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. 2023 Feb 11;14:783. doi: 10.1038/s41467-023-36421-3

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© The Author(s) 2023

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 5 — a Accuracy of GroupAcc in estimating the fold of increase in the substitution rate of foreground lineage given different portions of elements in a group under accelerated evolution. X-axis shows the scaling factor of accelerated lineage branch length in simulation setting, which is the real fold of increase in the substitution rate of accelerated lineage. Y-axis shows the accuracy of estimating the fold of increase in the substitution rate of the accelerated lineage. The weighted estimate of the fold of increase in the substitution rate of foreground lineage across the whole group of binding sites is calculated by ( $M \times \hat{r_{1}} / \hat{r_{2}} + 1 - M$ ). The accuracy of estimating the fold of increase is calculated as $\frac{The weighted estimate of the fold of increase in the substitution rate of foreground lineage}{r_{1} / r_{2} in the simulation setting}$ . b Comparison of performance estimating the number of elements under accelerated evolution between GroupAcc and element-level LRT method. Blue curves are the estimated numbers of GroupAcc. Red curves are the estimated numbers of element-level LRT.