Principled and constructive approaches and h-type indices

Gangan Prathap

CSIR National Institute for Interdisciplinary Science and Technology,

Thiruvananthapuram 695019, India; gp@niist.res.in

Abstract

We examine two classes of h-type indices; those that emerge

from a heuristic algorithm or construction and those that are

deduced from the citation sequence by ordered mathematical

operations using first principles. We shall call these

constructive approaches and principled approaches. These

indices are then compared using some real life examples. One

relates to papers on the topic of h-index itself; the other is

a bibliometric analyses of studies on the monsoon cycle over

South Asia and the forces that drive this annual cycle and its

variability from year to year and within the season. The

bibliometric analyses breaks down scholarly performance into

three primary components - quantity, quality and consistency

using citation data is retrieved from the Web of Science and

two h-type indices are derived from these primary components.

Keywords Bibliometrics ∙ Indicators ∙ Quality ∙ Quantity ∙

Consistency ∙ Constructed indices ∙ Principled indices

1

Introduction

Einstein (1954) had an interesting view on the nature of

scientific theory. Most scientific theories model phenomena

based on facts; these belong to a class he called the

constructive type originating from empirically observed

principles ordered into meaningful patterns using imaginary

or speculative conjectures and hypotheses (Chatterjee 2012).

These are accepted as accurate representations of the world of

experience, until falsified. However, constructive theories

lack unifying principles to explain why the world is what it

is (Gefter 2005). On the other hand, principled theories start

with first principles that underlie the theory and then work

down to deduce the facts; they are deductive (Gefter 2013).

"The supreme task of the physicist is to arrive at those

universal elementary laws from which the cosmos can be built

up by pure deduction," Einstein said (Gefter 2013). This

classification is very useful in building up a framework for

understanding the major problems in physics. In extending this

logic to rationalize how approaches to derive h-type indices

are founded, we must use this classification in a slightly

different way from what Einstein had intended. That is the

purpose of the present paper.

2

Informetric indicators and indices start with a basic sequence

called the citation sequence or distribution. If an actor

(author, group, institution or country, etc.) has P

publications and the k-th publication in this set has ck

citations, then ck, k = 1 to P is the citation

sequence. The earliest bibliometric indicators used for

research evaluation where naturally:

P, the number of papers in the publication set, which is a

proxy for quantity or size,

C = Σck, k = 1 to P, the total number of citations

and

i = C/P, the impact, which is taken as a proxy for the quality

of the publication set.

We can think of P and i as primary bibliometric indicators

of the parameter space and consider the information production

process as being two-dimensional in nature. Note that in this

conceptual scheme, C is derived as a product of a quantity

term and a quality term. While both P and i have the

dimensions of P (indicated as order of P, or O(P)), we see

that C has the dimensions of O(P2). One can think of P

as being generated from the operation P = Σ(ck)0, k = 1

to P while C is generated from the operation C = Σ(ck)1,

k = 1 to P. In this sense, P is a zeroth-order

indicator and C is a first-order indicator. To

simplify, if P is thought of as a zeroth-order measure of

scholarly performance, being a proxy for the quantity of

3

scholarly output, C = iP is the first-order measure of

scholarly performance. Thus impact i emerges at the

first-order level. Table 1 summarizes this. So far, all

these indicators are derived by basic mathematical operations

in a deductive way and can be called a "principled" approach.

The Hirsch index (or h-index) was introduced as a single

number informetric measure by using a heuristic construction

or algorithm on the citation sequence (Hirsch 2005). It was

intended to combine the quantity and quality dimensions into

a single number measure of performance. It is in this sense,

that it is a constructive approach as it does not start from

basic principles. Since then, more than a thousand articles

have used this metric in bibliometric, scientometric and

informetric research and also routine research assessment

exercises and several dozen copy-cat h-type constructions

have followed (Bornmann et al. 2011). However, only two indices

have been generated by purely deductive procedures, namely the

p- and z-indices. In the next section, we shall review these

three indices before we use them in a real life example.

The h-index and other h-type indices as constructed indices.

