D-plus: Beyond the D-statistic -- Accurate Detection and Characterization of Introgression Despite Among-Lineage Rate Variation

Ubiquitous in phylogenomic and population-genomic introgression work, the D-statistic has fallen victim to its own success. Its reliability is compromised by two major confounding factors: among-lineage substitution rate heterogeneity, which generates spurious signals of introgression, and unsampled "ghost" lineages, which mislead the characterization of introgression by erroneously attributing ghost ancestry to extant species. Here, we developed D-plus, a maximum-likelihood tool that explicitly accounts for both confounding factors to unpack the sources of significant D-statistic signals. Simulations demonstrate that D-plus effectively mitigates false introgression calls under among-lineage rate variation. Meanwhile, it retains high sensitivity for genuine introgression signals and correctly assigns detected signals to introgression involving ghost lineages, or solely between extant non-sister lineages, or a combination of the two. Our D-plus re-analysis of phylogenomic data from Jaltomata (Solanaceae) overturns prior interpretations of significant D-statistic signals, which were conventionally regarded as evidence for recurrent non-sister introgression. Instead, our results favor alternative explanations invoking artefacts driven by lineage-specific rate heterogeneity as well as two ghost introgression events. Collectively, D-plus addresses major methodological limitations inherent to traditional D-statistic analyses and enables accurate detection and characterization of introgression events.