Inferring admixture proportions from genetic data

We consider a single diploid individual from an admixed population. We assume two haplotypes H₁ and H₂ for this individual are given. Here, a haplotype is a binary vector of length n. n is the number of single nucleotide polymorphisms (SNPs) within the haplotype. Note that in real data H₁ and H₂ are usually inferred from the genotypes G and may have phasing errors. For the ease of exposition, we initially assume the absence of phasing errors in the haplotypes, and then extend the inference framework to allow phasing errors. The admixed population is assumed to be formed by an admixture of two ancestral populations (denoted as populations A and B) g generations ago. For simplicity we assume there are two ancestral populations, although the method can be extended to allow more than two ancestral populations. We further assume allele frequencies in the two ancestral populations are known for all SNPs. Note that allele frequencies from extant populations that are closely related to the ancestral populations are typically available, especially for recent admixed populations. For example, suppose the admixed individual has genetic ancestry in West Africa and Northern Europe. Then we may use the allele frequencies from the extant YRI and CEU populations, available from the 1000 Genomes Project [9], as approximations of the real ancestral allele frequencies. Here, CEU refers to Utah Residents with Northern and Western European Ancestry, and YRI refers to Yoruba in Ibadan, Nigeria. We also assume recombination fractions between every two consecutive SNPs are known. For human populations, recombination fractions are readily available (e.g. [9]).

Likelihood computation on the perfect pedigree model

The perfect pedigree model in [10] can be used to describe the segregation of admixture tracts. Here, an admixture tract is a segment of the genome which originates from a single ancestral population. This model differs from many of the models typically used for inferring admixture tracts of an extant individual (e.g. [11, 12, 13, 14]). This model directly models the segregation of admixture tracts within a pedigree. Most current models assume that the ancestry process follows a Markov chain along the chromosome. However, because of recombination between tracts from multiple ancestors, the exact process does not follow a first-order Markov process [10, 8]. The perfect pedigree model establishes a more accurate, but also much more computationally demanding, model that does not assume a Markov process for the ancestral process, especially for recent admixture events.

Fig 1 illustrates the perfect pedigree model for an extant observed haplotype H at a single site. A perfect pedigree is a perfect binary tree where each node represents a haplotype. All internal nodes in the pedigree are ancestors of H (the single leaf in the pedigree). We trace the ancestry of H backwards in time until reaching the time of admixture, g generations ago. The 2^g haplotypes at this time are called “founder” haplotypes (which themselves are unadmixed but may be from different ancestral populations). Under the assumption of no inbreeding, all ancestors are distinct. Notice that there is an assumption of a single admixture event. However, the model can easily be generalized to multiple admixture events.

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Fig 1. The perfect pedigree model for g = 3 for a single site.

H: extant haplotype. Pedigree is haplotype-based, which models ancestry changes along the genome. Ancestry origins of 8 founders are listed above the perfect pedigree. At one site, A and B indicate which of the two ancestral populations of each founder’s haplotype. The combined vector of these values is C. Here, C = (ABBAAABB). Arrows: the recombination vector R. Here, R = 0111101, where meiosis is ordered the in reverse time order and also from left to right. The population A shown in red: the ancestry of H as traced back by the recombination setting. Arrows can change direction at the next site. Founder ancestry is at a specific sites (say s₁, s₂, s₃, …). Note that founder ancestry at the founders of the pedigree remains the same at different genomic position: these founders are the founding members of the admixed population and they are not admixed themselves.

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There are two main aspects of the perfect pedigree: the ancestry vector C and the recombination vector R. C specifies which ancestral population each particular founder haplotype is from. For example, in Fig 1, C is a vector (ABBAAABB), of length 8. It indicates that the leftmost founder is from the ancestral population A while the rightmost founder is from ancestral population B. As founders are unadmixed, C does not change along the genome. R specifies from which of the two parental haplotypes each descendant haplotype inherits its DNA at a particular genomic position. R is the key component in the well-known Lander-Green algorithm [15]. As shown in Fig 1, one can visualize R as a set of arrows, one for each meiosis, pointing to the left or right. There are seven such arrows in R. We have a list of recombination vectors for n sites (R₁, R₂, …, R_n), where R_i is the recombination vector for site i.

