More equations, less citations?

We all knew it, but now it’s official: Tim W. Fawcett and Andrew D. Higginson show in a new PNAS article titled “Heavy use of equations impedes communication among biologists” that

The density of equations in an article [addition: in the field of ecology and evolution] has a significant negative impact on citation rates, with papers receiving 28% fewer citations overall for each additional equation per page in the main text. […] In contrast, equations presented in an accompanying appendix do not lessen a paper’s impact.

I suppose this doesn’t really come as a surprise to most people who observe the impact and citations of papers in ecology and evolution. The conclusion Fawcett and Higginson draw from this is that

  1. Equations negatively impact the success of a paper.
  2. Ideally, we would “enhance the technical understanding of biology graduates” so that equations do no longer hinder scientific communication, but because this is difficult to achieve, “a complementary and more immediate solution is for those doing theoretical work to describe their models in a way that can be more easily digested by a diverse audience”.

I think there is definitely some truth in both statements, but I have the feeling that there’s also some things that require a bit more thinking. Let me first reply to the first assertion.

1) Are equations really causal for the observed lower citation rates?

I’m sure equations have an effect, but I wonder how big this effect really is. The authors control for a few potentially confounding factors such as article length, but they leave out other factors that I would consider as strong candidates for confounding effects. Some of them are:

  • Article type and target audience – Even in the same journal, I might be writing a review, or a broad paper to give an overview about a topic or some general results, or I might write a more specialized paper in order to communicate to the experts in the field. Depending on that, I will adjust the complexity of the presentation and the number of equations. It might so happen that more specialized papers are less cited, but the reason for this lower citations could well be that they are more specialized, and not that they use more equations.
  • Topic – Nearly the same point could be made about the topic: some topics might require the use of more equations, and if it so happens that people are less interested in those topics, those papers get less cited, but the cause for the lower citations is the topic and not the equations
  • Journal preferences – When writing for higher impact journals, people tend to use less equations. Most high impact journals such as Nature and Science put nearly all equations in an online appendix – sure, those papers get cited well (this point is controlled to a certain extent by the study though, but not fully).

In conclusion, my gut feeling is also that many equations are not attractive for the average ecological reader, but we still have to ask ourselves:

Would a paper that examines the Lyapunov exponent of a oscillatory predator prey system in 27 spatial dimensions published in Theoretical Population Biology really massively increase in citations if they dropped the maths?

2) If equations are causal, does that imply we should use less equations?

Anyway, let’s assume for the moment that equations are really causal for citations, does that imply we should we use less equations? Again, I think we have to be careful.

I completely agree that, when communicating theory, we should try to maximize the information that is received by the reader. However, even when doing so, there are different classes of readers, and different appropriate levels of complexity associated with this readers. You can write very broad, nearly for a laymen audience, and you can write very dense, for experts. May be that the broad papers get more citations than the expert papers, but would we be really better off if everyone would only write broad papers?

Already now, this is turning into an arms race where everyone is trying to broaden his presentation at the cost of loosing clarity. Often, I have to read so far as the methods until I understand what a paper is exactly testing, because abstract and intro are filled with common places to make the paper appear interesting to the widest possible audience. The authors demonstrate this somewhat involuntarily themselves: in their title, they refer to “impedes communication among biologists”, but in the article, they consider only papers from ecology and evolution. Did they struggle with the title word limit, or did they think that there are simply more biologists to cite this paper than people in ecology and evolution? Words might boost your citation counts, but they may also be deceiving because they can be used to make a study or a model appear broader and more general than it actually is, and this is not a good thing.

Thus, I completely agree that we have to consider the mathematical abilities of our target audience. I’m also happy to conceit that many mathematical presentations are unnecessarily complex. However, what I missed in the paper completely is a bit of praise for equations, saying something like: our goal in science communication is clarity – sometimes equations may be in the way of clarity, and then they should be avoided, but often they are also absolutely essential because words are easily misunderstood. So, drop your unnecessary equations, but don’t think about dropping the essential ones!

Addition 25.7.12: somewhat related, a Nature feature defending the (wise) use of scientific jargon.

9 thoughts on “More equations, less citations?

  1. Excellent post, and quite a strange pattern these folks picked out (but not too surprising) — I agree with your final statement… and would add that well-written equations are an attempt to formalize clarity — but I agree that this is often lost when considering broad ecological audiences. Moreover, biological systems are physical systems, and I think there should be more resources put into training biologists in theoretical tools (which are often just as useful for empiricists)… I would imagine that the more we understand the underlaying principles of biological systems, the more important mathematical characterizations of these systems will become.


