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Bayesian inference of admixture graphs on Native American and Arctic populations [1]

['Svend V. Nielsen', 'Bioinformatics Research Centre', 'Aarhus University', 'Aarhus', 'Andrew H. Vaughn', 'Center For Computational Biology', 'University Of California Berkeley', 'Berkeley', 'California', 'United States Of America']

Date: 2023-02

The Methods section describes our implementation of a Markov Chain Monte Carlo (MCMC) algorithm, AdmixtureBayes, which samples admixture graphs from their posterior distribution. We summarize genetic data from multiple populations as a matrix that captures how allele frequencies in the data covary between populations. AdmixtureBayes samples graphs that explain this covariance matrix. The topology of any sampled graph captures the relationships between samples as a mixture of the graphically structured covariance matrices. Branch lengths capture the amount of genetic divergence between populations, measured by drift, and admixture events explain shared allelic covariance between otherwise independently evolving populations. As a property of the MCMC algorithm, each graph is sampled at a frequency corresponding to its posterior probability. AdmixtureBayes is available to use at https://github.com/avaughn271/AdmixtureBayes .

We do not label branches and nodes in general, meaning that even though the the leaves are given a unique label, the leaves themselves are not unique. For example, switching the labels of two leaves that form a cherry in the graph, would not change the graph topology. For a more formal definition, see the definition of topology in S1 Text . All branches have a length in the interval (0, ∞) and all admixture nodes are given an admixture proportion in the interval (0, 1).

We begin by presenting our formal definition of an admixture graph. An admixture graph consists of a topology and a set of continuous parameters. The space of topologies for a given number of leaves, L, consists of all uniquely labeled graphs of the set of all directed acyclic graphs which fulfill

Comparisons with TreeMix and OrientAGraph

We compared the accuracy of AdmixtureBayes to TreeMix and OrientAGraph on 4 distinct admixture graphs, shown in Fig 1. We simulated datasets from each of these admixture graphs in msprime [18] by using the add_population_split and add_admixture options and adjusting event times and population sizes until the allele frequency drift terms matched those of the admixture graph.

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TIFF original image Download: Fig 1. The graphs G1, G2, G3, and G4 used for the comparisons between methods. G1 and G2 are not based on any real dataset, but the branch lengths are chosen to have human-like values. Out was used as the outgroup for both graphs. G3 is based on M1 from Molloy et al. (2021), the graph that motivated the development of the MLNO approach of OrientAGraph. We have changed some of the branch lengths. popE was used as the outgroup. G4 is based on Model M7 from Fig 3 of Molloy et al. (2021), which is in turn based on Fig 7a from Wu (2020) [19]. The populations ITU, JPT, and ASW have been removed. The YRI population was used as the outgroup. For all graphs, as in Molloy et al. (2021), branch lengths are not shown to scale and are shown multiplied by 1000. Divergence nodes are shown as circles. Admixture nodes are shown as rectangles. The fractions inside the admixture nodes denote the contribution from the population represented by the dashed line. https://doi.org/10.1371/journal.pgen.1010410.g001

We then analyzed all simulated datasets with AdmixtureBayes, TreeMix, and OrientAGraph (see the section “Running AdmixtureBayes, TreeMix, and OrientAGraph” for details). Comparing their accuracy is not straightforward because TreeMix and OrientAGraph produce one graph whereas AdmixtureBayes produces posterior samples of graphs. In addition, TreeMix and OrientAGraph assume a fixed number of admixture events, whereas AdmixtureBayes samples graphs with different numbers of admixture events. We ran TreeMix and OrientAGraph conditioned on the true number of admixture events, while we considered all graphs produced by AdmixtureBayes, even those with the wrong number of admixture events. We note that this could increase the error of AdmixtureBayes. Furthermore, both TreeMix and OrientAGraph allow admixture involving the branch to the outgroup, which AdmixtureBayes does not. The extent to which this was a problem varied between simulation models, so we handled this on a case-by-case basis. We used three metrics to compare the graphs inferred by these methods to the true underyling admixture graph. The Topology Equality is a simple metric that is 1 if the inferred graph has the same topology as the true graph and 0 otherwise. The next metric we considered is the Covariance Distance, defined as the Frobenius distance between the allelic covariance matrix of the true graph and the allelic covariance matrix of the inferred graph (see Methods). Finally, we measured the Set Distance, which we defined as a topological distance measure similar to the Robinson-Foulds metric (S9 Fig; Methods section).

For each of the 4 admixture graphs we analyzed (see Fig 1), we performed the following analysis: 20 independent datasets were simulated using msprime and all three methods were run on each dataset. Then, each of the three metrics was calculated for the results of each method. For AdmixtureBayes, we measured both the accuracy of the sampled graph with the highest posterior (we call this the AdmixtureBayes Mode) and the mean accuracy of a graph sampled from the posterior (we call this the AdmixtureBayes Mean). We plot the values of these metrics across the 20 datasets as boxplots in Fig 2. We also highlight that an excellent comparison of TreeMix, OrientAGraph, and miqograph was done in Molloy et al. [3], which both illustrated OrientAGraph’s ability to infer topologies TreeMix could not and demonstrated that miqograph was unable to infer topologies with deep admixture events.

