indicators.Rmd

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title: "Formal definitions of field normalized citation indicators and their implementation at KTH Royal Institute of Technology"
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# Introduction

This document describes the calculation of bibliometric indicators based
on field normalization in the bibliometric database at KTH (Bibmet),
which is based on Web of Science data. The indicators are described in
Part&nbsp:1 and aspects regarding implementation in the KTH database are
addressed in Part 2.

The following indicators are defined in this document:

-   Mean field normalized citation rate (cf)

-   Proportion publication among the 10 % most frequently cited in the field (ptop10%)

-   Mean field normalized journal impact (jcf)

-   Proportion publications in the 20 % most frequently cited journals in
    the field (jtop20%)

# Part 1 Definitions

This document treats the case, in which fractional counts are used in
the calculations of indicator values. In case whole counts should be
used in the calculations, $a_i$ in the equation under subcase (1.1) below is set to unity.

Let $A$ be a unit of analysis, and $n$ the number of publications for $A$.
Let $r_i$ be the number of authors of the $i$:th publication for $A$. Let
$a_i$ be the fraction $A$ has of the $i$:th publication. We consider two
cases.

##### (1) $A$ is an organization.

We treat two subcases.

(1.1) $a_i$ is the *author* fraction A has of the $i$:th publication and is defined as

$$ a_i = \sum_{j=1}^{m_i} \frac{1}{r_i s_j} $$

where $m_i$ is the number of authors affiliated to $A$ regarding the
$i$:th publication and $s_j$ the number of affiliations of the $j$:th of
these $A$ authors. Note that the right-hand side is equal to $m_i / r_i$
when each $A$ author has exactly one affiliation.

(1.2) $a_i$ is the *organization* fraction $A$ has of the $i$:th publication and is defined as
the number of occurrences of $A$'s name in the address field of the $i$:th
publication divided by the total number organization name occurrences in
the address field in question.

##### (2) $A$ is an individual author.

In this case, $a_i$ is the *author* fraction $A$ has
of the $i$:th publication and is defined as $1/r_i$.

### 1.1 Mean field normalized citation rate

We define the *mean field normalized citation rate* for $A$, $mcf(A)$, as

$$ mcf(A) = \frac{\sum_{i=1}^n a_i \bar{x}_i}{\sum_{i=1}^n a_i} $$
$$ \bar{x}_i = \frac{1}{q_i} \sum_{i=1}^{q_i} \frac{c_i}{\mu_{iq}} $$
$$ \mu_{iq} = \frac{\sum_{j=1}^{m_{iq}} c_j / F_j}{\sum_{j=1}^{m_{iq}} 1 / F_j} $$

where $q_i (c_i)$ is the number of subject categories (the citation
rate) of the $i$:th publication for $A$, $m_{iq}$ is the number of
publications, with the same publication year and of the same document
type as the $i$:th publication for $A$, in the $q$:th subject category of
the $i$:th publication of $A$, and $c_j$ ($F_j$) the citation rate of the
$j$:th of these publications (the number of subject categories of the
$j$:th of these publications). $µ_{iq}$ is the *field reference value*
that the citation rate of the $i$:th publication, $c_i$, is normalized
against regarding the $q$:th subject category of the publication, and the
normalization gives rise to a *field normalized citation rate* for the
publication.

### 1.2. Proportion publication among the 10 % most frequently cited in the field

We define *ptop10%* for $A$, $ptop10\%(A)$, as

$$ ptop10\%(A) = \frac{\sum_{i=1}^n a_i \sum_{q=1}^{q_i} b_{iq}}{\sum_{i=1}^n a_i}$$
$$ b_{iq} = \frac{1}{q_i} \frac{\max(y_{iq}^{c_i+1} - max(0.9, y_{iq}^{c_i}), 0)}{y_{iq}^{c_i+1} - y_{iq}^{ci}} $$

where $q_i$ ($c_i$) is the number of subject categories (the citation
rate) of the $i$:th publication for $A$, $y_{iq}^{c_i}$ ($y_{iq}^{c_i+1}$) the proportion of
publications with the same publication year and of the same document
type as the $i$:th publication for A, and belonging to the $q$:th subject
category of this publication with less than $c_i$ ($c_i+1$)
citations.[^1] $\max(y_{iq}^{c_i+1} - max(0.9, y_{iq}^{c_i}), 0) / (y_{iq}^{c_i+1} - y_{iq}^{ci})$
is the fraction of the $i$:th publication assigned to the 10 % most cited publications.
Observe that this fraction is weighted by $1/q_i$, i.e., by the fraction of the
publication that belongs to the $q$:th subject category. The approach to
assign fractions of publications to the (for instance) 10 % most cited
publications is described and discussed by Waltman and Schreiber (2013).