Nature's editorial on the h-index (Ball, 2005) very quickly

brought international attention to Hirsch's (2005) proposal

that a single number could serve as an index for measuring

research achievement. Just as the h-index sums up the life-

4

time publication achievement or performance of an individual

scientist, nearly half-a-century earlier, the great

astrophysicist Sir Arthur Stanley Eddington had introduced a

single number entity called the cycling number n. Eddington

was an avid cycling enthusiast and he devised this criterion

for assessing lifetime cycling progress: the largest number n

such that one had cycled n or more miles on n different

days (Chandrasekhar, 1991). Edwards (2005) records that some

35 years before Hirsch's landmark paper, the geophysicist

Harold Jeffrey's was already recording his cycling prowess in

terms of the number n, having learned this idea from his

fellow cyclist Eddington. Note here the emphasis on words like

"prowess", "achievement" and "performance". Also on the use of

the qualifier "largest". This will be of significance later in

this discussion. Another aspect that is easy to ignore is the

key role the elementary units of measurement play in

determining the value of n. Eddington's algorithm is

effective because cycling distance is measured in miles and

the time windows are taken as days. Thus, Harold Jeffrey's n

which was 70 when he communicated this value to Edwards (2005)

made sense only in such units. If distances were measure in

centimetres or inches and time windows were taken in minutes,

or weeks, it may not have led to any meaningful number.

Hirsch’s (2005) definition relies on using papers and

citations as the units of measurement and life-time or

particularly chosen publication and citation windows as the

unit of time:

5

"I propose the index h, defined as the number of papers with

citation number ≥h, as a useful index to characterize the

scientific output of a researcher."

The calculation of the h-index proceeds thus: A scientist has

index h if h of her P papers during the chosen

publication window (say a life-time) have at least h

citations each during the chosen citations window, and the

other (P - h) papers have no more than h citations each. That

this value is the "largest" meeting this criterion is

implied. The h-index cannot be computed in a trivial manner if

the citation sequence is chronological or arranged in any

other fashion (e.g. in ascending order) and would need a

special algorithmic procedure. The simplest procedure to

compute h requires citations to be rearranged in descending

order according to rank. This need arises from the definition

of the h-index, as the highest number h of papers of a

scientist that have been cited h or more times. The

construction for h is facilitated by arranging citations in

descending order according to rank and displayed graphically

with citations on the y-axis and rank of papers on the x-axis.

That is, a paper at rank k that has ck citations is

displayed by a bar of unit width, and a height ck Schreiber

(2008, 2009). The h-index is then read off this sequence as

ck h c≥ ≥ k+1.

6

The h-index is a single, simple measure that combines papers,

a proxy of quantity and citations, actually a proxy of the

product of impact and quantity (Egghe, 2010). The advantages

and disadvantages have been neatly summarized by Egghe (2010).

Various generalizations and variants of the h-index, mostly

based on constructions, have emerged (Egghe, 2010). The h-

index is insensitive to the citations received in the h-core.

The most successful of these is perhaps the g-index which was

introduced by Egghe (2006a, 2006b, 2006c) to correct for this.

Most of these indices are highly correlated with each other

and in that sense, redundant.

The p- and z-indices as emerging from principled

approaches.

The p-index was originally introduced as a mock h-index

(Prathap 2010). The p-index was inspired by, and can be

thought of as a proxy of the h-index based entirely on

theoretical foundations which first appeared in Glanzel (2006)

and was empirically tested first in Schubert and Glanzel

(2007). This was a natural result of the attempts to relate h

to conventional scientometric measures like productivity

(total number of papers P), citation impact (total citations

C) and quality expressed as an impact (i = C/P). A composite

indicator appeared from this exercises, which was (C2/P)1/3. To

the physicist or engineer, this has the dimensions of h or P.

7

Interestingly, Glanzel had warned that ‘‘it is not intended

to substitute the h-index.’’ Prathap (2010) suggested that it

is profitable to disregard Glanzel’s well meaning caveat and

instead proposed that there may well be merit in treating this

as a substitute or mock h, say hm = (C2/P)1/3. Further studies

have indicated that the C2/P term is an second-order term which

is arguably the best measure of scholarly performance if only

two bibliometric dimensions are considered, namely quality and

quantity.

