statistics – Interaction Biology

Metrics for dominance interactions 2: Fighting success, Clutton-Brock et al. (1979)

This is the second in a series of blog entries exploring the metrics used for assessing dominance hierarchies: see the introductory post for the rationale behind doing this, with other metrics visible through the index page.

Clutton-Brock et al. (1979) were interested in giving a metric to fighting success in red deer stags, where individuals were studied over long periods of time. Studying any network system over longer periods of time is going to cause a problem, as the status of individuals may change during that period (see Rands 2014 for some discussion of this problem), and the authors of this paper were aware that a male’s dominance could change within a mating period as his energy levels flagged or he became injured. Simply counting the number of fights won and lost will not give a very accurate reflection of how an individual is placed within the herd, as his success is also going to depend upon the idntities of the individuals he beats: a male who consistently fights and wins against weak opponents is not necessarily going to be of similar quality to a male who consistently fights and wins against strong opponents. So, Clutton-Brock and colleagues designed a simple metric that takes account of the quality of opponents individuals win and lose against.

I’ll illustrate how this is calculated with by considering the fighting ability metric of two individuals (labelled black and blue) within the following group structure:

Figure 1: all winner/loser interactions recorded. As depicted in the box, the arrow denotes which individual is the winner (W) or loser (L) in a connected pair. — **Figure 1**: all winner/loser interactions within a group. As depicted in the box, the arrow denotes which individual is the winner W or loser L in a connected pair.

To gauge an individual animal’s fighting success, you need to calculate B, the number of other animals that the focal individual has won against, and note the identities of all the losers. For each of these marked losers, you also need to calculate the number of individuals that they in turn have beaten, and sum these to give Σb. Because we define one individual in an interacting pair as a winner, and the other a loser, this means that none of the summed interactions contributing to Σb are against the focal individual.

As well as assessing wins, you also need to calculate L, the number of other individuals that the focal loses against. These winning animals are marked and the summed number of animals that they themselves lose against is calculated, giving Σl.

Having collated these numbers, the fighting success of a focal individual (which I will refer to as D_CB) is calculated as

D_CB = (B + Σb + 1)/(L + Σl + 1),

where the “+1” term on both the top and bottom of the equation allows a meaningful metric to be calculated for individuals that are either never seen to win or lose.

Using the group interactions given in Figure 1, we calculate D_CB for the individual coloured black using the following reasoning:

Figure 2: Calculating black's win (left panel) and loss (right panel) statistics — **Figure 2**: Calculating black’s win (left panel) and loss (right panel) statistics

Following Figure 2, we see that B = 8, Σb = 2 + 2 + 2 + 1 + 1 + 0 + 0 + 0 = 8, L = 3, and Σl = 2 + 2 + 0 = 4. So, D_CB = (8 + 8 + 1)/(3 + 4 + 1) = 2.125 for the black individual. Similarly, using the reasoning given in Figure 3, D_CB =1.167 for the blue individual.

Figure 3: Calculating blue's win (left panel) and loss (right panel) statistics — **Figure 3**: Calculating blue’s win (left panel) and loss (right panel) statistics

A larger value of D_CB will notify a greater fighting ability, and the maximum size of the statistic within an observed group is going to depend on both the size of the group and the maximum number of other animals that each individual in the group interacts with. In their original paper, Clutton-Brock and his colleagues found D_CB for red deer ranged between 0 and a little over 3.

This is a simple statistic to compute, but I would caution that it should really only be used for comparing individuals within a group, given that it is dependent upon both group size and number of interactions recorded. The metric is also dependent upon observed relationships being fixed: an individual that wins an interaction will always win future interactions with the same opponent. This suggests that caution should be used if this metric were to be transferred to observed interactions where the dynamic between a dyad could change over time.

Metrics for dominance interactions 1: Zumpe and Michael’s ‘Dominance Index’ (1986)

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.

Metrics for dominance interactions: introduction

Many interactions between group-living individuals can be influenced by hierarchies that exist between the interactors. These interactions can be measured in lots of different ways, and once measured, whatever has been scored needs to be processed to give a reproducible estimate of the shape of these interactions.

What this means in practice if you’re starting a new project with a new study organism is that you spend a lot of time thinking about what behaviours to record, and how to record them, but don’t really give consideration to the means of crunching these numbers down to something meaningful at the end. Good experimental design implies that the analysis has been considered during the design of the experiment, but this intermediate stage of generating ‘raw’ information about any hierarchies that are in place may be left out, meaning that something has to be cobbled together post hoc after the work has been done. This is never ideal!

Having supervised a fair number of projects where exactly this has been done, I’ve decided to try and get my head around the various statistics out there that are designed for assessing and ranking hierarchies. Some of these are fairly straightforward, and some are slightly more involved, dipping into social network analysis and other emerging fields in animal behaviour. To make this a useful exercise, I’ll attempt to put together a how-to guide for using them, aimed at researchers with a mixed range of skills in manipulating numbers, and where time permits, I’ll try and add in some practice examples. How well this works depends upon both my own understanding, the time I have available, and the limitations of inputting maths into a WordPress blog!

What I won’t be doing (at least, initially) is being particularly critical about which techniques work best: this is a voyage of discovery for me too! I also won’t be focussing on what dominance is for, why it exists, and how it does or doesn’t drive particular group behaviours (but I do discuss how leadership decisions don’t necessarily depend upon the hierarchy present in Rands et al. 2008 and Rands 2011). This series of blog postings will take a little time to put together (An index page will give detailed links to other metrics within this blog), so if you’re looking for general advice on the sort of indices that are out there, I strongly recommend hunting down a copy of the excellent book by Hal Whitehead (pages 186-195 in particular).

Tag: statistics

Metrics for dominance interactions 2: Fighting success, Clutton-Brock et al. (1979)

Further reading

Metrics for dominance interactions 1: Zumpe and Michael’s ‘Dominance Index’ (1986)

Further reading

Metrics for dominance interactions: introduction

Further reading