Bayesian model

Consider a genotypic data set, , for a sample of diploid individuals genotyped at loci, and assume that there are clusters, each of which is characterized by a set of allele frequencies at each locus. Let be the vector of cluster labels of each individual in the sample, and let be the set of allele frequencies. In addition, assume that a set of covariates is measured for each individual, and stored in a design matrix, . The covariates represent the geographic and linguistic information that is available to build predictors of the population genetic structure that is encoded in vector . Regarding geography, predictors can be defined as linear or quadratic trend surfaces as proposed by Durand et al. [36]. Linear trend surfaces include two covariates, latitude and longitude, while quadratic surfaces also include squared and cross-product terms. Languages are coded as factors defined as binary dummy variables in the design matrix [37]. The factor levels will be dependent on the choice of the linguistic classifications considered further in this study. Remark that in regression models using factors, a linear constraint (or contrast) must be defined for identifiability reasons. In our study, we assumed that the sum of effects is null.

For algorithmic reasons, the latent regression model was implemented through a hidden multinomial probit model [38]. In the multinomial probit model, there are regression equations(1)each corresponding to a genetic cluster. The are “augmented” continuous variables defined for each individual and each cluster, is a column vector of regression coefficients, and denotes the identity matrix. For each individual , a cluster label can be obtained from the augmented variables as follows(2)

In the multinomial probit model the role of the clusters is not symmetric. The estimates of the regression coefficients are defined with respect to the cluster, called the reference cluster.

Given the above latent class model framework, we used a Markov Chain Monte Carlo (MCMC) algorithm based on Gibbs sampling to compute the joint posterior distribution on individual cluster labels, regression coefficients and allele frequencies

In this equation, the likelihood and the prior distribution on allele frequencies are computed in the same way as in the model without admixture of the program structure (equations (2) and (4) in [31]). The distribution is a noninformative prior distribution (see Appendix A), and corresponds to the distribution of cluster labels obtained from the multinomial probit model. The algorithm was implemented in the software POPS, and is described in more details in Appendix A.

For each subset of covariates, we additionally computed a matrix of posterior predictive membership probabilities using a Monte Carlo method. To perform the computations, we simulated cluster labels from the generative model described in equation (1) and (2) where the regression coefficients are sampled from their posterior distribution. To display predicted and inferred membership probabilities graphically, we used barplot representations. In these graphics, each individual is represented by aligned colored segments, and the segment lengths are proportional to their estimated or predicted membership probabilities.

Variable selection

To investigate whether a particular subset of covariates is a suitable proxy for genetic assignment, we used two distinct measures. Both measures are based on the posterior of regression coefficients and cluster labels. The first measure is a Pearson correlation coefficient, . For a given subset of covariates, the ability of the model to predict genetic structure was evaluated by computing the correlation between the matrix of predicted membership coefficients and the matrix of estimated membership coefficients. The second measure is based on cross-validation, a technique used in the field of machine learning [39], [40] and for latent class models [41]. In our analyses, a 2-fold cross-validation was implemented. More specifically, we divided the genotypic data set, , into two non-overlapping data sets containing complementary subsets of loci. We considered one of these data sets as the training set, , and the other one as the validation set, . The rationale of the cross-validation approach is that the demographic processes that shaped population genetic structure have affected all loci across the genome. Thus the training and validation sets are exchangeable, as they provide the same amount of information about population structure. We performed 500 runs of the Gibbs sampling algorithm using the training set, and retained the 50 runs having reached the highest likelihood values. For each of the retained runs, a predictive score was computed by averaging the log-probability of the validation set over the posterior distribution given the training set

The computation of predictive scores is detailed in Appendix B. Another series of 50 scores was computed after exchanging the role of the validation and training sets, and a cross-validation score was obtained by averaging the resulting predictive scores.