We have seen above that when we start with the citation

sequence ck, k = 1 to P, the impact i = C/P, appears

as an average or arithmetic mean. Since C is determined from

what we call a first-order process, i.e. C = Σ(ck)1, k =

1 to P, at first sight impact i is seen to emerge at

the first-order level through basic mathematical operations in

a deductive way in our "principled" approach. But why is this

the best measure for an average? For this to become obvious,

we have to take this up as a second-order process.

Given the citation sequence ck, k = 1 to P, let i be

that value that minimizes the error in the least-squares error

sense. Let S be the sum of the squares of the error as

S = Σ(ck - i)2

(1)

8

This is now a second-order measure. It is simple to show that

the "best" value of i is that which minimizes S and leads

to the equation

Σ(ck - i) = 0 (2)

or

i = Σck/Σ1 = C/P. (3)

In this interpretation, the impact i is a first-order measure

appearing through a second-order process. That is, i

appears as a deduction based on the principle that i

minimizes the sum of the squares of the errors. Using this

result, we can further show through a purely deductive

argument that

S = Σck2 - 2iΣ(ck - i) - i2 Σ1 (4)

Using the result from Eqn 2, we have,

S = Σck2 - i2 P (5)

The second-order process which has been invoked to rationalize

i now leads to two new second-order terms on the right-hand

side. This paradigm therefore leads to a trinity of second-

order terms (Prathap 2011a,b):

X = i2P

9

E = c∑ k2

S = (c∑ k – i)2 = E - X

We can then think of a hierarchy of indicators of performance

(Prathap 2011b):

Zeroth order indicator: P = i0P

First order indicator: C = i1P

Second order indicator: X = i2P = i1C.

Prathap (2011a,b) showed that the indicator X = i2P,

is a quantity which can be thought of as a second order

indicator of performance. The p-index is obtained by simply

scaling X down to the same dimensions as P or h to give

p = X1/3.

Ever since the h-index was proposed, it has become an

essential practice to rearrange the citation sequences in a

monotonically decreasing order so that by a simple algorithm

or inspection the value of the h-index could be determined.

The nature of the information production process is such that

very high skews are noticed. The highly cited articles are

found in a small core, implying a possible huge variation in

the quality of each paper in the publication set. Thus, two

different sets can have the same C, and one could have

achieved this with far fewer papers, with a higher quality of

overall performance, or with the same number of papers (i.e.,

same quality) but a higher degree of consistency or evenness.

10

Thus, C by itself, which is a first-order indicator may not

be the last word on the measurement of performance. The

product X = iC = i2P is a robust second-order performance

indicator (Prathap, 2011a,b) is arguably a better proxy for

performance. Apart from X, an additional indicator E also

appears as a second-order indicator as above . The coexistence

of X and E allows us to introduce a third attribute that

is neither quantity nor quality. In a bibliometric context,

the appellation “consistency” may be more meaningful. The

simple ratio of X to E can be viewed as the third

component of performance, namely, the consistency term η =

X/E. Perfect consistency (η =1, i.e., when X=E) is a case of

absolutely uniform performance; that is, all papers in the set

have the same number of citations, ck =c. The greater the skew,

the larger is the concentration of the best work in a very few

papers of extraordinary impact. The inverse of consistency

thus becomes a measure of concentration. This third dimension,

consistency η is a measure of the variability in the

quality of the individual papers in the publication set, or in

other words, the shape of the distribution curve.

Thus, for a complete 3D evaluation of publication activity, we

need P, i, and η. These are the three primary components of a

quantity–quality–consistency landscape. Using all three

components together, a z-index can be computed from the

composite indicator Z = ηX = η2E, as z = Z1/3, which has the same

dimensions as the number of publications, and therefore also

the h-index. This index combines quantity, quality, and

consistency (or efficiency) to give a 3D bibliometric

11

evaluation (Prathap 2013a,b,c). Thus, the h-, p- and z-

are secondary single number bibliometric indicators of

performance. This is summarized in Table 1. We can extend

Einstein's metaphor to imagine that the h-index is a

"constructed" index while the p- and z-indices are

"principled" indices.

