Understanding ARG statistics

Written by: Keaka Farleigh, Ph.D.
Date: September, 14th, 2026.
Date last modified: September, 15th, 2026

Purpose

To help you understand the different statistics calculated in ARGHelpR and determine which measure is appropriate for your question/objective.

Background

Ancestral recombination graphs (ARGs) can support robust inferences of many processes including natural selection and introgression (see this article for a background on ARGs). Inferring which processes may influence a particular genomic region or if they significantly influence a region (see this article for visualizations of different scenarios) at all requires that we use various statistics that evaluate the coalescence between and within populations or species (see figure below). In this article, we will explain each statistic, how to calculate it, and why it may be relevant for your study.

Figure 1. Cartoons of the ancestral recombination graph statistics that ARGHelpR can calculate. Colored tips represent individual haplotypes corresponding to different populations, colored nodes represent different statistics calculated by ARGHelpR. In some cases (center and right cartoons), ARGHelpR uses these statistics to calculate other statistics, which are represented by the formula in each cartoon.

Time to most recent common ancestor between (TMRCA_B)

The time to the most recent common ancestor between is the time (usually in generations) that it takes for coalescence to occur between two populations/species. This is used when trying to identify any genomic region with deeper coalescence than the genomic background (reproductive isolation, balancing selection) or when trying to identify signatures of within-population selection, recurrent selection, and introgression.

Time to most recent common ancestor within (TMRCA_W)

The time to the most recent common ancestor within is the time (usually in generations) that it takes for coalescence to occur within two populations/species. This is traditionally used to identify within-population selection, recurrent selection, and introgression. It can also be used to identify balancing selection and genomic regions involved in reproductive isolation and within-population selection.

ARGHelpR provides a modified calculation of this statistic, which is useful when there is incomplete lineage sorting, balancing selection, or introgression. These processes influence the ARG topology, biasing the traditional calculations. We address this by estimating the TMRCAW for seperate monophyletic clades of population haplotypes and averaging these values.

For example, if a population is paraphyletic and split into two groups (like in the figure above).

$$TMRCA_{W}= \frac{TMRCA_{W1} + TMRCA_{W2}}{2}$$ This can then be generalized for as many groups as exist in the dataset. If there are K groups, the formula becomes:

$$TMRCA_{W} = \frac{TMRCA_{W1} + TMRCA_{W2} + TMRCA_{W3} + ... + TMRCA_{WK}}{K}$$ Users can control the number of groups by setting the n.haps argument, which sets the number of haplotypes that must be present in a monophyletic clade for the TMRCA_W to be calculated. ARGHelpR also reports the minimum, median, and maximum TMRCA_W in case users feel like those values are more appropriate for their study.

Time to the most recent common ancestor 50 (TMRCA₅₀) and relative half life (RTH)

The time to the most recent common ancestor 50 is the time (usually in generations) that it takes for coalescence to occur for 50% of the haplotypes in an ARG. This is traditionally used to calculate the TMRCA relative half life (RTH), which provides a way to account for coalescence variation across the genome (Rasmussen et al., 2014). The RTH is calculated by dividing the TMRCA₅₀ by the TMRCA_B. Lower values of RTH indicate more recent coalescence, which may be indicative of a recent selective sweep. Higher values indicated a longer time to coalescence, which may be indicative of balancing selection or reproductive isolation. See identify_argcandidates_divergence and identify_argcandidates_shared for additional tests for selection.

$$RTH = \frac{TMRCA_{50}}{TMRCA_{B}}$$

Literature Cited

Rasmussen, M. D., Hubisz, M. J., Gronau, I., & Siepel, A. (2014). Genome-wide inference of ancestral recombination graphs. PLoS genetics, 10(5), e1004342.