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.

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 112 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 l/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. An important topic in social ethology is the analysis of dominance relationships in social groups of individuals. A recent paper by Drews (1993) presents an extensive review of the literature for the purpose of elucidating the concept of dominance. On the basis of the original definition of dominance given by Schjelderup-Ebbe (1922), Drews proposed the following structural definition: dominance is an attribute of the pattern of repeated, agonistic interactions between two individuals, characterized by a consistent outcome in favour of the same dyad member and a default yielding response of its opponent rather than escalation. The status of the consistent winner is dominant and that of the loser subordinate.In this paper I address the question of how to test for linearity in a set of observed dominance relationships, in particular if this set contains unknown or tied relationships. An unknown dominance relationship (or zero dyad) is the case when the two members of a dyad have not been observed to perform any agonistic interaction towards each other. This observational zero is to be distinguished from a structural zero. If it is structurally impossible for the members of a dyad to have agonistic interactions with each other, this dyad has a structural zero, and a fortiori a dominance relationship between two such individuals is absent. If, on the other hand, the members of a dyad could in principle show agonistic interactions towards each other but were not observed to do so during the observation period, the dyad has an observational zero (also called 'unknown relationship' or 'zero dyad' for short). A dominance relationship is tied if the two individuals in a dyad have performed...

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. Copyright 1998 The Association for the Study of Animal Behaviour. Copyright 1998 The Association for the Study of Animal Behaviour.

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The clinical population of aggressive children diagnosed as having an oppositional defiant disorder (ODD) or a conduct disorder (CD) is heterogeneous, both with respect to behaviour and aetiology. Recently, the following distinction has been proposed that might further clarify this heterogeneity: reactive aggression is an aggressive response to a perceived threat or provocation, whereas proactive aggression is defined as behaviour that anticipates a reward. In this article we examine various aspects of this distinction. We will [1] examine the evidence that reactive and proactive aggression are distinct phenomena by discussing the theories underlying the distinction between the subtypes in humans and we briefly review evidence for a similar distinction in animals; [2] we critically review the literature on the measurement in children via questionnaires and behavioural observations; we then point out that the correlation observed between the subtypes is due to the fact that many children show both types of aggression; [3] we review the literature on specific characteristics of the subtypes giving attention to social information processing, peer status, biological correlates and developmental history, and demonstrate that there is some evidence to suggest that reactive and proactive aggression are distinct dimensions; [4] we discuss the relevance of the distinction between reactive and proactive aggression for child and adolescent psychiatry.