Toward an alternative bibliometric

Impact factor isn’t great. A bibliometric based on entropy reduction may be promising.

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Impact factor

There are a variety of citation-based bibliometrics. The current dominant metric is impact factor. It is highly influential, factoring into decisions on promotion, hiring, tenure, grants and departmental funding (Editors 2006) (Agrawal 2005) (Moustafa 2014). Editors preferentially publish review articles, and push authors to self-cite in pursuit of increased impact factor (Editors 2006) (Agrawal 2005) (Wilhite and Fong 2012). It may be responsible for editorial bias against replications (Neuliep and Crandall 1990) (Brembs, Button, and Munafò 2013). Consequently, academics take impact factor into account throughout the planning, execution and reporting of a study (Editors 2006).

This is Campbell’s law in action. Because average citation count isn’t what we actually value, when it becomes the metric by which decisions are made, it distorts academic research. In the rest of this post, I propose a bibliometric that measures the entropy reduction of the research graph.


Claude Shannon codified entropy as \(H(X) = -\sum\limits_{i} P(x_i) \log_2 P(x_i)\) where \(x_i\) are the possible values of a discrete random variable \(X\) (Shannon 1948)(Cover and Thomas 2012). For example, the entropy of a 6-sided die is \[\begin{align} H(D) &= - P(⚀) \log_2 P(⚀) - P(⚁) \log_2 P(⚁) - P(⚂) \log_2 P(⚂) \\ & - P(⚃) \log_2 P(⚃) - P(⚄) \log_2 P(⚄) - P(⚅) \log_2 P(⚅) \\ &= -\left(6 \left(\frac{1}{6} \log_2 \frac{1}{6}\right)\right) \\ &= \log_2 6 \end{align}\].

If we next learn that the die is weighted and can only roll even numbers, this changes the entropy (our uncertainty).

\[\begin{align} H(D|\epsilon) &= - P(⚁) \log_2 P(⚁) - P(⚃) \log_2 P(⚃) - P(⚅) \log_2 P(⚅) \\ &= - \left(3 \left(\frac{1}{3} \log_2 \frac{1}{3}\right)\right) \\ &= \log_2 3 \end{align}\]

So the reduction in uncertainty is \(H(D) - H(D|\epsilon) = \log_2 6 - \log_2 3 = 1\).1


We can use these definitions to calculate the information provided by a research paper and assign an Infometric®™ score. We’ll start with a fairly classic example about cigarette smoking.

First study

Suppose we do a study on whether, in the normal course of smoking, cigarette smoke is inhaled into the lungs (we’ll call this proposition \(A\)). Prior to the study we use the (extremely) uninformative prior \(\cond{P}{A=t}{} = 0.5\). After the study (which we’ll call \(\alpha\)) we perform a Bayesian update and find that \(\cond{P}{A=t}{\alpha} = 0.8\). So our study has provided

\[\begin{align} H(A) - \cond{H}{A}{\alpha} &= -P(A=t)\log_2P(A=t) - P(A=f)\log_2P(A=f) \\ &+ \cond{P}{A=t}{\alpha}\log_2\cond{P}{A=t}{\alpha} \\ &+ \cond{P}{A=f}{\alpha}\log_2\cond{P}{A=f}{\alpha} \\ &= -0.5\log_20.5 \\ &- 0.5\log_20.5 \\ &+ 0.8\log_20.8 \\ &+ 0.2\log_20.2 \\ &\approx 0.278 \end{align}\]

bits of entropy reduction. Thus its score at the moment is \(0.278\). So far, so good?

Second study

Now, we wish to study whether smoking causes chronic bronchitis. Suppose the study we design pipes smoke directly into the lungs of experimental subjects. The validity of our conclusion (Smoking does (not) cause chronic bronchitis.) now depends on the truth of the claim that cigarette smoke is inhaled into the lungs. So this new study is dependent on the prior study and will cite it.

Graph depicting conditional dependencies

We are using uninformative priors.