Simulated data

We ran a first series of simulations using the generating model of the program POPS. Assuming three clusters, cluster labels of 300 individuals were simulated using the following regression equations(3)(4)where is a standard Gaussian noise. The interpretation of the above linear trend model is that latitude is the only variable that influences individual cluster labels. Biallelic genotypes were simulated at , 40, 100 loci. Allele frequencies were dependent on the population of origin, and were equal to and , and in each population respectively. We implemented four hidden regression models: one model without covariates, one with latitude, one with longitude and one with both covariates.

In the second series of simulations, we extended the model by including a factor with five levels representing five languages. The hidden regression equations were defined as(5)(6)where is equal to 1 if individual speaks the language and is 0 otherwise. When running POPS to predict population genetic structure, we considered three linguistic classifications. The first classification contained five languages corresponding to the indicator variables used in the simulation. The second classification contained seven languages obtained after splitting the second and the third languages of the first classification into two sublanguages. The last classification contained three languages because we merged two pairs of unrelated languages from the first classification.

In the third series of experiments, we studied two previously published data sets simulated from a five-island model [42]. The simulated data represented one population structured into five subpopulations differentiated at levels equal to and . Five hundred individuals (100 per subpopulation) were simulated using allele frequency distributions across 10 codominant unlinked loci. Spatial coordinates were simulated using Gaussian distributions. The subpopulations were adjacent to each other and arranged on a ring. We ran POPS using the spatial coordinates of each individual as covariates. In addition, we introduced a spurious noisy covariate independent on the subpopulation of origin. We considered the models defined by all the possible inclusions of those three covariates ( models). These data enabled us to compare the performances of POPS to other programs using spatial covariates [42]–[44].

Native American data

We applied POPS to 512 Native American individuals from the Human Genome Diversity Panel (HGDP) data set [15]. Individuals from 28 populations were genotyped at 678 microsatellite loci. Fourteen Siberian individuals from the Tundra Nentsi population were also included in the study. In the regression models we considered three linguistic classifications. The first and second linguistic classifications corresponded to Greenberg's classification at the stock level and at the group level [28], [45]. The third linguistic classification was given by the website The Ethnologue (www.ethnologue.com) [29], [46]. The three linguistic classifications were encoded with factors having 8, 14 and 16 levels respectively (see ). To account for geography, all models included quadratic trend surfaces. The combinations of geographic and linguistic variables resulted in the following four latent cluster regression models. Model A included geographic information only. Models B-D included geographic and linguistic information: Model B used Greenberg's classification at the stock level (8 levels), Model C used Greenberg's classification at the group level (14 levels), and Model D used The Ethnologue classification at the family level (16 levels).

MCMC parameters

For the simulated data, the runs of POPS used 2,000 sweeps with an initial burn-in period of 1,000 sweeps. For the human data, the runs used 5,000 sweeps with an initial burn-in period of 2,500 sweeps. These values ensured that the likelihoods stabilized around their stationary values. For the HGDP data and for each model, we ran a total of 500 MCMC runs. We retained the 50 runs with the largest likelihood values, and we averaged the resulting estimated and predicted membership coefficients using the computer program clumpp [47].

The number of clusters was set to [15]. Among these nine clusters, there were eight Native American clusters plus the reference cluster. For Native American population samples, we chose the Siberian population (Tundra-Nentsi) to represent the reference group. Individuals in the reference cluster were not allowed to switch to other clusters during the MCMC runs.

Simulation results

Using simulated data sets, we investigated whether including geographic and linguistic covariates can improve the estimation of membership probabilities or not, and we evaluated which subsets of variables best predict the estimated population genetic structure.

For the simulations where latitude was influential (equations (3) and (4)), we found that the true values of the regression coefficients were close to the mode of the posterior distributions (Figure 1). The influence of each covariate was thus correctly ascertained by POPS when the data were generated under its underlying statistical model. To further evaluate if missing the true set of covariates modifies the inference and the prediction of membership coefficients, we evaluated the performances of POPS using various hidden regression models. For all models, the misclassification rates were less than . The upper bound was obtained under a model without covariates and for the smallest number of loci (, Figure 2A). The misclassification rates never increased when we included a spurious longitude variable. With loci, the misclassification rate decreased to when the correct covariate (latitude) was used. With loci, the misclassification rates were less than for all hidden models. All individuals were perfectly assigned to their population of origin when latitude was included. For , the misclassification rate was equal to for all models. In the second series of simulations, linguistic covariates were added to the generating model (equations (5) and (6)). The misclassification rates were less than , a value obtained for loci in a model without covariates (Figure 2B). With , the misclassification rate decreased to when including latitude and a linguistic variable with five levels. With loci, the misclassification rate of the model without covariates was around . We conclude that when the data are generated from a hidden regression model, including covariates in POPS increases the performances of the program. This is particularly true when the number of loci is relatively small.

