Quartet-based computations of internode certainty provide accurate and robust measures of phylogenetic incongruence

Incongruence, or topological conflict, is prevalent in genome-scale data sets but relatively few measures have been developed to quantify it. Internode Certainty (IC) and related measures were recently introduced to explicitly quantify the level of incongruence of a given internode (or internal branch) among a set of phylogenetic trees and complement regular branch support statistics in assessing the confidence of the inferred phylogenetic relationships. Since most phylogenomic studies contain data partitions (e.g., genes) with missing taxa and IC scores stem from the frequencies of bipartitions (or splits) on a set of trees, the calculation of IC scores requires adjusting the frequencies of bipartitions from these partial gene trees. However, when the proportion of missing data is high, current approaches that adjust bipartition frequencies in partial gene trees tend to overestimate IC scores and alternative adjustment approaches differ substantially from each other in their scores. To overcome these issues, we developed three new measures for calculating internode certainty that are based on the frequencies of quartets, which naturally apply to both comprehensive and partial trees. Our comparison of these new quartet-based measures to previous bipartition-based measures on simulated data shows that: 1) on comprehensive trees, both types of measures yield highly similar IC scores; 2) on partial trees, quartet-based measures generate more accurate IC scores; and 3) quartet-based measures are more robust to the absence of phylogenetic signal and errors in the phylogenetic relationships to be assessed. Additionally, analysis of 15 empirical phylogenomic data sets using our quartet-based measures suggests that numerous relationships remain unresolved despite the availability of genome-scale data. Finally, we provide an efficient open-source implementation of these quartet-based measures in the program QuartetScores, which is freely available at https://github.com/algomaus/QuartetScores.