The most obvious method for computing the likelihood P(H|M) of the given haplotype H on the perfect pedigree model is using the Lander-Green algorithm [15] to compute the probability of H for a given C. Then we sum these probabilities over all possible C to obtain P(H|M), assuming each C is equally likely. Here M is a vector of admixture proportions for ancestors of interest in the pedigree. However, computation of P(H|M) directly using the Lander-Green algorithm is not practical for most datasets. This is because first we need to determine the ancestral setting, C, which specifies the ancestral population for each founder. Moreover, the number of possible R grows very fast with the number of generations in the pedigree. Note that the Lander-Green algorithm needs to enumerate all possible R values. Even considering just a single site i, there are possible values of C and possible values of R_i (1 ≤ i ≤ n). These numbers are prohibitively large for e.g. g = 10. To circumvent this problem, we adopt a two-stage model as described below.

A two-stage Markovian pedigree model for genotypes

Our objective is to infer the admixture proportions of ancestors in the perfect pedigree at the K^th generation in the past. Here, K is usually much smaller than the number of generations since admixture. For example, 1^st generation inference (K = 1) is for parents and 2^nd generation inference (K = 2) is for grandparents. The first phase of the two-stage model involves modeling the first K generations in the past using the perfect pedigree model. In the second phase, starting at the K^th generation in the past, there are 2^K ancestors, which are assumed to have ancestry distributions following the standard Markovian model. The ancestry of these 2^K founders can change along the genome following the standard Markovian process. This allows us to model the admixture of recent ancestors (e.g. parents and grandparents) without explicitly considering the entire pedigree.

The model defined so far concerns haploid genomes/chromosomes. However, most real data are from diploid individuals, possibly with unknown or relatively poorly estimated haplotype phasing. We extend the two-stage pedigree model by assuming that each of the two haplotypes from the extant individual has been estimated, but with phasing errors that occur at a constant switch error rate. This leads to a genotype-based perfect pedigree model.

Fig 2A illustrates the genotype-based perfect pedigree at a single position. It consists of two perfect pedigrees, one for each of the two haplotypes H₁ and H₂. Each node in the outline tree denotes an ancestral genotype of the extant genotype G. The two haplotypes H₁ and H₂ of G follow different pedigrees independently. For simplicity, we use a single haplotype with “average” admixture tracts to represent a diploid founder, which works well in practice. Note that the estimated admixture proportion of a founder is the average of the admixture proportions of its two haplotypes. One can view this “average” haplotype as one with the admixture proportion equal to the diploid founder.

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Fig 2. The perfect pedigree model for genotype G = (H₁, H₂) at a site, and K = 2.

Two sources populations: A and B. (A) Outline pedigree in black: the perfect pedigree for genotype G. Two pedigrees embedded in red are for haplotype H₁ and H₂ respectively. Ancestral settings and recombination settings with the same label have the same meaning. (B) The simplified perfect pedigree for genotype G. Ancestral vector C: (ABAB). The arrows without label define recombination vector R. Each Rⁱ is for a meiosis in the pedigree. P: the phasing error setting. Note that different from Fig 1, the founders in this pedigree can be admixed themselves (i.e., ancestry of these founders can change along the genome) based on the two-stage Markovian pedigree model.

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To allow phasing errors between H₁ and H₂, we introduce the phase-switching indicator P. It indicates whether at this position the two haplotypes switch or not. One can visualize P as the arrow labeled by P in Fig 2. A P-arrow pointing to the left indicates that H₁ traces to the left half of the pedigree and H₂ traces to the right half of the pedigree. A P-arrow pointing to the right indicates the opposite. When moving along the diploid sequence (genotype), the direction of P changes when a phasing error occurs. Thus we can combine the two pedigrees for H₁ and H₂ and let the two haplotypes from a single individual collapse into one node, as illustrated in Fig 2B.

The full information regarding the ancestry of a genotype, G = (H₁, H₂), in a fixed pedigree is then given by the ancestral configuration AC = (P, C, R). The sampling probability of G can be computed naively by summing over all possible ACs. The ancestral configuration AC naturally leads to an Hidden Markov Model (HMM) that can be used for efficient calculation of the likelihood.