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Figure 1. Posterior distributions of the regression coefficients for a data set simulated with the hidden regression model ().

The dashed vertical lines correspond to the regression coefficients used for generating the data. Two spatial covariates (latitude and longitude) are included in the regression model but only the first one influences genetic structure.

https://doi.org/10.1371/journal.pone.0016227.g001

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Figure 2. Misclassification rates for simulated data as a function of the covariates included in the clustering algorithm.

A. The cluster memberships are influenced by latitude but not by longitude. B. The data are generated using latitude and a 5-level linguistic classification. C. The data are generated in a five-island model for which or .

https://doi.org/10.1371/journal.pone.0016227.g002

Finally, we studied the variable selection criteria for the data where latitude was influential (equations (3) and (4)) as well as linguistic covariates (equations (5) and (6)). Whatever the number of loci we considered, the increase of the correlation coefficient was larger when including latitude rather than longitude in the regression model. Figure 3 shows that the correlation coefficient and the cross-validation score reach a plateau when the true predictors are included in the hidden regression model. This plateau was found when latitude was the sole determinant of genetic structure and when linguistic covariates had an additional contribution to genetic differentiation.

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Figure 3. Variable selection for simulated data.

The correlation coefficients correspond to the correlations between the estimated and predicted membership probabilities. Confidence intervals of the correlation coefficients are estimated by assuming that the Fisher's transform follows a Gaussian distribution [65]. The validation scores are estimated with the 2-fold cross-validation method. Their standard deviations are estimated by using a non-parametric bootstrap method. A. The cluster memberships are influenced by latitude but not by longitude. B. The data are generated using latitude and a 5-level linguistic classification. C. The data are generated in a five-island model for which .

https://doi.org/10.1371/journal.pone.0016227.g003

For the five-island data with a level of differentiation of , the misclassification rates were less than (Figure 2C). The worst performances were obtained for a model without covariates. When latitude (or longitude) was included in the hidden regression model, the misclassification rate decreased to . When both latitude and longitude were included in the model, the misclassification rate decreased to . The addition of a spurious noisy covariate did not impact the performance of the program. Regarding variable selection, Figure 3C shows that the correlation coefficients and the validation scores reach a plateau when longitude and latitude are included in the hidden regression model. For the five-island data with a level of differentiation of , a model including latitude and longitude was also selected. In this case, the misclassification rate was equal to 2.8%. For these data, POPS compared favorably to the spatial versions of BAPS (misclassification rate = 3.9%) and TESS (misclassification rate = 4.4%) [42]–[44].

Native American HGDP data

To investigate the relationships between geography, languages and genes in Native American populations, we applied POPS to a multilocus genotype data set including 512 individuals from the HGDP. We compared the posterior membership coefficients predicted by four different models that use distinct linguistic classifications and we computed two variable selection criteria in order to discriminate among models (see Material and Methods).

The four clustering models resulted in highly similar patterns of estimated membership coefficients, and these patterns were also similar to the pattern found with structure (Figure 4, Figure S1, Wang et al. [15]). As we used a large number of microsatellite loci, these results are not surprising, and they warrant that the predictive power of the three linguistic classifications will be ascertained consistently.

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Figure 4. Estimated and predicted population genetic structure for 28 Native American populations.

