simPhy-ILS: On the simPhy-ILS dataset where the number of species is fixed to 201, differences between various summary methods were generally small (Fig. 1 b), but overall, ASTRAL was significantly better than DISTIQUE-8 (p<0.001). However, DISTIQUE-8 and ASTRID were indistinguishable (Additional file 1: Table S3). The magnitude of the difference between ASTRAL and DISTIQUE-8 significantly depended on the level of ILS (p=0.001), where with low or medium ILS levels, the two methods had a similar error, but with increased ILS, ASTRAL outperformed DISTIQUE; the differences were more pronounced when we had fewer gene trees (significant: p=0.039; Additional file 1: Table S3).