Let denote a set containing all possible ancestral configurations at site i and AC_i denote an element that belongs to (). In a perfect pedigree of K generations, AC_i = (P_i, C_i, R_i) is a binary vector of 2^K+1 − 1 bits and represents a state at the site i. For each state, P_i has exactly one bit where a “0” (respectively “1”) represents the phasing arrow pointing to the left (respectively right). R_i is a binary vector of 2^K − 2 bits indicating the recombination states associated with all 2^K − 2 meiosis in the pedigree, where “0” (respectively “1”) represents a recombination arrow pointing to the left (respectively right). C_i is a binary vector that indicates the ancestry of each of the 2^K ancestors and contains 2^K bits when there are two ancestral populations. Also, if C[j] = 0 (respectively C[j] = 1) the j-th founder is from the population A (respectively B) at the current site. In the example in Fig 2B, AC = (P, C, R) at this site can be expressed as the binary vector (1, 0101, 10).

We define h(AC_i) as the joint probability of the length-i prefix of G (i.e. G[1..i]) and the ancestral configuration AC_i at the site i. Given a genotype G with n sites, the likelihood . The critical step is the computation of h(AC_i) for each configuration AC_i at the site i. This can be carried out in a recurrence for i ≥ 2 (which resembles the Lander-Green algorithm on the high-level): (1) where p_t(AC_i|AC_i−1) is the transition probability from AC_i−1 at the site i − 1 to AC_i at the site i and p_e(AC_i) is the emission probability of an allele given the ancestral configuration AC_i at the site i. This is the standard forward algorithm for HMMs. Details are given below. Transitions in the HMM may occur between adjacent sites and we assume, for generality, that the configurations at sites i − 1 and i are fully connected as illustrated in Fig 3.

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Fig 3. An example of -based HMM with 128 ACs as states for K = 2 generations (i.e. grandparents).

Arrows: possible transitions along the Markov chain from site i − 1 to i. The vectors under each pedigree provide the binary representations of P, C, and R, respectively, for the pedigree. The two top thick arrows and the lower thick arrow indicate the settings of R and P, respectively.

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Fast computation of sampling probability in PedMix.

The main computational burden in the evaluation of Eq 1 is that the calculation of h(AC_i) requires a multiplication of the transition probability matrix and the vector h(AC_i−1). This leads to a computational complexity of , where N_K is the number of possible states in . This is a significant burden on computation: for example if K = 3, the run time is on the order of 2³⁰. To address this problem, we have developed a divide and conquer algorithm for computing the probability of ACs, which runs in O(N_K log(N_K)) time.

Let P_i denote the probability vector that contains all h(AC_i) for at the site i. Let T_i−1,i denote the transition probability matrix containing the transition probabilities p_t(AC_i|AC_i−1) that one AC at the site i − 1 transits to another AC at the site i. To obtain P_i, we need to compute T_i−1,i P_i−1. Direct computation leads to quadratic complexity.

For simplicity, we omit the site index notation i or i − 1 in T_i−1,i and P_i−1. Let T^b denote the transition probability matrix for AC that has b bits. The AC is represented as a binary vector of length b. Let P^b denote the probability vector for the previous site (i − 1). A bipartition of a matrix is a bipartition of each dimension, which divides a matrix into four sub-matrices with equal size. A bipartition of a vector is a division that equally cuts the vector into two sub-vectors. Fig 4 shows an example of a transition probability matrix T³ and a probability vector P³ for AC with 3 bits. For example, the (2, 3) element in T³ is the transition probability p_t((001)|(010)). The bipartition for T³ and P³ is shown as red lines.

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Fig 4. Faster calculation of the probabilities of ACs.

Red lines break the transition probability matrix into four smaller pieces (for ACs with length 2). The probability vector at the previous site is broken into two pieces. Multiplication of the matrix and the vector is faster due to shared parts between these pieces.

