Analyses of hierarchy structure

Two-way tables (contingency tables) evaluate whether the succes in a particular dyadic interaction is independent of other interactions. For this analysis, each cell of the matrix represents an interaction between two individuals - the winner (rows) and the loser (columns). Analyses of dominance matrices cannot use the usual tests of independence (Chi2, G) as they violate asumptions of independence (i.e. rows are not independent of column values). Instead the Kc-statistic is used here in combination with a permutation approach (Hemelrijk 1990 Anim. Behav. 39: 1013-1029.

We should really ask two separate questions, namely "Who takes part in fighting?" and "Who fights with whom?". The solution to these questions rely on iterative proportional fitting (Bishop et al. 1975 Discrete multivariate analysis. Cambridge, MA; MIT Press) as described by Freeman (Freeman et al. 1992 Animal Behaviour 44: 239-245). The respective analyses are included for both questions with expected values in the upper and Freeman-Tukey deviates in the lower triangle. Freeman-Tukey deviates permit a cell-wise examination of the table and identify cells which are particularly "large" or "small".

Linearity of dominance structure is present when all component triads are transitive (if animal A dominates B, and animal B dominates C, then A must also be dominant over C). With an increasing number of circular triads, the hierarchy becomes progressively less linear. Appleby's/Kendall's Test of Linearity ("K") compares the actual number of circular triads to those maximally possible Appleby 1983 Anim. Behav. 31: 600-608. If several dyadic relationships within this matrix are either unknown or tied, a correction may be applied as proposed by deVries 1995 Anim. Behav. 50: 1375-1389.

Ordinal ranks and dominance indices are calculated according to Theraulaz et al. 1992 Ethology 91: 177-202, and dominance activity indices following Bartos 1986 Aggressive Behavior 12: 175-182

Estimates for cardinal dominance ranks are based on Thurstone's method of paired comparisons. Two iterative procedures are used to solve these models - the Boyd & Silk method (Boyd and Silk 1983 Anim. Behav. 31: 45-58) and the Batchelder-Bershad-Simpson (BBS) method (Jameson et al., 1999 Anim. Behav. 57: 991-998). The Boyd & Silk method, where low values correspond to high ranks, will not result in stable rank values if we lack information about interactions between some pairs of individuals or if the group includes individuals who are never dominanted by others. The BBS method, however, will converge on stable ranks even if both of these conditions are not met.

Analysis of hierarchical social structures using Applet "HierarchyMatrixApplet" Version 3.1 (updated on 1/6/09).

Thanks to Christoph Goessmann for explaining the iteration procedure for dropping diagonals and Lin Freeman for help with the iterative proportional fitting. Alisdair Daws worked out the test for independence of social interactions (using Kc). Don Edwards helped with the calculations for Boyd & Silk cardinal ranks and Kimberly Jameson who straightened out our implementation of the BBS method. Back to Java grinders behavior stats page.

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last updated 1/11/11