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212 Becoming Metric-Wise
7.3.3 Independence
The most significant disadvantage of the h-type index is that it is not
independent in the sense of Bouyssou and Marchant (2011). If A and B
represent sets of publications and citations, then strict independence for
an indicator F means that if F(A) , F(B) and one adds to A and to B the
same publications (with the same number of received citations) then still
F(A) , F(B). This does not hold for the h-index. For instance:
A 5 4; 4; 4; 4; 0; then h AðÞ 5 4
½
B 5 5; 5; 5; 2; 1; then h BðÞ 5 3 and h BðÞ , hAðÞ
½
Adding two publications with 5 citations (one may even assume that
A and B were coauthors on these two publications) gives:
0 0
½
A A with added publicationsð Þ 5 5; 5; 4; 4; 4; 4; 0; then h AðÞ 5 4
0 0
B B with added publicationsð Þ 5 5; 5; 5; 5; 5; 2; 1; then h BðÞ 5 5
½
and consequently h AðÞ , hðBÞ:
This example is based on a similar one presented in (Waltman & van
Eck, 2012a).
7.4 SIMPLE VARIATIONS ON THE H-INDEX
Besides using different publication and citation windows one may restrict
publications to those where the scientist is either first author or correspond-
ing author (Hu et al., 2010), or take only the first author into account.
Note that we do not support the use of these two methods as they might
not give credit to some collaborators. The first of these methods might,
however, be used in addition to the standard h-index. Another simple vari-
ation is using complete-normalized fractional counting for citations (if an
article is written with three authors, then each of them receives only one
third of the citation credit), see Subsection 5.6.3.
The original h-index is a natural number, leading to many scientists
with the same value for their h-index. Two proposals have been made to
refine this.
7.4.1 The Rational h-index
The rational variant of the h-index, denoted as h rat , was introduced by
Ruane and Tol (2008) in the context of publications and citations (whole
counting). It is defined as follows.
