### Post provided by Julia Fischer

Anyone who studies social animals in the wild (or human groups, for that matter), will soon find that some individuals threaten or attack others frequently, while others try to get out of the way or signal their submission in response to aggression. Observers tally the outcome of such aggressive interactions between any given two individuals (or ‘dyads’) and try to deduce the rank hierarchy from such winner-loser matrices. One drawback of this approach is that all temporal information is lost.

Imagine Royal, a baboon, dominating over Power, another baboon, 20 times, and Power dominating over Royal 20 times as well. If we just look at these data, we might think that they have the same fighting ability and similar ranks. But, if we know that Royal beat Power the first 20 of the interactions, then Power beat Royal in all further interactions, we’d come to a totally different conclusion. We’d infer that Power had toppled Royal and a rank change had taken place.

# How do Rank Hierarchies Change Over Time?

One prominent method that takes the temporal dynamics of winner-loser interactions into account was originally developed to calculate the relative skill level of chess players. This method was introduced by Arpad Elo and is hence known as Elo-Rating. Elo-Rating has also been applied to rate the relative skills in a variety of competitive fields, including Major League Baseball, video games, and Scrabble.

The general idea is that each player has a certain rating. If a highly rated player wins against a lowly rated player, a few points will be transferred from the lowly rated player to the highly rated player. If the lowly rated player unexpectedly beats the higher ranked one though, then more points will be transferred from the loser to the winner. How many points are transferred depends on a constant, the so-called ‘shift co-efficient’. In this way, we can track the trajectory of individual players over time. And for any given point in time, we can figure out the rank hierarchy of the players by comparing their Elo-scores.

# What Animal Researchers Learned from Chess Tournaments

A few years ago, Paul Albers and Han de Vries suggested using Elo-rating to assess rank hierarchies among animals. The idea really caught on – especially among primatologists – after Christoph Neumann and colleagues developed the idea further. By now, several packages in the “R” statistics environment are available to run such analyses (a tutorial can be found here).

But how do we determine the starting scores of individuals when we begin to study a group of animals and have no information about their current rank hierarchy? Traditionally, we just started with an arbitrary value that was assigned to all animals. By tracking the outcome of dyadic interactions, some animals would gain points, while others would lose. The phase until a certain stability was reached is called the ‘burn-in phase’ during which the rank hierarchy cannot be discerned. One aim of our contribution was to improve this arbitrary assignment of the starting scores, by applying a method based on Bayesian statistics, called ‘partial pooling’. We found that estimates of the start ratings based on ‘partial pooling’ were more robust than other approaches.

# How Certain Can We be about Inferred Rank Relationships?

There is a growing consensus that we need to estimate the certainty with which we can determine the current rank position of a given individual. For instance, if you don’t have much data or you have a lot of undecided interactions, you may arrive at a highly unreliable rank hierarchy. To address this, we’ve also introduced a method to estimate the certainty we can infer a given rank hierarchy with .

Using previously published data on the rank relationships of female chimpanzees by Steffen Foerster and colleagues, we found that the estimates for the ranks were in fact highly uncertain. In some cases, uncertainty may not be a bad thing. It can tell us a lot about the type of society the animals live in. For instance, in the Guinea baboons that our group studies in Senegal, we aren’t able to discern a rank hierarchy with certainty, no matter how many interactions we use. It seems that overt displays of dominance do not explain the quality of the males’ social relationships well – unlike in most other primate species studied to date.

# Data Simulations Help to Test the Robustness of Your Method

To probe the robustness of our method, we developed a number of scenarios and simulated the data accordingly. In the first, we assumed a stable rank hierarchy, where a new individual joins the groups and takes the top position (“external take-over”). This is often the case in highly competitive primate species. At the beginning of the simulated period, the uncertainty with which the Elo-scores could be discerned was relatively large, but rapidly decreased. After the simulated take-over, about 200 interactions were necessary to work out the new rank hierarchy with certainty. Similarly, in a scenario where we simulated the death of an animal followed by social upheaval, it took about 200 interactions until the rank hierarchy was stable again.

Overall, such simulations allow us to test whether our method is sensitive enough to pick up important changes in the social dynamics of the animals we’re studying. Our study also found that variation in the shift-coefficient may greatly contribute to variation in the results. The higher the shift-coefficient, the greater the volatility. Yet, using a very small shift coefficient might prevent the scores from tracking actual rank changes. So the choice of the appropriate shift coefficient is critical.

We believe that Elo-Rating and other approaches are of great value, because they take the temporal dynamic of relationships into account. But, it’s also clear that we need to develop a better understanding of the limitations of these methods and how the settings we use may affect the outcome of our results.

You can find out more about our contributions to the improvement of these approaches in ‘Bayesian inference and simulation approaches improve the assessment of Elo‐ratings in the analysis of social behaviour‘. We also recommend an excellent How To… Paper in Journal of Animal Ecology by Alfredo Sánchez-Tojár and colleagues, in which they analyze the strengths and weaknesses of other approaches to estimate rank relationships in animals. And if you want to know more about the Guinea baboons we are studying in Senegal, you can find an overview here.