Rosenberg Impact Factors

EM′-index

The EM′-index (Bihari and Tripathi 2017) is an extension of the EM-index which includes all publications, rather than just those from the core. Like the EM-index, we begin by creating a vector (E) where the first value is E₁ = h. Subsequent values of the vector, E_i+1, are determined by subtracting E_i from the citation count for all publications in the core defined by E_i, and recalculating h from these new citation counts, reranking all publications by these new citation counts as necessary (i.e., some of the publications previously in the tail of the citation distribution may advance beyond publications in the core as citations representing earlier calculations of h are “used up”). This process continues until one runs out of citations, all of the remaining publications have only a single remaining citation, or there is only a single publication left to be considered. From this vector, one calculates the index as:

$$EM^\prime=\sqrt{\sum\limits_{i=1}^{n}{E_i}},$$

where E_i and n are the i^th element and length of E, respectively.

Example

Publications are ordered by number of citations, from highest to lowest. After each step, E_i is subtracted from the citations of the top E_i publications and all publications are re-ranked by this adjusted citation count for the next step.

Citations (C_i)	59	26	16	11	11	10	4	3	3	2	1	1	1	0	0	0	0	0
Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
						E₁ = 6

Adjusted Citations (C_i)	53	20	10	5	5	4	4	3	3	2	1	1	1	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
					E₂ = 5

Adjusted Citations (C_i)	48	15	5	4	4	3	3	2	1	1	1	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
				E₃ = 4

Adjusted Citations (C_i)	44	11	4	3	3	2	1	1	1	1	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
			E₄ = 3

Adjusted Citations (C_i)	41	8	3	3	2	1	1	1	1	1	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
			E₅ = 3

Adjusted Citations (C_i)	38	5	3	2	1	1	1	1	1	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
			E₆ = 3

Adjusted Citations (C_i)	35	2	2	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
		E₇ = 2

Adjusted Citations (C_i)	33	2	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
		E₈ = 2

Adjusted Citations (C_i)	31	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₉ = 1

Adjusted Citations (C_i)	30	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₁₀ = 1

Adjusted Citations (C_i)	29	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₁₁ = 1

Adjusted Citations (C_i)	28	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₁₂ = 1

Adjusted Citations (C_i)	27	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₁₃ = 1

Adjusted Citations (C_i)	26	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₁₄ = 1

Adjusted Citations (C_i)	25	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₁₅ = 1

Adjusted Citations (C_i)	24	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₁₆ = 1

Adjusted Citations (C_i)	23	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₁₇ = 1

Adjusted Citations (C_i)	22	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₁₈ = 1

Adjusted Citations (C_i)	21	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₁₉ = 1

Adjusted Citations (C_i)	20	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₂₀ = 1

Adjusted Citations (C_i)	19	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₂₁ = 1

Adjusted Citations (C_i)	18	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₂₂ = 1

Adjusted Citations (C_i)	17	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₂₃ = 1

Adjusted Citations (C_i)	16	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₂₄ = 1

Adjusted Citations (C_i)	15	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₂₅ = 1

Adjusted Citations (C_i)	14	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₂₆ = 1

Adjusted Citations (C_i)	13	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₂₇ = 1

Adjusted Citations (C_i)	12	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₂₈ = 1

Adjusted Citations (C_i)	11	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₂₉ = 1

Adjusted Citations (C_i)	10	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₃₀ = 1

Adjusted Citations (C_i)	9	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₃₁ = 1

Adjusted Citations (C_i)	8	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₃₂ = 1

Adjusted Citations (C_i)	7	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₃₃ = 1

Adjusted Citations (C_i)	6	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₃₄ = 1

Adjusted Citations (C_i)	5	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₃₅ = 1

Adjusted Citations (C_i)	4	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₃₆ = 1

Adjusted Citations (C_i)	3	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₃₇ = 1

Adjusted Citations (C_i)	2	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₃₈ = 1

Adjusted Citations (C_i)	1	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
New Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
	E₃₉ = 1

The sum of the 39 E values is 59. The EM′-index is the square-root of this sum, thus EM′ = 7.6811.

History

Year	EM′
1997	1.0000
1998	2.6458
1999	4.5826
2000	5.6569
2001	7.6811
2002	9.6437
2003	11.3137
2004	13.0384
2005	14.6969
2006	16.1864
2007	18.3576
2008	20.0749
2009	21.9089
2010	24.0832
2011	26.0384
2012	27.8388
2013	29.3598
2014	31.2730
2015	32.6956
2016	34.4674
2017	36.1109
2018	37.3229
2019	38.5097
2020	39.4715
2021	40.5832
2022	41.9404
2023	43.0232
2024	44.0000
2025	44.4635

References

Bihari, A., and S. Tripathi (2017) EM-index: A new measure to evaluate the scientific impact of scientists. Scientometrics 112(1):659–677.