Anim. Behav., 1992, 44, 1164-1165

Dominance and mating success: avoiding incest and artefact

A D R I A N J. S I M P S O N & R O B E R T A. B A R T O N * Department of Psychology, University of Sheffield, Sheffield SIO 2TN, U.K.

(Received 26 February 1992; accepted 23 March 1992; MS. number: sc-741)

Three matters raised in Cowlishaw & Dunbar's reply are briefly discussed below: (1) The import- ance of dominance for mating success; (2) undesir- able characteristics of Cowlishaw & Dunbar's new statistic; (3) the likelihood that the new statistic is subject to bias.

(1) Cowlishaw & Dunbar agree that mating success is not solely determined by dominance rank. The thrust of our point, however, is not that there are alternative strategies for obtaining the resource (matings), but that the value of the resource is not equal for all males in a group, and therefore that there is a danger of confusing differ- ences in resource-holding potential (RHP) with differences in the propensity to compete. Using mating fi'equency as a measure of success assumes that each male is trying to obtain matings. In essence, the confounding factor is incest avoidance. Young pre-emigration males (in matrilocal groups) may simply not be competing for access to females to whom they are related. That is why they should always be excluded from the analyses. Similarly, older males, which have been in the group long enough for some of their daughters to be reaching sexual maturity (4~5 years for baboons; Altmann et al. 1988) may be less 'successful' in obtaining matings simply because they are not competing for them, or at least not for so many of them (if they concentrate on older females). It is these considerations which lead us to suggest caution in simplistically equating rank with RHP and mating frequency with success. We do not doubt the role of aggressive competition for mates, we simply suggest that analyses which lump all males together may be glossing over behavioural and life-historical complexity, and consequently overestimating the quantitative importance of dominance.

(2) What are the characteristics of Cowlishaw & Dunbar's new index measuring the relationship between mating success and dominance rank? Their statistic (proposed to circumvent problems

*Present address: Department of Anthropology, Univer- sity of Durham, 43 Old Etvet, Durham DH 3HN, U.K.

associated with Spearman's coefficient, rs) is calculated by taking each successive dyad in the hierarchy of Nmales within a group and calculating the difference between the proportion of the total number of matings achieved by one member of the dyad and the proportion achieved by the other. The median difference is then calculated from the set of N - 1 differences. We argue below that this measure has three deficiencies: it is unstable; it can result in an apparently strong relationship where only a weak relationship is present in the data; and it can, on the other hand, potentially fail to reveal a relationship that is in fact present.

First, we suggest that any measure based on comparisons only between adjacent group mem- bers, as opposed to all group members, is likely to be unstable, and that this instability will be exacerbated whenever small values are involved since slight and unimportant changes in the values can then give large changes in proportions. Furthermore, when the median is employed as the measure of location, the moderating influence of the full set of values is minimized and one, possibly misleading, value is taken as the statistic. As an illustration of these points, consider the two sets of mating success values consisting of 2, 2, 2, 2, 2, 20 and 0, 2, 0, 2, 0, 20 and let dominance ranks run from 1 to 6. The first set gives the value zero for the statistic, whereas the second, though very similar, gives the maximum value of 1.0. (Calculation of Kendall's rank correlation coefficient, r, between each of the two sets and the dominance ranks gives values of 0.577 and 0.389, respectively.) In short, a minor change in the data creates a major change in the index, a change from no relationship to maximum relationship.

Second, we consider the case where high and low scores alternate: for example, 2, 18, 2, 18, 2, 18. Here a high value of 0.8 for the index would occur, although all the even-numbered males have equally large success scores and Kendall's ~ is only 0.258. In this case the index is subject to shortcomings derived from calculations based on adjacent

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Short Communications 1165

scores, compounded by employment of the median. Obviously this example, and those above, are carefully chosen to support our argument, but to engender confidence a statistic should provide an acceptable result for any set of data.

Third, the Cowlishaw & Dunbar index can fail to reveal a relationship actually present in the data. This is illustrated by the set 1, 1, 1,9, 9, 9, for which the index gives a value of 0 although there clearly is a relationship, albeit discontinuous, between the two variables. Using Kendall's ~ as a measure of the relationship we obtain the value 0'775.

(3) If the Cowlishaw & Dunbar index is unstable and potentially misleading, why does it consistently give significant negative correlations with group size? The answer may be that the index is subject to systematic bias, as was the correlation measure rs, though for a different reason. To appreciate this point, let us assume (for simplicity) that an even number of mating success scores are uniformly distributed between zero and some value R, and that rank and mating success are perfectly corre- lated. The two middle values of the mating success variable will then be (R-- i)/2 and (R + i )/2, where i is the interval separating successive values. Given the assumption of a uniform distribution, clearly i = R / ( N - 1 ) , and the difference between the two middle proportions will be the median difference. This difference is given by

R + i R - i R

2 2 i N - I 1 d = - - -

R + i R - i R R N - I ' +

2 2

which obviously decreases as N increases. Under the same assumptions, Cowlishaw & Dunbar's alternative indices behave similarly: the differ- ence between the two highest ranking males is

d= 1/(2N-3), and the mean difference is approxi- mated by d= ( t + log e (N - 1))/(N - 1). Both indices decrease as N increases. Clearly the assumptions of a minimum of zero and a uniform distribution are arbitrary, but the principle is likely to apply to some extent when the mating success variable is distributed in other ways, and with imperfect correlations between rank and success. Indeed, we have simulated the behaviour of the index when mating success scores were allowed to vary from 0 upwards but were sampled from a Gaussian, rather than a uniform, population distribution correlating 0.74 with dominance rank. Taking 70 samples of sizes running from N = 2 to N-= 30, we found significant negative regressions of the index (Fisher z-transformed) on log N in 80% of cases. Moreover, in these simulations the slope of the function relating the index to log N asymp- totically approached zero as group size increased. Cowlishaw & Dunbar note this effect in their own data and comment that, 'This itself would seem to be an important result ( . . .) since it suggests that high-ranking males are only able to monopolize matings when they have fewer than four rivals in the group'. Since it appears in simulated data where the relationship between dominance rank and mat- ing success is constant, we suggest that Cowlishaw & Dunbar's effect is likely to be at least partly an artefact arising from employment of an index subject to bias by group size.

R E F E R E N C E S

Altmann, J., Hausfater, G. & Altmann, S. A. 1988. Determinants of reproductive success in savannah baboons (Papio cynocephalus). In: Reproductive Success (Ed. by T. H. Clutton-Brock), pp. 403-418. Chicago: University of Chicago Press.