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TIFF original image Download: Fig 2. We here plot the results of our method comparison with TreeMix and OrientAGraph. For each of the graphs in Fig 1, we simulated 20 datasets and ran each method on each dataset. We compared the accuracy of each method with the 3 statistics discussed in the section Comparisons with TreeMix and OrientAGraph. For AdmixtureBayes, we examined both the Mode graph (the sampled graph with the highest posterior) and the mean value of the statistics when 100 graphs are sampled from the posterior (we refer to this as the AdmixtureBayes Mean). TreeMix and OrientAGraph allow admixture involving the outgroup, an error which AdmixtureBayes is not allowed to make. For fairness, we only plot the results for the graphs not involving admixture with the outgroup. We have listed the number of datasets that resulted in such graphs in parentheses next to the method name on the x-axes. The Topology Equality statistic for TreeMix, OrientAGraph, and the AdmixtureBayes Mode can only be 0 or 1, so we plot a horizontal line at the mean value over the datasets, rather than a true boxplot. https://doi.org/10.1371/journal.pgen.1010410.g002

On graph G1, which contains 1 admixture event, all methods perform similarly well. The correct topology was inferred by all methods on all datasets (giving a Set Distance value of 0), and the accuracy of the covariance matrix implied by each of the inferred graphs (as measured by the Covariance Distance) is quite similar.

On graph G2, which contains no admixture events, TreeMix and OrientAGraph are able to infer the correct topology for all 20 datasets. The Mode estimate of AdmixtureBayes also infers the correct topology in all cases. For all datasets, the AdmixtureBayes Mean topologies are highly concentrated on the true topology, though there is some variation. This is to be expected given the inherent noise in the data. It is also worth noting that the incorrectly inferred topologies sampled by AdmixtureBayes may include graphs with an admixture event, an error which we do not allow TreeMix and OrientAGraph to make as we run them with the correct number of admixture events (zero). We note that the AdmixtureBayes Covariance Distance is slightly larger than the TreeMix and OrientAGraph distances. This is to be expected as both of those methods explicitly perform optimization on branch lengths and admixture proportions, which will likely result in a better model fit than the graph AdmixtureBayes samples that happens to have the highest posterior.

On graph G3, which has one admixture event, TreeMix does quite poorly. This is by design, as G3 is based on Model M1 from Molloy et al. [3], which motivated the development of OrientAGraph. In particular, TreeMix incorrectly infers an admixure event involving the outgroup in 17 out of the 20 datasets. Of the 3 remaining datasets, TreeMix was only able to infer the correct topology for 2 of them. We only plot the accuracy statistics for the 3 graphs that do not involve admixture with the outgroup as these are the only graphs that exist in the same state space as AdmixtureBayes. However, we highlight that the boxplots in Fig 2 do not necessarily represent all simulated datasets.

In contrast to TreeMix, OrientAGraph never infers admixture involving the outgroup and infers the correct topology in almost 80% of all datasets. AdmixtureBayes, however, outperforms both methods by inferring the correct topology for all datasets, both using the Mode estimate and the Mean estimate. We attribute this to a superior framework for exploring the state space of topologies. We still note that TreeMix and OrientAGraph provide better estimates of branch lengths and admixture proportions, which we again attribute to the fact that AdmixtureBayes is not designed for optimizing the likelihood function for branch lengths but instead provides posterior distributions. If point estimates for branch lengths are of interest, we recommend that users optimize the branch lengths using other methods with the AdmixtureBayes Mode topology fixed.

Graph G4 represents a very complicated topology and is based on a model used by Molloy et al. [3] to represent the shortcomings of OrientAGraph. TreeMix incorrectly infers an admixture involving the outgroup for all datasets, so we do not plot the results from running TreeMix. OrientAGraph incorrectly infers an admixture involving the outgroup for 14 datasets, leaving 6 datasets to compare with AdmixtureBayes. We see that OrientAGraph never infers the correct topology and never has a Set Distance of less than 4. In contrast, the AdmixtureBayes Mode estimate represents the correct topology for more than half of all datasets, which we again attribute to a superior framework for exploring the state space of topologies. The AdmixtureBayes Mean estimates are fairly noisy, but still represent a posterior distribution that is often concentrated on the true topology. The optimization employed by OrientAGraph results in a lower Covariance Distance than AdmixtureBayes, even in the presence of an incorrect topology. Performing a similar optimization on the AdmixtureBayes Mode topology will likely yield a smaller Covariance Distance if a point estimate of an admixture graph with branch lengths is desired. From these results, we conclude that the MCMC framework of AdmixtureBayes provides an effective algorithm for searching through the topology space of admixture graphs and often infers the correct topology when other methods do not. All of the scripts used to run these simulations can be found in the SimulationStudy folder on the AdmixtureBayes GitHub.

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[1] Url: https://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1010410

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