### 1.3 Mean field normalized journal impact

We define the *mean field normalized journal impact* for $A$, $mjcf(A)$,
as

$$ mjcf(A) = \frac{\sum_{i=1}^n a_i\ jcf_i}{\sum_{i=1}^{n} a_i} $$
$$ jcf_i = \frac{\sum_{j=1}^{p_i} \bar{x}_j}{p_i} $$
$$ \bar{x}_j = \frac{1}{F_{ij}} \sum_{q=1}^{F_{ij}} \frac{c_j}{\mu_{jq}} $$
$$ \mu_{jq} = \frac{\sum_{k=1}^{m_{jq}} c_k / F_k}{\sum_{k=1}^{m_{jq}} 1 / F_k} $$
where $jcf_i$ is the *mean field normalized citation rate of the journal*,
say $J_i$, of the $i$:th publication, $p_i$ the number of
publications in $J_i$, $c_j$ the citation rate of the $j$:th
publication in $J_i$, say $P_j$, $F_{ij}$ the number of subject
categories of $p_j$, $m_{jq}$ the number of publications, with the same
publication year and of the same document type as $P_j$, in the $q$:th
subject category of $P_j$, and $c_k$ ($F_k$) the citation rate (the
number of subject categories) of the $k$:th of these publications.
$\mu_{jq}$ is the field reference value that the citation rate of $P_j$,
$c_j$, is normalized against regarding the $q$:th subject category of
$P_j$, and the normalization gives rise to a field normalized citation
rate for $P_j$ (cf. the definition of mean field normalized citation
rate above). If $J_i$ is a non-multidisciplinary journal,
$F_{ij} = F_{i(j+1)} (j = 0, 1, ...,\ p_i - 1)$, since the number of
subject categories of a publication in $J_i$ is then equal to the
number of subject categories of $J_i$. (Cf. Section 2.5 below)

### 1.4 Proportion publications in the 20 % most frequently cited journals in the field

We define the *proportion publications in the 20 % most frequently cited
journals in the field* for $A$, $jtop20\%(A)$, as

$$ jtop20\%(A) = \frac{\sum_{i=1}^n a_i \sum_{q=1}^{F_i} b'_{iq}}{\sum_{i=1}^n a_i} $$
$$ b'_{iq} = \frac{1}{F_i} \frac{(max(min(0.2, y_{iq}^{r_{iq}}) - y_{iq}^{r_{iq}-1}, 0)}{y_{iq}^{r_{iq}} - y_{iq}^{r_{iq}-1}} $$ 
where $F_i$ is the number of subject categories of the journal, say
$J_i$, of the $i$:th publication of $A$, $r_{iq}$ the rank of $J_i$ in the
ranking of the journals in the $q$:th subject category of $J_i$, where
the journals are ranked descending after their mean field normalized
citation rates[^2], $y_{iq}^{r_{iq}}$ ($y_{iq}^{r_{iq}-1}$) the proportion publications appearing in the
journals in the $q$:th subject category of $J_i$ with a rank less than or equal to $r_{iq}$
($r_{iq}-1$).[^3] The rightmost factor in $b'_{iq}$ is the fraction of $J_i$ with
which $J_i$ is assigned to the 20 % most frequently cited journals in
the $q$:th subject category of $J_i$. Observe that this fraction is
weighted by $1/F_i$, i.e., by the fraction of $J_i$ that belongs to
the $q$:th subject category. The approach to assign fractions of journals
to the (for instance) 20 % most cited journals is basically the same as
the assignment approach used in the definition of *ptop10%* (Section 1.2).

# Part 2 Implementation at KTH Royal Institute of Technology

### 2.1 Database contents

The bibliometric database at KTH (Bibmet) contains the following
indexes:

-   Science Citation Index Expended (SCIE)
-   Social Sciences Citation Index (SSCI)
-   Arts & Humanities Citation Index (AHCI)
-   Conference Proceedings Citation Index - Sciences (CPCI-S)
-   Conference Proceedings Citation Index - Social Sciences & Humanities
    (CPCI -SSH)).

SCIE, SSCI, AHCI from 1980 and CPCI-S and CPCI-SSH from 1990.

### 2.2 Document types included in calculations

In Bibmet, calculations are made for all combinations of document types,
publication years and Web of Science categories. However, the default
presentation of field normalized citation indicators concern only
articles and reviews. The reason for excluding other document types is
the risk for anomalies caused by a low number of publications in the
reference groups and question marks regarding data quality and citation
matching (this especially applies to proceedings papers).

