Beyond infinite sites: Generalized ABBA-BABA statistic for deeper phylogenies

The Patterson’s D statistic detects gene flow from ABBA-BABA site patterns, but its biallelic site patterns fail under deeper divergences where multiple hits cause false positives. We propose two extensions, D ⁺ and D ^* . Both incorporate multiallelic site patterns to reduce saturation bias under JC and F84 model. Simulations show that D ⁺ and D ^* both remain correctly null under all conditions and detect gene flow effectively, with distinct advantages: D ⁺ guarantees non-negativity of the denominator, while D ^* provides greater robustness when mutation rates vary across genomic regions. The source code and binary files are publicly available at https://github.com/chaoszhang/ASTER .