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We observe that each bit in an AC_i−1 transits to a bit in AC_i independently (i.e. the transition probability of each bit in AC_i doesn’t depend on other bits). We use to denote the transition probability of the bth bit from x to y (x, y ∈ {0, 1}). We can adapt the divide-and-conquer approach in [16] to our problem as follows. With bipartition, T^b can be viewed as four sub-matrices, and P^b can be divided into two sub-vectors. The key of the divide and conquer approach is given in the Eq 7. (7)

Each sub-matrix of T^b is equal to T^b−1 multiplied by . Here, T^b−1 is a transition probability matrix where the bth bit of T^b is masked off. For example, the top left sub-matrix of T³ (Fig 4) is equal to and the top right sub-matrix is equal to . Let P^b = (P^b,0, P^b,1) denote the bipartition of the probability vector. Then T^b P^b can be computed by computing T^b−1 P^b,0 and T^b−1 P^b,1. In general, T^b−1 P^b,0 and T^b−1 P^b,1 can then be divided in a similar way until we reach T¹ (masking off b − 1 bits in AC). For the K^th generation inference, each AC has b = 2^K+1 − 1 bits, which leads to N_K = 2^B = 2^2K+1−1 possible states at each site. The divide and conquer scheme reduces computational complexity from to O(N_K log(N_K)). For more detailed explanation of this divide-and-conquer approach, we refer the readers to a related approach in [16].

Probabilistic inference

Maximum Likelihood (ML) inference of admixture proportions can be obtained by maximizing the sampling probability P(G|M) of the -based HMM model: (8)

Let and denote the admixture proportions of the populations A and B respectively for the ancestor j. These admixture proportions are then given by the stationary frequencies of the Markov chain, which according to standard theory are given by (9) and (10) respectively. From the invariance principle of ML, it follows that if and are estimated by ML, the resulting estimates of and are also ML estimates.

To obtain ML estimates of and we apply the Boyden-Fletcher-Goldfarb-Shanno (BFGS) method of optimization. We use an implementation of the limited-memory version of the algorithm, L-BFGS [17], from http://www.chokkan.org/software/liblbfgs. We use the finite difference method for estimating derivatives. We transform bounded parameters using the logit function to accommodate bound constraints.

Preprocessing

There are several aspects of real data that are not considered by our models and may affect the inference accuracy, in particular background Linkage Disequilibrium (LD) and phasing errors. Background LD refers to non-random association between alleles not caused by admixture. Background LD may mislead HMM methods which assume conditional independence among SNPs. As a consequence it may confuse the background LD with the admixture LD. Phasing errors may also introduce an extra layer of noise.

The traditional approach for addressing the problem of background LD is to trim the data sets by removing SNPs. We compare two possible strategies for doing this:

1. Data trimming based on allele frequency differences (frequency-based pruning).
2. Data trimming based on LD patterns (LD pruning).

Frequency-based pruning relies on a trimming threshold d_f, which specifies the minimum allele frequency difference in the two source populations. A SNP site is trimmed if the absolute difference between the allele frequencies in the source populations is smaller than d_f. See Section S1.1.1 in S1 Text for more details on this approach.

In LD pruning, SNPs are removed in order to minimize the LD among SNPs located in the same region. This is the more commonly used strategy implemented in programs such as PLINK [18]. See Section S1.1.2 in S1 Text for more details on LD pruning. The advantage of the first approach is that it keeps the most ancestry informative SNPs in the data set. The advantage of the second approach is that it more directly reduces LD in the data. Both approaches improve inference accuracy and reduce computational time. For frequency-based trimming, for example, the parental inference error without trimming is about 6.5%, while with trimming, the error rate can be about 5% (with 50% threshold). However, our implementation of frequency-based pruning leads to slightly better performance (see Table C in S1 Text), and we, therefore, use this method as the default unless otherwise stated.

Results on simulated data

Simulation settings and evaluation.

We perform extensive simulations to evaluate the performance of our method. The parameters we use in the simulations are listed and explained in Table 1 together with their default values. We first simulate a number of haplotypes using macs [19] from two ancestral populations which diverged from one ancestral population at 4N_et generations in the past. Here N_e is the effective population size. An admixed population is then formed by merging the two ancestral populations and simulating the process of random mating, genetic drift, and recombination using a diploid Wright-Fisher model for g additional generations. We model recombination rate variation using the local recombination estimates from the 1000 Genomes Project [9]. The hotspot maps of the 22 human autosomal chromosomes are concatenated for a single string of 3 × 10⁹ bp and subsequently simulated genomes are divided into 22 chromosomes of equal length to facilitate clearly interpretable explorations of the relationship between accuracy and the amount of data. Haplotypes are paired into genotypes and phasing errors are then added stochastically, by placing them on the chromosome according to a Poisson process with rate p_p. By default, no phasing error is included in the simulations. For the default setting, the approximate total number of SNPs simulated by macs is ∼14.7M. Here we apply frequency-based pruning to trim data. Frequency-based pruning removes SNPs with a minor allele frequency difference in two ancestral populations less than the pruning threshold d_f. After pruning with the default d_f, each of the 22 chromosomes contains ∼26, 000 SNPs. In some cases, the default simulated length L = 3 × 10⁹ bp results in high computational burden. Therefore, in some simulations we also use a shorter length of L = 5 × 10⁸ bp, divided into 3 chromosomes. If not otherwise stated, we use L = 3 × 10⁹ bp to be the default setting.