### 2.3 Citations included

Citations from all records in the database are included in the
calculations. The indicators are calculated both with self-citations
included and excluded. The default presentation is made with
self-citations excluded, since the intention when calculating citation
indicators is to see what impact a publication has had on other
researchers than those who wrote the publication. Furthermore, one
should avoid giving incentives to systematic self-quotation.

### 2.4 Retroactive changes of the Web of Science subject category assigned to journals

If a journal is reclassified from one Web of Science subject category to
another by Clarivate, no retroactive changes are made in the
delivered raw data. However, in Web of Science changes the
classification retroactively. Changes of the classification affect the
field reference values and consequently the outcome of the calculations
described in this document. For Bibmet to be consistent with Web of
Science, retroactive changes of the Web of Science subject categories
assigned to journals are made in Bibmet.

### 2.5 Reclassification of journals categorized as Multidisciplinary in Web of Science

Some journals are classified by Clarivate as multidisciplinary.
When field normalization is applied the classification of these
journals into the same category can results in skewed field reference values
for this field. By reclassifying publications in journals within the multidisciplinary
subject category the publications are instead compared to other publications within the same subject field.
The Swedish Research Council has developed and applied a methodology for
reclassification of publications within the multidisciplinary Web of
Science subject category into other categories based on citations
(Gunnarsson, Fröberg, Jacobsson, & Karlsson, 2011). It enables a higher
degree of like-to-like comparison. A very similar method is used at KTH.

### 2.6 Exclusion of publication fractions with low field reference values

For all the four indicators defined in Part 1, publication fractions
with field reference values less than 0.5 are excluded.[^4]

##### **Example 2.1** (*mcf* and *ptop10%*).

Assume that the $i$:th publication of $A$, say $P_i$, belongs to three
subject categories and that exactly one of these categories has a field
reference value less than 0.5. Then, under Section 1.1, $a_i$ in the
denominator and $\bar{x}_i$ in the numerator are multiplied by $2/3$,
and $q_i$ is equal to 2 (and not to 3). Thus, the sum of $\bar{x}_i$ concerns
two field normalized citation rates for $P_i$, and the sum is multiplied by
$1/2$ (and not by $1/3$).

For the equation under Section 1.2, under the assumptions given, $a_i$ in
the denominator and the rightmost sum in the numerator are multiplied by $2/3$,
and $q_i$ is equal to 2 (and not to 3). Thus, the sum concerns two ratios, both of
which are weighted by $1/2$ (and not by $1/3$).

##### **Example 2.2** (*mjcf* and *jtop20%*).

Assume that the journal $J_i$ of the $i$:th $A$ publication belongs to four
subject categories. Assume that for the $j$:th publication in $J_i$, say
$P_j$, published a given year and of a given document type, two of the four
subject categories have a field reference value less than 0.5.
For the equations in Section 2.3, $2/4$ is subtracted from the denominator of
$jcf_i$, and $\bar{x}_j$ in its numerator is multiplied by $2/4$.
$F_{ij}$ is equal to 2 (and not to 4). Thus, the sum of $\bar{x}_j$
concerns two field normalized citation rates for $P_j$, and the sum
is multiplied by $1/2$ (and not by $1/4$).

For *jtop20%* (Section 1.4), under the assumptions given, two of the four
rankings in which $J_i$ occurs are such that $P_j$ does not contribute
to the mean field normalized citation rates of $J_i$ in the rankings.

# References

Gunnarsson, M., Fröberg, J., Jacobsson, C., & Karlsson, S. (2011).
*Subject classification of publications in the ISI database based on
references and citations* (No. 4).

Waltman, L., & Schreiber, M. (2013). On the calculation of
percentile-based bibliometric indicators. *Journal of the American
Society for Information Science and Technology, 64*(2), 372-379.

[^1]: Weights are used for the citation distributions at stake. Each
    citation value in a given distribution is assigned the weight $1/k$,
    where $k$ is the number of subject categories of the corresponding
    publication. The weight is the fraction with which the publication
    contributes to each of its subject categories. The proportion
    publications with less than $c$ citations is then the sum of the
    weights for the citation values that are less than $c$, divided by
    the sum of weights for all the citation values in the distribution.

[^2]: Note that for a given journal in the ranking, these rates may vary
    across the different rankings, corresponding to different subject
    categories, in which the journal occurs.

[^3]: A non-multidisciplinary journal in the ranking contributes, if
    each of its publications has a field reference value, with respect
    to the $q$:th subject category of $J_i$, greater than or equal to
    0.5 (see Section 2.6), with $(1/k)m$ publications to the
    ranking, where $k$ is the number of subject categories of the
    journal and $m$ the number of publications of the journal.

[^4]: Such field reference values might give rise to very large field
    normalized citation rates in spite of few citations.