001 373

where Ea is a constant, decreasing witha. Equation (4) isthe decreasing graph at the right side in Figure 1. The“fading away” effect, observed in Figure 1, when goingfrom low values ofm andA to high values ofm andA isgiven by the different values ofa and corresponding prob-abilities (12 a). Of course, the left part of (1) is usually cutoff by the requirement thatm $ 0 andA $ 1.

Note also that, from (4), we have limm3`

A 5 1, limm30

A5 1`, all in agreement with the graph in Figure 1.

In short, Figure 1, but with “median” replaced by“mean”, is explained by the high variances of small samples(and conversely by the small variances of large samples).

Note: there is one “informetric” element in the graph ofFigure 1, although this is not determinant for its shape; thefact that the cloud of points is thicker for lowA than for highA, follows from the law of Lotka on the number of journalswith A articles, being proportional to 1/Ab, whereb $ 1(see, e.g., Egghe & Rousseau, 1990).

We think this gives a rationale for Figure 1 (where“median” is replaced by “mean”), and sheds light on thisregularity as being an effect of statistics and probabilityrather than being the consequence of an informetric law orregularity. We leave open to explain the “median” case, butare convinced that the given explanation more or less showsthat also the graph of the relation between number ofarticles and median citation age is noninformetric in natureas well.

In the next section we will review a few other regularitiesthat have been explained so far.

Explanations in Informetrics

The Arcs at the End of a Leimkuhler Curve

One of the simplest regularities ever found in informet-rics, but which is not an informetric regularity at all, is thefact that, at the end of a Leimkuhler curve, one detects“arcs.” One obtains a Leimkuhler curve when graphing thecumulative numberR(r) of items in the first (largest)rsources, versus logr. The graph looks as in Figure 2, andcan be found, for example, in Warren and Newill (1967),

Brookes (1973), Praunlich and Kroll (1978), Wilkinson(1973), Summers (1983).

Although the graph (without the arcs) has an equation ofthe form

R~r ! 5 a log~1 1 br! (5)

which is certainly an informetric regularity (see Egghe,1989,1990; Egghe & Rousseau, 1990), the arcs, apparentlydeviating from (5), are not informetric of nature. Indeed,there are frequently several high-ranking sources that pro-vide the same number of items: there might be a largenumber of sources with three items, a larger number withtwo items, and an even larger number of sources with onlyone item each. Because increases ofR(r) at these ranks arelinear in r (per group of equal productivity), the graph ofRversus logr is exponential (per group of equal productivity).These exponential graphs become more visible as thegroups of sources with equal productivity become longer.This explains these arcs near the end of a Leimkuhler curve.

Hence, this phenomenon is a purely mathematical con-sequence and has nothing to do with informetric aspectssuch as (5) or the so-called Groos droop (see Groos, 1967).

This first example was a case of mathematical explana-tion. The graph of Wallace (Fig. 1) was explained with astatistical-probabilistic argument. The next example yields apurely probabilistic explanation of another regularity.

The Relation between the Fraction of MultinationalPublications and the Fractional Score of a Country

In Nederhof and Moed (1993), a slightly different prob-lem as in its title is studied. The relation between thefraction of multinational publications of a countryc (i.e., thefraction of publications in which, besides an author ofcountryc also at least one other author appears, belongingto another countryc9 Þ c) and the country’s fractionatedscore. In this scoring scheme a country receives a score 1/bin a paper ifb is the total number ofdifferentcountries inthis article and ifb Þ1. If b 5 1, the fractionated score is 0.Only in this last element, the fractionated score is differentfrom the fractional score; here, the score is 1 ifb 5 1.

The regularity found in Nederhof and Moed (1993) isseen in Fig. 3 (reprinted with permission from the publish-er). The concave decrease is clear, and is explained inEgghe (1999), but with “fractionated” replaced by “frac-

FIG. 3. The relation between the fraction of multinational publications ofa country and its fractionation degree.

FIG. 2. A Leimkuhler curve, with arcs for larger.


tional” (leaving open the other problem). Using the frac-tional score of a countrryc, it is possible to determine thefraction of multinational publications ofc, using elementarytechniques of probability theory (such as independence andconditionality). The obtained relation between the fractionof multinational publications ofc and its fractional score isa certain average of functions of the type

y 5 1 2xa

1 2 ~1 2 x!a (6)

where a [ N and x [ [0,1]. All of these functions areconcavely decreasing.

The next example deals with the relation between thePrice Index and the mean or median reference age. This lastvariable is, essentially, the same as the citation age (i.e., thesynchronous one) as studied in the previous section. Also inthis case, no clear graph but a cloud of points is found (seebelow). But, interestingly, here an informetric-probabilisticexplanation is needed, contrary to the case in the previoussection.