The h-, p- and z-indices as defined in amplitude or

frequency spaces.

Table 2 is a useful exercise to show the mathematical unity

of the p- and z- indices in both amplitude space and frequency

space representation. Although this is straightforward with

regard to the p- and h-indices, it is not easy when the h-

index is concerned. Consider a publication set with ten

papers. Let the counter be k in the amplitude space and an

index I in the frequency space (Table 2). Assume that when

the papers are ordered chronologically (or any other

meaningful ordering system), the citations (amplitudes) are 6,

10, 4, 9, 6, 0, 2, 4, 0, 4 as shown in the second row of

Table 2. The indicators P, C, X and E are directly

computable from this sequence and from this i, η, and the

p- and z- indices can be easily computed. However, to

visualize the h-index, it is necessary to order the numbers in

descending order as shown in the third row of Table 2. The

lower half of Table 2 shows how typically a statistician will

approach the problem using the frequency domain. To simplify

12

the discussion, an index I is used for citations

(amplitudes) increasing from a value of zero. For each

citation value x, there are f(x) instances where such

citations are realized. There is no difficulty here to

determine the same indicators P, C, X and E from this

representation and from this to compute i, η, and the p-

and z- indices (Table 2 shows how an excel sheet calculation

is performed.) It seems almost impossible to obtain the h-

index from such an approach. The indices obtained from a

"principled" approach have a mathematical unity that is not

found for the h-index.

Real life exercises

Case 1 - Research on the topic, h-index.

The precise computation of η requires the knowledge of the

complete citation sequence (i.e. the distribution curve) for

each individual scientist (or aggregation like institute,

journal or country). This is obtained directly from the Web of

Science for each country, Web of Science category and journal

taken up in the present analysis and the methodology to obtain

this is discussed below.

Consider the scientific output in the area described by

Topic=(h-index) as indexed in the Web of Science (a Thomson-

Reuters product). We choose the period 1986-All Years

(updated 2014-01-08 ) for which subscription was available.

13

All articles P, and citations C gathered by these P

articles, are counted. Then the impact i is computed for this

period. From the citation sequence for each entity (author,

country, organization or journal), consistency η can be

computed using simple Excel spread sheet functions.

The search strategy

Topic=(h-index)Timespan=All years. Databases=SCI-EXPANDED, CPCI-S, CPCI-SSH, CCR-EXPANDED, IC.

using the Web of Science database (accessed on 10 January

2014) picked up 1012 records matching the query of the

32,942,465 in the data limits selected for the period 1987-

till date. The citation sequence is downloaded as an Excel

file and can be easily processed to extract the impact, and

the h-, p- and z-index. Table 3 displays the primary and

secondary bibliometric indicators and indices. We notice the

curious fact that the h- and p-indices are very close to each

other. We shall find that this is largely true where i < h

and i << P, leading one to conclude that in such instances,

the h-index is a mock of the p-index, or vice versa. However,

where there is extraordinary scholarly performance and i > h,

both p- and z-indices can be larger than the h-index.

The Web of Science database allows us to refine the results in

terms of countries, Web of Science categories, and journals

(source titles), etc. Our focus in this paper will be to see

14

how the constructed and principled indices perform relative to

each other for these three aspects.

We follow up with the address option to identify the share of

various countries in this area. A typical search is of the

type:

Topic=(h-index)Refined by: Countries/Territories=( USA )Timespan=All years. Databases=SCI-EXPANDED, CPCI-S, CPCI-SSH,CCR-EXPANDED, IC

Table 4 shows the primary and secondary bibliometric

indicators and indices for country-wise performance in the

area of h-index research. Also shown are the correlation

coefficients between the various indicators and indices.

