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de VRIES, H. A. N. (1998). Finding a dominance order most consistent with a linear hierarchy: a new procedure and review. Anim. Behav., 55(4), 827–843.
Abstract: A procedure for ordering a set of individuals into a linear or near-linear dominance hierarchy is presented. Two criteria are used in a prioritized way in reorganizing the dominance matrix to find an order that is most consistent with a linear hierarchy: first, minimization of the numbers of inconsistencies and, second, minimization of the total strength of the inconsistencies. The linear ordering procedure, which involves an iterative algorithm based on a generalized swapping rule, is feasible for matrices of up to 80 individuals. The procedure can be applied to any dominance matrix, since it does not make any assumptions about the form of the probabilities of winning and losing. The only assumption is the existence of a linear or near-linear hierarchy which can be verified by means of a linearity test. A review of existing ranking methods is presented and these are compared with the proposed method.
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de Vries, H., Stevens, J. M. G., & Vervaecke, H. (2006). Measuring and testing the steepness of dominance hierarchies. Anim. Behav., 71(3), 585–592.
Abstract: In the analysis of social dominance in groups of animals, linearity has been used by many researchers as the main structural characteristic of a dominance hierarchy. In this paper we propose, alongside linearity, a quantitative measure for another property of a dominance hierarchy, namely its steepness. Steepness of a hierarchy is defined here as the absolute slope of the straight line fitted to the normalized David's scores (calculated on the basis of a dyadic dominance index corrected for chance) plotted against the subjects' ranks. This correction for chance is an improvement of an earlier proposal by de Vries (appendix 2 in de Vries, Animal Behaviour, 1998, 55, 827-843). In addition, we present a randomization procedure for determining the statistical significance of a hierarchy's steepness, which can be used to test the observed steepness against the steepness expected under the null hypothesis of random win chances for all pairs of individuals. Whereas linearity depends on the number of established binary dominance relationships and the degree of transitivity in these relationships, steepness measures the degree to which individuals differ from each other in winning dominance encounters. Linearity and steepness are complementary measures to characterize a dominance hierarchy.
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De Vries, H., & Appleby, M. C. (2000). Finding an appropriate order for a hierarchy: a comparison of the I&SI and the BBS methods. Anim. Behav., 59(1), 239–245.
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de Vries, H. (1995). An improved test of linearity in dominance hierarchies containing unknown or tied relationships. Anim. Behav., 50(5), 1375–1389.
Abstract: Appleby (1983, Anim. Behav., 31, 600-608) described a statistical test, based on the work of Kendall (1962, Rank Correlation Methods), for the significance of linearity in dominance hierarchies. He suggested that unknown relationships should be assigned the value 1/2 and that subsequently the same test procedure can be used. In this paper it is shown that incorrect results are obtained by this method whenever there are unknown relationships. Values of the linearity index are systematically too low. P-values can be too high (underestimating the significance) or too low (overestimating), and seem to differ by not much more than a factor two (respectively a half) from the correct P-value. An improved method is developed for testing linearity in a set of dominance relationships containing unknown relationships. Furthermore, it is argued that, if one admits the possibility of tied dominance relationships, which should indeed be assigned the value 1/2, Landau's linearity index is to be preferred to Kendall's index. A randomization test is developed for assessing the significance of linearity or non-linearity in a set of dominance relationships containing unknown or tied relationships. The test statistic employed in this testing procedure is based on Landau's linearity index, but takes the unknown and tied relationships into account.
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Bouskila, A., de Vries, H., Hermans, Z. M., & van Dierendonck, M. (2012). Leadership roles in movements of free-roaming Konik horses (Equus caballus) in a Dutch reserve. In K. Krueger, & (Ed.), Proceedings of the 2. International Equine Science Meeting (Vol. in press). Wald: Xenophon Publishing.