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Table 1. A list of parameters and their default values used in the simulation.

https://doi.org/10.1371/journal.pcbi.1008065.t001

Default settings are chosen to be some reasonable values. For example, for g (the number of generations since admixture), ten generations roughly correspond to 200 to 300 years in humans, where there are some significant events related to admixture in human history within this time frame. We acknowledge that the choice of parameters can affect the accuracy. To explore this issue further, we have made additional simulations, where we show that simulation parameters may indeed affect accuracy. For example, when g increases, accuracy gets somewhat better.

PedMix is applied to the simulated genotype data from the admixed population for inference of admixture proportions of ancestors in the 1st generation (parents), the 2nd generation (grandparents) and so on. To evaluate accuracy, we use the mean absolute error (MAE) between the estimated admixture proportion, , and the true admixture proportion, mⁱ, for the ith ancestor in the K^th generation, as the metric of estimation error. If there are multiple individuals, we further take the average over all individuals, i.e. the mean error for n individuals is defined by Eq 11. Without loss of generality, we only consider the estimate of the proportions of the ancestral population A. Because we assume two ancestral populations, the expected mean errors of admixture proportions for two ancestral populations are identical. (11)

As the admixture proportions inferred by the method are unlabeled with respect to individuals, this leads to ambiguity on how to match the inferred proportions to the true proportions for ancestors. We address this problem using a “best-match” procedure by rotating the parents for each internal node in pedigree to find the best match between the inferred and the simulated admixture proportions. For example, for inference in parents, we have true admixture proportions for two parents (m¹, m²) and estimated admixture proportions . We match both and to (m¹, m²) and choose the one with smaller mean error. For the case of grandparent inference, there are eight possible matchings and we explore all eight to obtain the best match.

Evaluation of ancestral inference accuracy.

Fig 5 shows the mean error when inferring admixture proportions of parents, grandparents, and great grandparents under the default simulation settings (as shown in Table 1). We compare the performance of PedMix to what is expected from random guess based on a Bayesian model. The random guess is described in Section S1.3 of S1 Text. We sample 10 genotypes from simulated admixed population for g (g ≥ 3) generations.

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Fig 5. Comparison between the accuracy of PedMix and a random guess for parents, grandparents and great grandparents.

About 570,000 SNPs are used for parent and grandparent simulations. About 26,000 SNPs are used for great grandparent simulations since this case needs more computing resources.

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Note that inference of great grandparents admixture proportions is computationally demanding in the current framework. Therefore, we use a more extreme trimming threshold, d_f = 0.9, when inferring great grandparent admixture, resulting in only 26,638 SNPs.

As expected, it is easier to estimate admixture proportions of more recent ancestors. This is because, as we trace the ancestry of a single individual back in time, the genome of the extant individual contains progressively less information about an ancestor.

Comparison of PedMix to existing methods.

Although there are no existing methods for inferring the admixture proportions of grandparents and great grandparents that we can compare PedMix to, there is a method called ANCESTOR [7] that infers admixture proportions of parental genomic ancestries given ancestry of a focal individual. And there are many methods (e.g., ADMIXTURE [6] and RFMix [20]) for inferring admixture proportions of individuals of the current generation. In this section we first compare estimates of the admixture proportions of a focal individual obtained from ADMIXTURE and RFMix, arguably the state-of-the-art methods for ancestry inference, to the average of parental or grandparental admixture proportions inferred using PedMix. Here, we use the average of the estimated admixture proportions from ancestors as the proxy for the admixture proportion of the focal individual. If the admixture proportions of ancestors inferred by PedMix are accurate, we would expect the average of these admixture proportions of ancestors can serve as a good approximation for the focal individual. And this average should be approximately as accurate as the admixture proportions inferred by RFMix and ADMIXTURE. This is verified with simulation data in Section S1.6.1.1 of S1 Text. We note that the high accuracy of PedMix in inferring the admixture proportion of a focal individual from the average of parental or grandparental proportions does not necessarily imply that the parental and grandparental admixture proportions themselves are accurately inferred. However, if the admixture proportion of a focal individual is poorly estimated from the inferred admixture proportions of the ancestors, this may suggest that the admixture proportion estimates for the ancestors also are not accurate.

