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226   Becoming Metric-Wise


          number of documents in A belonging to class k. Then the percentile rank
          score of A, denoted as R(A), is defined as:
                                          K
                                         X     n A ðkÞ
                                 RðAÞ 5     x k                       (7.17)
                                                 N
                                         k51
          Proposition: The percentile rank score is a strict independent indicator
          for average performance.
                                                 X  K     n A ðkÞ
             Proof.    Suppose    that   RðAÞ 5        x k       . RðBÞ 5
                                                    k51     N
             K      n B ðkÞ
          X
                 x k     (for average performance the sets A and B must have
             k51     N
          an equal number of elements). We take a new element from class j and
          add it to A as well as to B,leading to sets A and B .Then
                                                           0
                                                                    0
                    N              1               N              1
              0
          RðA Þ 5       RðAÞ 1 x j     . RðB Þ 5       RðBÞ 1 x j     .This
                                             0
                  N 1 1          N 1 1           N 1 1          N 1 1
          proves that R is a strict independent indicator for average performance.
             We note that the independence property does not hold if sets A and B
          have a different number of elements. Indeed, let A consist of 2 elements, one
          with score 1 and one with score zero. Then its percentile rank score is /2.Let
                                                                     1
          B consist of 10 elements, one with score 3, one with score 2, one with score
          1 and 7withscore zero.Thenits percentile rank scoreis 6/10, which islarger
          than A’s. Now we add an element with score 1. This leads to a score of 2/3
                               0
               0
          for A and for 7/11 for B , reversing the order of the scores for A and B.
          Definition: The integrated impact indicator (I3) (Leydesdorff &
          Bornmann, 2011)
             Leydesdorff and Bornmann argued that impact means: many publica-
          tions and many citations. In this sense the term JIF is a misnomer. The
          nonnormalized percentile rank score, denoted I3(A) (the integrated
          impact indicator) score satisfies this definition. Indeed, I3(A) is defined as:
                                          K
                                         X
                                 I3ðAÞ 5    x k n A ðkÞ               (7.18)
                                         k51
             Clearly, the I3 indicator too is a strict independent indicator, not only
          for average performance but even in general.



          7.10 CITATION MERIT

          It is always difficult to compare sets with a different number of elements.
          A proposal by Crespo et al. (2012) tries to remedy this. These colleagues
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