The Price Index and Its Relation between the Mean andMedian Reference Age

In a fixed literature set, the Price Index PId is the pro-portion of the references that are to the lastd years ofliterature. De Solla Price (1970) usesd 5 5, Glanzel andSchoepflin (1995) used 5 2. In this last article, the relationbetween PI2 and the mean reference age is studied. A graphas in Figure 4 is obtained [reprinted with permission fromInformation Today, Inc. (former name: Learned Informa-tion) 143 Old Marlton Pike Medford, NJ 08055].

At first glance, it resembles a bit the graph in Figure 1. Inboth cases we have a cloud of points (rather than a clear

graph of a function, as, e.g., previously), and in both casesthe overall impression is that the graph is convexly decreas-ing. Here, however, not the entire first quadrant, below acertain decreasing function, is filled up as is the case inFigure 1.

We can report here that both regularities are completelydifferent in nature, and that they have completely differentexplanations. The explanation given for the Price Indexversus the mean (and also for the median) reference age isgiven in Egghe (1997), and is based on the relation

PId 5 E0

d

c~t! dt, (7)

wherec is the age distribution of the references. Using forc(t) the exponentially decreasing distribution

c~t! 5 d z at, (8)

whered anda are constants and 0, a # 1, (7) yields anexplanation for the convexly decreasing aspects but not forthe cloud of points, with the apparent thickness in themiddle part. The explanation for this is obtained using themore realistic lognormal distribution

c~t! 51

tÎ2pse2~1/2!~~lnt2m!/s!2

(9)

where lnt denotes the Neperian logarithm andm ands areconstants denoting the mean and standard deviation of lnt.This distribution itself is used and explained, for example,in Egghe and Rao (1992a), and is now generally accepted asage distribution of references (or citations). The fact that it

FIG. 4. Plot of the Price Index (d 5 2) versus mean reference age.


has one parameter more than (8) yields the explanation ofthe cloud of points (see Egghe, 1997).

We, hence, have here an informetric-probabilistic expla-nation, and hence, this regularity has more informetric valuethan the ones described above in the sense that the latterbelongs more to the “informetric theory” than the onesabove. Yet, in none of the above arguments, the classicallaws of informetrics (Lotka, Bradford, etc.) are used toexplain regularities. This will be the case in the next expla-nation.

An Explanation of the First-Citation Distribution

In a fixed literature set one can look, for each article, atthe time t1 at which this article receives its first citation.Here, t1 is interpreted as “time after publication of thearticle.” Over the whole literature set one can then wonderwhat is the underlying first-citation distribution. Based ondata collected in Gupta and Rousseau (1999), Motylev(1981), and Rousseau (1994), we know already that thecumulative first-citation distribution can be of two types—concavely increasing, or S-shaped, i.e. convexly increasingfollowed by a concave increase.

Rousseau (1994) tries to explain these regularities bydetermining two differential equations, which yield twocumulative distributions, fitting each type of curve verywell. As noted in Egghe (2000), the drawbacks of such amethodology (although it is better than only statistical fit-ting) are (a) two different rationales are needed to “explain”both models; (b) such types of “explanation” start fromunexplained dynamics (through the simple formulation ofthe differential equations); (c) they do not involve anypreviously established informetric results.

In Egghe (2000), we proposed a model in which we usethe simple decreasing exponential distribution (8) for theage distribution of the citations in combination with Lotka’slaw

w~A! 5D

Ab , (10)

whereA is the number of citations per article, andw(A) isthe fraction of articles withA citations.

This combination yields for the cumulative first-citationdistributionF(t1) the following formula:

F~t1! 5 g~1 2 at1!b21, (11)

whereg is a constant. In Egghe (2000), one can see that (11)is capable of fittingboth types of first-citation distributions,and that the fits are very good. Moreover, the case 1, b# 2 takes care of the concave case, andb . 2 takes care ofthe S-shaped case. This link between shapes of first-citationdistributions and Lotka’s exponentb was new, and gives anintrinsic informetric explanation. In this sense we considerthe discovery of (11), among the other explanations re-

viewed in this article, the one with the highest informetricvalue.

Concluding Remark

We note the importance of detecting regularities ingraphs of (clouds of) points. We also underline the impor-tance of giving a rationale for these regularities and further-more to determine if the rationale is informetric in nature ornot. If so, the link with previously known informetric dis-tributions must be established which gives then, in turn, adeeper explanation of the observed phenomena.

We want to stimulate the reader to find other regularitiesin the existing literature or (which is of course more diffi-cult) to find new ones. Then we pose the open problem ofexplaining them (at least partially) in the sense describedabove.

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A noninformetric analysis of the relationship between citation age and journal productivity