Φ values and match rates.

The turning rate is fixed at 0 rad/s. All right-side values are average distance parameter values for all samples and range from 300 mm to 1000 mm. VF: visual field parameter; 〈Φ(N)〉S: mean Φ for a given set S, which is the set of single-OFF collective states such as {0111, 1011, 1101, 1110}; Φ11…1: Φ values for the all-ON collective state. The p-values for 〈Φ(N)〉S and Φ11…1 were computed using the Wilcoxon signed-rank test. MR: match rate, computed by Eq (3) for a corresponding parameter set; MR2π: match rate computed by Eq (3) for fixed VF = 2π. The p-values for MR and MR2π were computed using the Mann—Whitney U test. Values in bold correspond to the parameter settings for which the positional leader highly correlates with a single OFF-state fish in the group and where this fish also yields lower Φ values than those of Φ11…1.