    • I agree. Let’s hope that your last point turns out to be true – compared to other sciences, we don’t really have such a good track record in finding simple underlying explanations, but I also nurture the hope that this may improve in the next decades, and that might indeed give a boost to the appreciation of mathematics.


  2. It’s great to see that our study is generating some debate. This is an interesting post, but I’d like to add one important clarification: if you read our PNAS article carefully, you’ll see that we are not suggesting that authors start cutting equations from their papers. Mathematical equations are indispensable to research in evolution and ecology. We do not conclude that equations negatively impact on the success of a paper, but rather that a *high density of equations per page* negatively impacts on the success of a paper. Mathematical equations may be off-putting to a broad audience *if they are not explained in sufficient detail*. The solution we propose is that the authors of theoretical articles add more explanatory text to take the readers carefully through the assumptions and biological implications of the mathematics. In a nutshell: not ‘use fewer equations’, but ‘use more words’. Unfortunately this might be resisted by short-format journals, in which case one alternative solution is to present most of the mathematical details in an appendix — but by no means are we suggesting that it is a good idea to do away with the maths altogether.


    • Dear Tim,

      it’s great to have you commenting here, much better of course to have the authors responding directly than others speaking about them, so I greatly appreciate your response.

      I believe we both agree on the general point that equations are important, but too many, and particularly also overloaded equations impede scientific communication, so no problem there.

      I also agree that it was oversimplified to speak in my post about “using fewer equations” when you are indeed suggesting that, because improving the math capacities and increasing the amount of words is often difficult, the most achievable solution would be to put the equations in the online supplementary (does that sound fair now?). Anyway, I apologize for the somewhat sloppy summary of your conclusions.

      Nevertheless, if you replace all my “use fewer equations” under point 2) by “use fewer equations in the main text ”, I still stand by my opinion that there is a cost associated with making the main paper appealing to the broadest possible audience and moving all the details to the online supplementary because I believe that many results, particularly when it comes to the results of theoretical models, cannot be understood without understanding the definitions, assumptions and the maths that lead to the conclusions.

      Because, let’s be honest, if we did move most details to the online supplementary, do you think that the additional readers and citations we gain from that will read the online supplementary? Clearly not, because they don’t like equations, and they will not like them better if we present them in unreadable word typesetting in an online supplementary. So, those readers will read and cite a theoretical paper without really understanding and being able to judge the assumptions, the scope and the generality of the results. It’s great to have more readers on the one hand, but I still think we must recognize that there is a trade-off at some point between the amount of people I reach and level of understanding that my average readers has after reading the paper.

      Thus, I applaud your point that theorists need to reach out more to the empirical people. Also, at a given level of detail, they should maximize the clarity and understandability of their papers. However, while doing so, I believe there is at some stage a trade-off between general understandability and depth of the presentation, and I had the feeling that your paper was a bit too enthusiastically embracing the benefits of presenting research “in a way that can be more easily digested by a diverse audience” while neglecting that there may also be a cost associated with this route.


  3. That’s a fair point, Florian. We did write in our article that “the risk of moving equations to an appendix is that the main text then glosses over the fine details of the model’s assumptions, which can have a big impact on how the predictions are interpreted. Authors should avoid this potential problem by clearly stating any assumptions in the main text” — but perhaps we should have made that part more prominent.

    Our focus in the article was on potential ways to facilitate communication between theory and experiments, and to that end we would hope to stimulate three debates: (1) among those doing theoretical work, about how to communicate their research most effectively; (2) among those running scientific journals, about what format best suits theoretical work; and (3) among science educators, about how to prepare biology graduates for a research career in which mathematical literacy is increasingly important.

    I wholeheartedly agree with you that clarity of communication should be the goal, and that mathematical equations can be an important part of this.


    • I’m sure your paper will stimulate debate, and that is a good thing indeed. And again, I see your point and I also have some papers in mind that would probably have been improved by a few equations less. Just as long as we don’t throw out the baby with the bathwater – citations are not everything, and I still prefer the main equations of an important theoretical work to be placed in the text, even if this puts a few people off.

      Btw., the media coverage your paper created is really hilarious (which doesn’t seem your fault at all). points to some English links, but it’s not getting better when translated into German. The Spiegel, for example, has this great headline “Forscher fürchten Formeln” (Researchers fear formulas), and they state that you confirm an old hypothesis of the Physics genius Stephen Hawking 😀

      Reminded me of an old post:


  4. Pingback: On equations, citations, and mathematical understanding (UPDATEDx2) | Dynamic Ecology

  5. Pingback: Interactions: Foggy with a Risk of Mathematics |

  6. Pingback: Interactions: Here in the Blogosphere… |

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s