(In this and subsequent graphs, we follow the conventions of

Bayesian networks (i.e. a cited paper is the parent rather than the child—the arrow runs from rather than to the cited paper) rather than the conventions of citation graphs.)

Now we carry out our study. It provides evidence that cigarette smoking does lead to bronchitis (conditional on the supposition that cigarette smoke is inhaled into the lungs). So we update our \(\cond{P}{B=t}{\beta}\). The entropy reduction from this study, considered in isolation, is \(H(A,B) - \cond{H}{A,B}{\beta} \approx 0.266\).

Graph depicting conditional dependencies
We have integrated data from the second study.

But what if we don’t consider it in isolation? First, we look for the total entropy reduction from both studies and find \(H(A,B) - \cond{H}{A,B}{\alpha,\beta} \approx 0.703\). Note that this is not simply the sum of the isolated reductions.2

Graph depicting conditional dependencies
We have integrated data from both studies now.

How do we apportion this gain into Infometric®™ scores then? We can decompose the aggregate gain into a sum like

\[\begin{align} H(A,B) - \cond{H}{A,B}{\alpha,\beta} &= \cond{H}{A,B}{\beta} - \cond{H}{A,B}{\alpha,\beta} \\ &+ H(A,B) - \cond{H}{A,B}{\beta} \end{align}\]

where \(\cond{H}{A,B}{\beta} - \cond{H}{A,B}{\alpha,\beta} \approx 0.437\) represents \(\alpha\)’s score and \(H(A,B) - \cond{H}{A,B}{\beta} \approx 0.266\) represents \(\beta\)’s score.

(the general form is \(H(S_1,S_2,\cdots,S_n) - \cond{H}{S_1,S_2,\cdots,S_n}{\sigma_1,\sigma_2,\cdots,\sigma_n} = \sum\limits_{i=1}^n I(\sigma_i))\) where \(I(\sigma_i) = \cond{H}{S_1,S_2,\cdots,S_i}{\sigma_{i+1},\sigma_{i+2},\cdots,\sigma_n} - \cond{H}{S_1,S_2,\cdots,S_i}{\sigma_i,\sigma_{i+1},\sigma_{i+2},\cdots,\sigma_n}\)

We can see that \(\beta\) citing \(\alpha\) has increased \(\alpha\)’s score (\(\alpha\) now reduces our uncertainty not only about \(A\), but also about \(B\)), a “citation bonus”. Or, if you prefer, you can think of it as \(\alpha\) capturing the externalities it generates in \(B\).

Fourth study

We’ll now jump to a fourth study so we can examine a fuller set of interactions (i.e. multiples studies citing one study, one study citing multiple studies).

Graph depicting conditional dependencies

The decomposition

\[\begin{align} H(A,B,C,D) - \cond{H}{A,B,C,D}{\alpha,\beta,\kappa,\delta} &= \cond{H}{A,B,C,D}{\beta,\kappa,\delta} - \cond{H}{A,B,C,D}{\alpha,\beta,\kappa,\delta} \\ &+ \cond{H}{A,B,C,D}{\kappa,\delta} - \cond{H}{A,B,C,D}{\beta,\kappa,\delta} \\ &+ \cond{H}{A,B,C,D}{\delta} - \cond{H}{A,B,C,D}{\kappa,\delta} \\ &+ H(A,B,C,D) - \cond{H}{A,B,C,D}{\delta} \\ \end{align}\] leads to scores of \(I(\alpha) = 0.547\), \(I(\beta) = 0.387\), \(I(\kappa) = 0.123\), and \(I(\delta) = 0.434\).