We randomly sample 20 individuals from an admixed population and run ADMIXTURE, RFMix and PedMix on the same datasets. The genotypes are preprocessed with LD pruning (see Section S1.1.2 of S1 Text) and contain phasing errors simulated with rate p_p = 0.00002 per bp. We deduce the ancestry of each individual using PedMix from the inferred admixture proportions of either parents or grandparents, by using the average of the inferred admixture proportions of the ancestors. ADMIXTURE and RFMix infer the admixture proportions of extant individuals directly. More details on how ADMIXTURE and RFMix are applied are given in Section S1.6.1.2 of S1 Text. Table 2 shows the mean error as defined in Eq 11 and the error’s standard deviation. Our results show that the admixture proportions inferred from the average of ancestral admixture proportions in PedMix are comparable to those of RFMix and ADMIXTURE. The estimate by parents matches (within one standard deviation) the results of RFMix and ADMIXTURE. The estimate by RFMix and ADMIXTURE is slightly better than the estimate by grandparents. Note that the simulation setting tested here is only one out of many possible simulation settings.

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Table 2. Mean and standard deviation of the error in the estimate of admixture proportions for ADMIXTURE, RFMix and PedMix (in units of %).

Note the PedMix results are the average proportions from the estimated admixture proportions of parents (denoted as par.) or grandparents (denoted as grandpar.).

https://doi.org/10.1371/journal.pcbi.1008065.t002

We further compare the estimates of admixture proportions of parents from PedMix to those from ANCESTOR. ANCESTOR requires that the ancestry states and the tract lengths are provided as input for the algorithm. To accomplish this, we use the ancestry tracts inferred by RFMix when running ANCESTOR. More details on how ANCESTOR is applied are given in Section S1.6.1.2 of S1 Text. Mean error is computed between the true admixture proportions of parents and the estimates from ANCESTOR or PedMix (Table 3). Estimates by PedMix are more accurate than ANCESTOR (approx. 6.5% error versus 9.6% error).

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Table 3. Mean and standard deviation of the error in the estimate of admixture proportions of parents from ANCESTOR and PedMix (in units of %).

https://doi.org/10.1371/journal.pcbi.1008065.t003

Impact of simulation parameters.

We perform additional simulations to investigate the impact of various simulation parameters on the accuracy of our method. To investigate the effect of mutation rates and recombination rates, we use the default setting with a shorter genome of length L = 5 × 10⁸ (Table 1) to reduce the computational time (Fig 6). The expected number of SNPs simulated in a region increases linearly with the mutation rate. This leads to a reduction in the mean error with increased mutation rates, as more informative markers are available for analysis (Fig 6A). However, the reduction is modest because the statistical accuracy is mostly limited by the number of admixture tracts and not by the number of markers. In contrast, recombination rate has a much stronger effect on the accuracy than mutation rate because increased recombination rates introduce more admixture tracts (Fig 6B, and also Fig D of S1 Text). The mean error for both parental and grandparental inferences decreases as recombination rate increases. When the recombination rate increases to more than 5 × 10⁻⁸, the improvement in accuracy becomes smaller, especially for parental inference (also see Section S1.6.3 of S1 Text). As the length of each tract decreases, the information regarding the ancestry for each tract also decreases. Even with very high recombination rates, there may still be some error determined by the degree of genetic divergence between populations and the number of generations since admixture.

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Fig 6. Mean error for different simulation parameter settings.

