Page 222 - Becoming Metric Wise

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Indicators

Definition: Consider a researcher with h-index h. Let n be the smallest
possible number of citations necessary to reach an h-index equal to h 1 1,
then the rational h-index, denoted h rat , is defined as:
n
h rat 5 h 1 1 2 (7.8)
2h 1 1
We next explain formula (7.8). If a researcher has h-index h,then
one may ask about the minimum number of citations necessary to reach
an h-index equal to h 1 1. This number is denoted here as n.If, for
instance, a scientist has the following series of citations [7, 4, 3, 1, 0]
and hence has an h-index of 3, then she needs at least 4 more citations
in order to reach an h-index of 4. Indeed the third article needs one
more citation and the fourth needs 3 more. Of course, there is no guar-
antee that her next four citations are exactly for these two articles and
with exactly these numbers of citations. If the next 4 citations go to the
article ranked first then her h-index stays the same and if the “right”
articles receive these four citations but equally (each two), then she is
still one citation short. That is why we use the term “minimum.” The
next question is: If you only know that this scientist’s h-index is 3, and
no further details, what is then the largest possible value of this
minimum? The answer is 7 which happens when the scientist has the
following series of citations [3,3,3,0] (or with more uncited publica-
tions). In general, this largest minimum is 2h 1 1for ascientist with
h-index h, corresponding with the “worst case scenario” that there are
h publications with h citations each and the publication at rank h 1 1
has 0 citations. This explains the occurrence of the factor 2h 1 1inthe
formula for the rational h-index.

Some examples: for [7,4,3,1,0] the rational h-index is 4 4/7 5
24/7 3.43.
For [4,4,3,1,1,0] it is also 4 4/7 3.43; while for [6,5,4,3,2,1,0] it is
4 1/7 3.86. Finally for [10,3,3,0,0] it is 4 6/7 3.14 and for [10, 2, 2, 2]
(same number of citations and higher average number of citations as the
previous case) it is 3 2/55 2.6.

7.4.2 The Interpolated h-index
The interpolated h-index, denoted as h int , was introduced by Rousseau
(2006d) under the name of real-valued h-index. This variant of the
h-index can be used in similar situations as the rational h-index, but