You can try it out below. Maybe look for:

  • Two studies that, when considered in isolation, have the same score but have different scores when placed in the network context
  • A citation that doesn’t give any “citation bonus”
  • Two networks with the same topology but different scores


Desirable properties

Score depends on study design
Tends to encourage Bayesian optimal designs
Aggregates sensibly
The impact factor is often used in inappropriate circumstances (Editors 2006) (Agrawal 2005) (Moustafa 2014). That is, the warning that impact factor is a metric for journals, not authors or departments, is seldom heeded. The proposed metric can used in such cases trivially. For example, if an author has published studies \(\eta\) and \(\theta\), their score is simply \(I(\eta) + I(\theta)\), the total reduction in entropy they contribute.
Handles replications appropriately
Impact factor tends to undervalue replications (Neuliep and Crandall 1990) (Brembs, Button, and Munafò 2013). With a simple extension of the proposed metric, if \(\iota\) is a replication of \(\eta\) about proposition \(E\) it shares the “citation bonus” in proportion to how much it increases our certainty in \(E\).
Gradated citations

With impact factor, a citation to study \(\alpha\) essential to the validity of study \(\gamma\) is given the same weight as a citation to study \(\beta\) providing some minor context for \(\gamma\). With the proposed metric, if \(\gamma\) only depends minorly on \(\beta\), \(\gamma\) will only boost \(\beta\)’s score minorly. This should counteract the inflated value of review articles.

Additionally, being cited by an “important paper” (one that provides great certainty or occupies an important position in the research network) provides a larger boost than being cited by a peripheral paper.

Undesirable properties

Not incentive compatible

For example, if study \(\beta\) depends on study \(\alpha\) it will receive a better score by hiding that dependence and marginalizing. \(\beta\) receives a higher score when presented as

 | A P
 | t 0.8
 | f 0.2

 | B P
 | t 0.82
 | f 0.18

than when presented as

 | A P
 | t 0.8
 | f 0.2

 A | B P
 t | t 0.9
   | f 0.1
 f | t 0.5
   | f 0.5

. However, impact factor also theoretically discourages citation ( e.g. boosting the impact factor of someone that might compete against you come hiring time). This problem does not seem to be devastating (Liu 1993).

The proposed metric is more calculationally complicated than impact factor. (Though the actual impact factor calculation procedure is more complicated than one would suppose.)
Requires assessment of degree of dependence
The “degree of dependence” (e.g. \(\cond{P}{B}{A=t}\) vs. \(\cond{P}{B}{A=f}\)) occupies an important role in the procedure. It’s not obvious to me how this should be determined other than by discussion between authors, editors and reviewers.

Future work

  • Improve interface of demonstration
  • KL divergence and Shapley value approach
  • Extend to replications and multi-proposition studies
  • Extend to richer outcome spaces (i.e. not just studies about a single discrete value)
  • Compare with impact factor on real corpus
  • Compare with expert evalution on real corpus

Agrawal, Anurag A. 2005. “Corruption of Journal Impact Factors.” TRENDS in Ecology and Evolution.

Brembs, Björn, Katherine Button, and Marcus Munafò. 2013. “Deep Impact: Unintended Consequences of Journal Rank.” Frontiers in Human Neuroscience.

Cover, Thomas M, and Joy A Thomas. 2012. Elements of Information Theory. John Wiley & Sons.

Editors, The PLoS Medicine. 2006. “The Impact Factor Game.” PLoS Med.

Liu, Mengxiong. 1993. “Progress in Documentation the Complexities of Citation Practice: A Review of Citation Studies.” Journal of Documentation.

Moustafa, Khaled. 2014. “The Disaster of the Impact Factor.” Science and Engineering Ethics.

Neuliep, James W, and Rick Crandall. 1990. “Editorial Bias Against Replication Research.” Journal of Social Behavior & Personality.

Shannon, Claude Elwood. 1948. “A Mathematical Theory of Communication.” Bell Systems Technical Journal.

Wilhite, Allen W, and Eric A Fong. 2012. “Coercive Citation in Academic Publishing.” Science.

  1. The intuition behind this result is something like our uncertainty is halved (1 bit) because one half of the fair die states are no longer possible.

  2. This accords with the intuition that value of two facts considered together is not simply the sum of their separate values (e.g. learning that Fido is small is largely redundant once you’ve learned that Fido is a Chihuahua).