An improved test of linearity in dominance hierarchies containing unknown or tied relationships

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. 0 1995 The Association for the Study of Animal Behaviour 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 litera- ture 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 defi- nition: dominance is an attribute of the pattern of repeated, agonistic interactions between two indi- viduals, 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