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

The Pattersons 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.