This is the first in a series of blog entries exploring the metrics used for assessing dominance hierarchies. The previous introductory post explains the rationale behind doing this. An index page will give detailed links to other metrics within this blog.

I’ll start with giving details for how to calculate the *Dominance Index* metric described by Zumpe & Michael (1986), which uses counts of agonistic encounters to generate individual scores, which can then be used to suggest a hierarchy. Using this technique is possible with pen and paper, so I may be giving a bit more detail of the nuts and bolts than I will with some of the more complex metrics. This technique is intended to give the user a ‘cardinal ranking’ – rather than just sorting the interacting individuals into a ranked order, this technique provides a way of assigning each individual a weighting statistic. The authors suggest that this could be useful for assessing how individual dominance changes over time, if different datasets are used.

The agonistic data required for this statistic are the counts of aggressive/dominant and submissive behaviours between all possible pairings of the group members. These should be collected in two tables. So, for an example group with four individuals (identified as A, G, H, and K), we tally the number of aggressive behaviours committed by each individual to each of its three group members:

For example, A directs aggression towards H on seven occasions. Note also that no aggressive interactions are observed between G and K.

Similarly, we tally the number of submissive behaviours directed towards each individual by the other three group members:

For example, H displays submissive behaviours to G on 11 occasions. Note also that no submissive behaviours were recorded between H and K.

Having collected this information, we now calculate the percentage of the aggressive actions between pairs of individuals that each individual directs at the other. For example, within the pairing of A and G, nine aggressive acts are recorded (five by A, and four by G). A is aggressive towards G in 55.6% of their aggressive interactions (= 5 / 9), and G is aggressive towards A for 44.4% (= 4 / 9). If we work out these two percentages within each pairing, we can build up a table giving the percentages of aggressive behaviours given by each individual. If no aggression is seen within a pair, the two corresponding table entries for the pair should be marked as ‘null’, as is given for the two entries between G and K here:

Similarly, the percentages of submissive actions received by individuals within each pairing should also be calculated. Again ‘null’ values should be recorded where no submissive actions within a pair were observed, as seen between H and K here:

The aggression/submission percentages are then combined by calculating an average aggression/submission score for each possible pairing of group members. For example, the average score for A when it interacts with G is

65.3% = (55.6% + 75.0%) / 2.

If no aggressive actions are recorded for a pair, this average is simply given the value of the percentage of submissive actions (calculated in table 4). So, the average score for G when it interacts with K is 90.9%. Similarly, if no submissive actions are recorded between pair members, the average is assumed to be the percentage of aggressive actions committed by an individual (recorded in table 3). So, the average score for H when it interacts with K is 71.4%. Calculating all possible pairing, we get:

Finally, the ** Dominance Index** for each of the group members is calculated as the mean of the averages calculated for each focal individual, as given in table 5b. For example, the dominance index for A is calculated as 73.4% = (65.3% + 69.2% + 85.7%) / 3.

From this, we can use the Dominance Index rankings to construct a hierarchy. In this case, A > G > H > K.

The metric falls apart when there are no aggressive or submissive acts recorded within a pairing, which means that no average score can be recorded in table 5a. This could potentially be remedied by observing the interacting individuals until some agonistic interaction is recorded, but it may be that the non-interacting individuals are able to assess each other without needing to interact (using alternative cues, or through recognising each other from earlier unrecorded interactions). This metric is therefore not ideal if some individuals do not interact with others.

Similarly, a dataset which records few interactions between individuals may be biased by a few anomalous recorded encounters. However, using mean percentages (as calculated in table 5b) removes biases that could be introduced by simply scoring the overall number of ‘wins’ in dyadic agonistic encounters for each individual, which may be incorrectly inflated by many interactions with a small subset of the group members. I’m also curious to see what happens when a group consists of two or more subgroups where interactions tend to be within rather than between subgroups.

#### Further reading

Zumpe D & Michael RP (1986). Dominance index: a simple measure of relative dominance status in primates. *American Journal of Primatology* **10**: 291-300 | doi: 10.1002/ajp.1350100402