Case 2 - Research in the area of monsoon studies

The monsoon system is a unique geophysical phenomenon that

determines the weather and climate over most parts of Asia,

mainly the Indian sub-continent, and over South-east Asia and

large parts of China. It influences the lives of nearly one-

third of the world population sirectly, and indirectly the

lives of nearly three-quarters of the world population.

The emergence of global satellite data and images and

computational models that integrate the global earth and

15

atmospheric systems has permitted the monsoon to be studied as

a very complex global phenomenon and research work is

performed in many countries outside South Asia and China

(Turner & Annamalai, 2012). The bibliometric evidence from the

Web of Science (accessed on 30 December 2013) shows that from

1987 to 2012 the number of records in database has increased

from 48 to 1833 when the search was restricted to

Topic=(monsoon). In all, there were 19603 results for the

period 1987-till date.

We will like to studied critically the relationship of the h-

index with the derived indices like the p-index and the z-

index. For this, we need to investigate large-scale systems

where there are a large number of papers and the impact is

considerably lower than these indices. For this, we shall

restrict attention to publications refined according to the

Web of Science categories. For example, there are 7700 papers

in the category "METEOROLOGY ATMOSPHERIC SCIENCES" and the impact is

a modest 18.05 and the h-, p- and z-indices are 135.88,

132 and 75.66 respectively (Table 5). Table 5 and Figure 1

shows the primary and secondary bibliometric indicators and

indices for twelve web of science categories. We find from the

correlation coefficients between the various indicators and

indices, and visually from the figure that the h- and p-

indices are very close to each other. In all the twelve

categories listed here, i < h and i << P. In such

instances, the h-index is a mock of the p-index, or vice versa.

16

Concluding remarks

We compare the h-index, which belongs to a class of

heuristically or algorithmically constructed indices with the

p- and z-indices which are systematically deduced from the

citation sequence by ordered mathematical operations using

first principles. Loosely following a terminology introduced

by Einstein, we shall call these "constructive" approaches and

"principled" approaches. For the "principled" approach, the

bibliometric analyses breaks down scholarly performance into

three primary components - quantity, quality and consistency.

The citation data is retrieved from the Web of Science and the

p- and z-indices are derived from these primary components.

Two real life examples are used to see the relationship

between these indices. One relates to papers on the topic of

h-index itself; the other is a bibliometric analyses of

studies on the monsoon cycle over South Asia and the forces

that drive this annual cycle and its variability from year to

year and within the season. In many instances, the h- and p-

indices are close to each other. This is largely true for

large datasets where i < h and i << P, leading one to

conclude that in such instances, the h-index is a mock of the

p-index, or vice versa. However, where there is extraordinary

scholarly performance and i > h, both p- and z-indices can be

larger than the h-index. The h-index therefore considerably

understates performance of high achievers.

17

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20

Table 1. Schedule for computation of the three primary

indicators, and three secondary indices from the citation

sequence ck with summation from 1 to P.

Order of

indicator or

index

Name Formula Dimensionality

Zeroth Papers P = Σ1 O(P)First impact i = Σck/Σ1 O(P)Second consistenc

y

η = (Σck)2/((Σ1)(Σck2)) O(1)

Second p-index p = ((Σck)2/(Σ1))1/3 O(P)Second z-index z = ((Σck)4/(Σ1)2/(Σck

2))1/3 O(P)unspecified h-index Unspecified;

determined by

construction

O(P)

21

Table 2. Amplitude space and frequency space computation of

the h-, p- and z-indices.

Amplitude space k   1 2 3 4 5 6 7 8 9 10 P= 10Chronological x   6

10 4 9 6 0 2 4 0 4 C= 45

Ordered x  10 9 6 6 4 4 4 2 0 0 E= 305

In amplitude space approach, h-index can be computed easily only when the amplitudes are ordered from highest value to lowest. The p- and z- indices can becomputed easily even when the amplitudes are ordered in any fashion.

i= 4.5

X=202.5

η=0.664

Frequency space I 0 1 2 3 4 5 6 7 8 9 10    Amplitude x 0 1 2 3 4 5 6 7 8 9 10    Frequency f(x) 2 0 1 0 3 0 2 0 0 1 1 Σf(x)= 10

  xf(x) 0 0 2 012 0

12 0 0 9 10 Σxf(x)= 45

 x2f(x) 0 0 4 0

48 0

72 0 0

81

100 Σx2f(x)= 305

In frequency space approach, it is not easy to visualize the idea of the h-index. The p- and z- indices can be computed easily even in this case.

i= 4.5

X=202.5

η=0.664

22

Table 3. The primary and secondary bibliometric indicators and

indices for research publications in the area of h-index.