(A) Varying mutation rates. L = 5 × 10⁸. (B) Varying recombination rates. L = 5 × 10⁸. (C) Varying the time since admixture. (D) Ancestral population split time. For t = 0.01, there are no SNPs left after using the d_f = 0.5 cut-off. As a result, we use d_f = 0.2, leaving 74,227 SNPs for analysis. Default parameters are used except for the variable indicated by the X axis of each plot.

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The simulations assume a model of two ancestral populations that diverged 4N_e t generations ago and then admixed g generations ago. The performance of the method clearly depends on these parameters. If g is small, the number of admixture tracts is also small, complicating inferences, particularly in the grandparental generation. As g increases from 4 to 10, the mean error reduces from 7.13% to 4.47% for parent inference and from 15.34% to 7.36% for grandparents respectively (Fig 6C). There is also a strong effect of t on the accuracy. As t increases, allele frequency differences between the admixing populations increase and it becomes easier to distinguish admixture tracts from two ancestral populations (Fig 6D). When t > 0.5 the mean error for parental inferences drops to below 1%.

Phasing error.

Real data may contain phasing error. We have implemented a preprocessing approach for reducing the phasing error. See Section S1.2 of S1 Text for details. To evaluate the effect of phasing error, we simulate data with the phasing error rate 2 × 10⁻⁵ per bp. We then compare the mean error by PedMix using genotype data without phasing errors, genotype data with phasing errors, and genotype data with phasing errors preprocessed to remove some phasing errors. As for the data generated with phasing error rate 2 × 10⁻⁵, we run PedMix directly without preprocessing. Then we use the technique described in Section S1.2 of S1 Text to preprocess the data.

As shown in Fig 7A, phasing error reduction by preprocessing increases the inference accuracy. Note that at the default setting, the phasing error rate is nearly 2,000 times larger than the recombination rate, which can affect the accuracy of the method significantly. Thus, in real data analyses, it is important to reduce the phasing error in some way.

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Fig 7. Inference error vs phasing error: Comparison among three different datasets.

These include: dataset without phasing errors, dataset with phasing errors and preprocessed datasets. (A) Samples from an admixed population. (B) Samples from an unadmixed population.

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We also consider the case of unadmixed individuals. In that case, phasing errors may be interpreted as recombination events during inference, but the overall admixture proportion estimates should be relatively unaffected. To illustrate this point, we sample 10 individuals from the same population and run PedMix to infer their admixture proportions. The performance of PedMix is very stable as shown in Fig 7B, with an inference error of approximately 3% for parents and grandparents.

Phasing error adds noise to the model, especially in the region where the two haplotypes have different ancestral states. As we decrease the effect of the phasing error in the data using preprocessing, the inference error decreases significantly.

Results on semi-simulated data

We now show results on semi-simulated data. Here, we use genotypes of CEU/YRI/ASW populations from the 1000 Genomes Project as the founders of a fixed pedigree topology as shown in Fig 8. Here, ASW refers to Americans of African Ancestry in SW USA. This way, the genotypes are closer to the real data and we know the origin of the founders. For this pedigree of two generations, we select four genotypes from one or more populations among CEU, YRI and ASW populations as grandparents. We assume there is no phasing error along these grandparental genomes. Then we simulate two genotypes as parents and one genotype as the focal individual based on the pedigree with recombination. Recombination rate is modeled from the hotspot maps of the 1000 Genomes Project as in the other simulations. To assess the impact of phasing errors, we also create data with phasing errors by adding phasing errors stochastically with the rate p_p = 0.00002 per bp for the focal individuals. We run PedMix on the genotype of the focal individual genotype with or without phasing error to infer the admixture proportions of parents and grandparents. RFMix is run to estimate the admixture proportions of parents (respectively grandparents) using the genotypes of parents (respectively grandparents). We use the estimates from RFMix with the ancestral genotypes as the ground truth on the admixture proportions of ancestors. This is because these ancestors are from real data and their true admixture proportions are not known. Here we examine six cases with different ancestral origins of the grandparents: CCCY, CCYY, CYCY, AAAA, AAAC and AACY (where C is for CEU, Y is for YRI and A is for ASW). As an example, CCCY stands for the four grandparents from CEU, CEU, CEU and YRI respectively. Fig 8 shows the estimates by RFMix and PedMix. Mean error is computed from the six inferred admixture proportions (two parents and four grandparents) in the pedigree and their estimates by RFMix. Although the focal individuals in the pedigrees CCYY and CYCY both have around 50% admixture proportion, PedMix is able to tell the difference in the parents by estimating the parental admixture proportions being 82.41% and 16.43% for CCYY and 47.05% and 43.78% for CYCY. This largely agrees with the true admixture proportions of the parents, which are 99.91% and 0.09% for CCYY and 50.04% and 49.96% for CYCY. Note that the true admixture proportions for the two parents are known for a pedigree with the known grandparental origin. For example, in the CCYY case the true parental admixture proportions are 99.91% and 0.09%. This is because one parent has two CEU grandparents and thus this parent is about 100% CEU. Similarly, the other parent is around 100% YRI. This indicates that PedMix is able to collect useful information from the admixture tract lengths in the focal individual. Results on genotypes without phasing errors tend to be more accurate than those with phasing errors. Our results indicate that phasing errors can indeed lead to larger inference error for some cases. Thus, it is useful to use haplotypes with less phasing errors. Estimates for parents are more accurate than those of grandparents.