Abstract: We observed the activity and movements of Konik horses (Equus caballus) in order to determine the initiators of movements and the individuals taking the lead in movements of the main groups. We conducted our observations between March-June 2010 along the shores of the Rhine river, in the Blauwe Kamer reserve, in the Netherlands. The horses were introduced to the reserve alongside with cattle to prevent the growth of the forest and maintain the grass habitat. We videotaped all observations on two digital video cameras, one providing the general view of the group and the other scanning and focusing on the individuals, to aid with identification. Horses were recognized based on individual profiles that were created for each one, consisting of photos of both sides and notes of the main characters, such as orientation and coloration of the mane, prominent scars and markings, etc. Twenty three horses (not counting foals) were organized in two harem groups with 11 individuals (two of which were dominant stallions) in the large group and six individuals (one of which was a stallion) in the second group. These two main groups were always within sight of each other, and two bachelor males moved usually in their vicinity. An additional group of three young bachelor males roamed elsewhere in the reserve. We divided the movements of the horses to local movements while grazing and to long-distance movements, in which the horses moved to a different grazing location, to a pond of water, resting area or groups of trees that were used by the horses for scratching themselves. During the local movements, any two of the three oldest females in the large harem group were enough to cause the whole group to follow them and gradually change position within the pasture. The smaller harem always followed the large harem’s movements. The long-distance movements of the large harem were sometimes initiated by one of the harem stallions that herded their group and at other times – by the oldest females. Soon after the movement was clearly initiated, the dominant stallions positioned themselves in the back of the group or in the center and had no influence on the direction of move that was only determined by the leading females. In the long-distance movements too, the small harem followed the large harem group, and the two bachelor males usually followed behind them. Social interactions included aggressive interactions between the two dominant males within the large harem or between dominant stallions and the two bachelor males accompanying the two harems. In addition, dominant males courting or attempting to mate with mares sometimes caused a turmoil that eventually initiated movement of the harems. KW -
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Bouskila, A.,, Lourie, E., de Vries, H., Hermans, Z. M.., & van Dierendonck, M. (2015). Sex, but not relatedness nor age, affect the social network of horses in a semi-natural reserve. In Proceedings of the 3. International Equine Science Meeting.
Abstract: For some time now, social network analysis provides tools to describe concisely the social structure of animals. Nevertheless, the factors that shape the social network and determine the frequency of different types of affiliations are often unclear. For example, the effects of relatedness on social networks have been only studied on relatively few species. Both among social or sub-social species the effects ranged from no effect to a substantial effect. Here we test the hypothesis that relatedness has an effect on the social network of horses (Equus caballus) that live freely in semi-natural conditions, and specifically, that individuals with high values of relatedness to their neighbors in the network will have fewer links Wolf et al, 2011). This hypothesis is based on the notion that related individuals have common interests and are likely to assist each other, and may need less links to other, unrelated individuals. In addition, individuals of similar age and sex are likely to have common needs, and thus are more likely to be associated. We thus tested a second hypothesis: homophily to individuals with similar age and sex will have a role in determining the associations within the social network. The field study was conducted on 27 horses in the Blauwe Kamer reserve in the Netherlands (1.1 sq km). We videotaped horses and their groups and used the information from 22 horses, after excluding the five foals from the analysis. Relatedness was calculated from the pedigree, which was based on parentage, determined by DNA analysis. The social network was constructed based on spatial proximity data. We assessed the influence of relatedness, age-homophily and sex-homophily on the network structure with Multiple Regression Quadratic Assignment Procedure (MRQAP) (Krackhardt 1988), with the R package sna. The results show that there was no significant effect of relatedness on the network, nor an effect of age-homophily. Nevertheless, we found a significant effect of sex-homophily, the tendency of individuals to associate with individuals of the same sex. We argue that the lack of a relatedness effect is not likely to have been caused due to the inability to detect who is kin. The structured social system in horses includes strong associations between often unrelated individuals, alongside with young individuals leaving their natal harem as part of the inherent inbreeding avoidance (Linklater & Cameron 2009; Boyd et al. in press). The significant effect of sex-homophily could stem from the protection females get from associations with other females, e.g., in the reduction of harassment from males. Previous studies on feral horses showed that mares that were better connected with other females in their harem benefited from higher survival rates to their foals (Cameron et al. 2009). The associations among bachelor males could also contribute to the strength of the sex-homophily effect.
In order to generalize from our results, one needs to examine additional populations of horses, because the conditions in the Blauwe Kamer reserve may not be representative, mainly due to the limited opportunity for dispersal in a restricted area.
Keywords: Long-term affiliation; spatial proximity, kin detection
Boyd, L., Scorolli, A. L., Nowzari, H., & Bouskila, A. (2016). Chapter 2: Social organization. In J. I. Ransom, & P. Kaczensky (Eds.), Wild equids. Baltimore, Maryland (in press): The Johns Hopkins University Press.
Cameron, E. Z., Setsaas, T. H., & Linklater, W. L. (2009). Social bonds between unrelated females increase reproductive success in feral horses. Proceedings of the National Academy of Sciences, 106(33), 13850-13853
Krackhardt, D. (1988). Predicting with networks: Nonparametric multiple regression analysis of dyadic data. Social Networks, 10(4), 359-381.
Linklater, W. L., & Cameron, E. Z. (2009). Social dispersal but with philopatry reveals incest avoidance in a polygynous ungulate. Animal Behaviour, 77(5), 1085-1093.
Wolf, J. B., Traulsen, A., & James, R. (2011). Exploring the link between genetic relatedness r and social contact structure k in animal social networks. The American Naturalist, 177(1), 135-142.
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Albers, P. C. H., & de Vries, H. (2001). Elo-rating as a tool in the sequential estimation of dominance strengths. Anim. Behav., 61(2), 489–495.
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