# Michael S. Rosenberg’s Laboratory

Computational Evolutionary Biology & Bioinformatics

E-mail: msr@asu.edu
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## q-index

The q-index (Bartneck and Kokkelmans 2011) is an attempt to measure whether an author is manipulating their own h-index through self-citation. It calculates a potential cost of a self-citation for each publication whose citation count is equal or less than h, sums these costs for all self-citations, then normalizes by the total number of publications. Essentially, publications with a citation count equal to h generate a cost of 1 per self-citation, with each subsequent publication in rank order generating costs of 1/2, 1/3, 1/4, etc., with ties generating equal values. The potential costs are calculated as:

$$q_i=\left|\begin{matrix} 0 & \text{if }C_i > h \text{ or } i < h \\ \frac{1}{i+1-a_i-h} & \text{if }C_i \leq h \text{ and } i \geq h \end{matrix}\right.,$$

where

$$a_i=\left|\begin{matrix} 0 & \text{if }i \leq h \\ a_{i-1} & \text{if }i>h \text{ and } C_i \neq C_{i-1}\\ 1+a_{i-1} & \text{if }i>h \text{ and } C_i = C_{i-1} \end{matrix}\right. .$$

The total self-citations for any publication where Ci ≤ h are multiplied by the associated cost and these values are summed and averaged for all publications.

$$Q=\frac{1}{P}\sum\limits_{i=1}^{P}{c_i q_i}.$$

where ci is the number of self-citations to the ith publication.

Simulations indicate that selective citing to boost h may lead to a q-index, greater than one, while lower values are found when self-citation is random or fair.

### Example

Publications are ordered by number of citations, from highest to lowest.

 Citations (Ci) Self Citations (ci) Rank (i) ai qi ciqi 42 36 14 11 9 9 3 2 2 2 1 1 1 0 0 0 1 0 0 0 2 2 2 2 0 0 1 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 h = 6 0 0 0 0 0 0 0 0 1 2 2 3 4 4 5 6 0 0 0 0 0 0 1/2 1/3 1/3 1/3 1/4 1/4 1/4 1/5 1/5 1/5 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.667 0.000 0.000 0.250 0.000 0.000 0.000 0.000 0.000

The sum of ciqi is 1.9167, thus Q is 0.1198.

YearQ
19970.4000
19980.0556
19990.1190
20000.2500
20010.1198
20020.1443
20030.2854
20040.1651
20050.2070
20060.2630
20070.2200
20080.1843
20090.1772
20100.1185
20110.1739
20120.0968
20130.0671
20140.0611
20150.0425
20160.1414
20170.1385

## References

• Bartneck, C., and S. Kokkelmans (2011) Detecting h-index manipulation through self-citation analysis. Scientometrics 87(1):85–98.