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Fig 8. Six pedigrees used for the simulations scheme of semi-simulated data.

The percentage of CEU origin is shown in the pedigrees. Admixture proportions in black are estimates by RFMix using the ancestors’ genotypes directly and are assumed to be the true proportions. Admixture proportions in blue and red are estimates by PedMix using genotypes of the focal descendant individual with (red) or without (blue) phasing error. Mean error is the average over two parents and four grandparents.

https://doi.org/10.1371/journal.pcbi.1008065.g008

Results on real data

To evaluate the performance of PedMix on real genetic data, we run PedMix on the haplotypes of ten trios from the ASW population from the HapMap data [21]. Phased genotypes of the parents are available from parents, while children’s genotypes are unphased. We use the program Beagle [22] to phase children’s genotypes from the phased haplotypes of the parents. Note that the exact ancestry of parents and grandparents in these trios are not known. Here, for parents, we use the inferred admixture proportions by running RFMix on the given haplotypes of parents as the true admixture proportions of the parents. The case of grandparents is more difficult because genetic data of grandparents is not available from the HapMap project. In order to evaluate the performance on grandparents, we adopt the following indirect approach: for each trio, we run PedMix on the child’s haplotypes to infer grandparents’ ancestry; we also run PedMix on the parents’ haplotypes to infer grandparents’ ancestry; we then examine whether the two inference results are consistent. That is, we view the inferred grandparents’ admixture proportions from parents as proxy of the true grandparental proportions. While this is not a direct evaluation, we believe it at least offers some hints on how well grandparental inference is likely to perform on real data.

We use the YRI and CEU populations as the two ancestral populations. Allele frequencies of these two ancestral populations and genetic maps from the HapMap project are used. We preprocess the data using the frequency-based trimming (default threshold is 0.4). We apply the phasing error removal procedure as described in the Methods Section. We use the reduced phasing error rate (see Section S1.2 of S1 Text for details) when running PedMix. Fig 9 shows the average admixture proportion inference error for parents and grandparents over ten trios. We show results for individual trios and also the average error. The average inference error for parents is about 3.4%, while the difference between the two inference results for grandparents is about 13%. Accuracy for most trios is relatively small, especially for parental inference. But for grandparent inference, the difference between the two estimates is relatively large for some trios. Fig 9 shows the parental/grandparental inference error with different amount of data (in terms of the number of chromosomes used). Our results show that inference error decreases when the amount of data increases, although sometimes the decrease in inference error is not very large. Overall, our results show that PedMix performs reasonably well on the real data.

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Fig 9. Admixture proportion inference error for parents and grandparents for ten HapMap trios from the ASW population.

Left: inference error for ten HapMap trios individually or on average. Right: inference error with different amounts of data. X-axis: the number of chromosomes used. Y-axis: parental/grandparental inference error. Parental: the difference between PedMix’s parental inference for the child and RFMix for the parents’ haplotypes. Grandparental: the difference between PedMix’s grandparental inference for the child and PedMix’s parental inference for the parents’ haplotypes in the trio.

https://doi.org/10.1371/journal.pcbi.1008065.g009

We also compare PedMix with ANCESTOR on real data. The results are given in Section S1.6.7 of S1 Text.