A. The membership coefficients are estimated in a model that includes spatial information (longitude, latitude). Inference of genetic structure is unchanged when we include additional linguistic covariates (Supporting Information Figure S1). The main differences between predictions obtained with or without linguistic information are framed in red. B-D. Membership coefficients predicted by Models B–D. The membership coefficients are averaged over individuals within the same linguistic unit.

https://doi.org/10.1371/journal.pone.0016227.g004

Using a quadratic trend surface to correct for geographic effects, we compared the predictions of a model without languages (Model A) to the predictions of a model using Greenberg's classification at the stock level (Model B), a model using Greenberg's classification at the group level (Model C), and a model using The Ethnologue classification (Model D). Figure 4A compares the predictions of Model A and Model D. For many population samples, the membership probabilities predicted by Model A were close to the estimated coefficients (, Figure 5A). The predictions of Model A for every geographic location in the American mainland are displayed in Figure 6. The value of the correlation coefficient and the map of predicted membership coefficients confirmed that geography is a good predictor of genetic structure in Native American populations. When including linguistic covariates (Models B–D), the predictions of cluster membership were closer to the estimates of the MCMC algorithm than those obtained without languages (Model A) except for the Pima. The correlation coefficient increased from to (Figure 5A), and the predicted genetic structure changed substantially (Figure 4A and Figure S1). For several populations the predictions obtained from linguistic covariates (Models B–D) differed from the predictions obtained with the geographic covariates only: Model A predicted that the Kaqchikel and the Wayuu samples shared substantial ancestry with a group comprising Cabecar, Guaymi, Kogi, Arhuaco, Waunana and Embera populations; Model A also predicted that the Kaingang and Guarani samples clustered with the Ache population, and that the Inga and Piapoco samples were grouped with the Ticuna sample.

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Figure 5. Variable selection for the Native American HGDP data.

Geographic information includes longitude and latitude. Green. stands for Greenberg and Geog. stands for geography. The best model uses The Ethnologue linguistic classification.

https://doi.org/10.1371/journal.pone.0016227.g005

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Figure 6. Genetic structure of Native American populations as predicted by geographical covariates.

Geographical covariates include latitude, longitude, quadratic terms and an interaction term. Locations for which there is a cluster with a predicted membership coefficient larger than are colored with the cluster color. Locations for which there is no cluster that reaches the threshold or that are too distant from a sampled population are colored in grey. The barplot displays the membership probabilities as predicted by geographical covariates.

https://doi.org/10.1371/journal.pone.0016227.g006

Figure 4 B–D displays the membership coefficients predicted by POPS using Greenberg's and The Ethnologue classifications (Models B–D), grouping populations with the same linguistic taxon. At the exception of the Andean and Ge-Pano-Carib stocks, Greenberg's linguistic stocks were associated with multiple clusters (Figure 4B). Refining Greenberg's classification at the group level improved the characterization of genetic clusters by linguistic taxa (Model C, Figure 4C). At the group level, the Northern Amerind stock split into Almosan-Keresiouan and Penutian groups that correspond to genetically divergent clusters. Similarly, the Central Amerind stock split into Uto-Aztecan and Oto-Mangue groups which are also genetically divergent. However, the split of the Equatorial-Tucanoan stock into the Macro-Tucanoan and Equatorial groups, and the split of the Chibchan-Paezan stock into the Chibchan and Paezan groups, did not improve the prediction of genetic clusters. In The Ethnologue classification (Model D), the Equatorial group split into the Arawakan and Tupi families. This separation improved the prediction of genetic clusters since the Arawakan family was associated with a unique genetic cluster. In contrast, the separation of the Penutian group into the Mixe-Zoque and Mayan families did not improve the characterization of genetic groups. Overall The Ethnologue classification provided better predictions of genetic groups than Greenberg's classification. Among the 16 families of The Ethnologue classification, only the Tupi, Choco and Chibchan families were not associated to a unique genetic cluster (Figure 4D). Supporting these comparisons, Figure 5B shows that the cross-validation score increases when using The Ethnologue (Model D). The values of the cross-validation scores are approximately equal to for Models B and C, and around for Model D. These scores provide quantitative evidence that the classification of The Ethnologue leads to better predictions of genetic structure than Greenberg's classification at the stock or group levels.