  P i η p h zTOPIC = h-index 1012 10.28 0.15 47.47 45 25.17

23

Table 4. The primary and secondary bibliometric indicators and

indices for country-wise performance in the area of h-index

research.

Countries P i η p h zUSA 215 13.24 0.20 33.52 27 19.67PEOPLES R CHINA 110 6.98 0.12 17.50 15 8.67BELGIUM 76 17.79 0.12 28.87 16 14.22GERMANY 65 10.17 0.41 18.87 14 14.07ENGLAND 62 12.05 0.30 20.80 15 13.88ITALY 62 8.32 0.12 16.25 10 8.11SPAIN 60 9.15 0.32 17.13 14 11.71BRAZIL 47 7.28 0.09 13.55 8 6.04AUSTRALIA 45 13.16 0.26 19.82 12 12.62INDIA 43 6.51 0.08 12.22 6 5.33NETHERLANDS 40 15.08 0.16 20.87 11 11.37CANADA 33 8.00 0.30 12.83 10 8.62TAIWAN 19 4.74 0.28 7.53 4 4.94FRANCE 15 8.80 0.40 10.51 7 7.73HUNGARY 20 20.00 0.45 20.00 11 15.28CORREL P i η p h z

P 1.00 0.11 -0.31 0.75 0.90 0.59i 0.11 1.00 0.22 0.71 0.44 0.76

η -0.31 0.22 1.00 -0.13-

0.04 0.31p 0.75 0.71 -0.13 1.00 0.91 0.89h 0.90 0.44 -0.04 0.91 1.00 0.86z 0.59 0.76 0.31 0.89 0.86 1.00

24

Table 5. The primary and secondary bibliometric indicators and

indices for performance in various Web of Science categories

in the area of monsoon research.

WEB OF SCIENCE CATEGORIES P i η p h z

METEOROLOGY ATMOSPHERIC SCIENCES 7700 18.05 0.17135.8

8 132 75.66

GEOSCIENCES MULTIDISCIPLINARY 4681 16.45 0.23108.2

0 98 66.13ENVIRONMENTAL SCIENCES 2018 10.45 0.24 60.38 56 37.64OCEANOGRAPHY 1908 19.19 0.18 88.90 82 50.60GEOGRAPHY PHYSICAL 1803 21.28 0.24 93.45 81 57.89MULTIDISCIPLINARY SCIENCES 1191 26.54 0.09 94.31 87 42.08WATER RESOURCES 1118 8.10 0.23 41.85 40 25.71GEOCHEMISTRY GEOPHYSICS 876 16.29 0.15 61.48 60 32.41PALEONTOLOGY 650 23.44 0.38 70.95 61 51.37ECOLOGY 643 15.52 0.29 53.71 46 35.62MARINE FRESHWATER BIOLOGY 490 11.24 0.30 39.56 36 26.41REMOTE SENSING 485 5.47 0.08 24.40 26 10.36CORREL P i η p h z

P 1.00 0.16 -0.14 0.82 0.86 0.79i 0.16 1.00 0.09 0.67 0.62 0.64

η -0.14 0.09 1.00 -0.09-

0.16 0.19p 0.82 0.67 -0.09 1.00 0.99 0.95h 0.86 0.62 -0.16 0.99 1.00 0.93z 0.79 0.64 0.19 0.95 0.93 1.00

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Figure 1. Scatter plot showing how the p- and z-indices are

related to the h-index for the leading Web of Science

categories under which monsoon research can be classified.

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