The IBDNe method

We first summarize the key points of the IBDNe method for estimating the effective population size of homogeneous populations (further details may be found in [14]), and then describe how this method can be applied to ancestry-specific effective size in admixed populations.

Consider a population with an effective size of N_g diploid individuals g generations before the present (generation 0 is the current generation, generation 1 is the most recent previous generation, and so on). If q_g is the probability that a pair of haplotypes randomly sampled from the current generation coalesce (have most recent common ancestor) at generation g given that they have not coalesced by generation g − 1, then N_g = 1/(2q_g) by definition (see Introduction). Suppose we have a sample of individuals from the current generation, and we identify long segments of identity by descent between pairs of haplotypes drawn from different individuals, finding all those segments of identity by descent that exceed some length threshold C. In most settings, we use C = 2 cM, since we have high power to detect such segments using existing methods such as Refined IBD [17]. Then, if the past trajectory of effective size N = {N_g: g ≥ 1} was known, we could calculate E_g, the expected total length of detected IBD genome-wide that is attributable to most recent common ancestry at generation g. Also, if the effective size trajectory N was known, we could calculate the probability that an IBD segment of length l is due to most recent common ancestry at generation g, and hence obtain O_g, the expected total length of detected IBD genome-wide that is attributable to most recent ancestry at generation g, conditional on the observed IBD segments. Both E_g and O_g are functions of N. Finding values of N_g that give equal values of E_g and O_g provides a methods of moments estimate of N. We iterate estimating N and re-calculating E_g and O_g until convergence of our estimate of N. During the iterative estimation process we also impose a smoothness requirement on N_g as a function of g to aid in the estimation.

Now consider an admixed population, in which local ancestry has been determined. The ancestry-specific effective size for ancestry a considers only those haplotypes that are descended from ancestry a. If is the probability that a pair of such haplotypes randomly sampled from the current generation coalesce at generation g given that they have not coalesced in generations 1 to g − 1, then . For generations prior to the admixture event, this ancestry-specific effective size represents the total effective size of the ancestral population that contributed to ancestry a in the admixed population. For example, if considering European ancestry, and the European ancestors came from some population in Spain, the pre-admixture European-specific effective population size will be the effective size of that population in Spain. Our interest in the ancestry-specific effective population size is mainly for the pre-admixture effective population sizes, but we also obtain estimates of post-admixture ancestry-specific effective population sizes. If the post-admixture population is randomly mating, and has proportion p^(a) of its ancestry being of ancestry a, then it is straightforward to show that where N_g is the overall effective size of the admixed population. If there is assortative mating or ongoing migration, this relationship will not hold. We now discuss how to estimate using the IBDNe framework.

The IBDNe framework needs the following information: the IBD lengths, in order to obtain O_g; and the number of pairs of sampled haplotypes, which are needed to obtain E_g. With a homogenous population, we obtain the IBD lengths directly from the detected IBD segments, and we calculate the number of pairs of sampled haplotypes from the number of sampled individuals. With ancestry-specific analysis, there are differences because we are only interested in IBD between haplotypes of the given ancestry, but ancestry is not constant along the genome. This affects both the way in which the IBD lengths are handled, and the way in which the number of pairs of sampled haplotypes is calculated. Some IBD segments in an admixed population will have a very recent common ancestor (from the generations post-admixture), and since this ancestor is admixed, the IBD segment may include more than one ancestry. Only those parts of the segment that are derived from the ancestry of interest will contribute to the total length of detected IBD for this ancestry, but we still need to know the length of the whole IBD segment in order to calculate the probability that the most recent common ancestor lived in generation g (see Methods). Further, the number of pairs of sampled haplotypes of the given ancestry now varies from one genomic position to another, because the ancestry of each individual’s DNA varies along the genome, but the expected number of pairs of sampled haplotypes of the given ancestry can be calculated from the genome-wide ancestry proportions (see Methods). Apart from these differences, estimation of ancestry-specific effective population size is the same as for estimation of overall effective population size, and the existing IBDNe software may be used. An example analysis pipeline is provided at http://faculty.washington.edu/sguy/asibdne/.

Accuracy in estimating ancestry specific effective population size

We simulated an admixed population with ancestry from three continental groups in order to test the accuracy of our methods. The simulation includes a low rate of genotype error (0.1%) and the simulated data have a marker density that is similar to a 1M-feature SNP array. A full description of the data simulation can be found in Methods (in the “Simulated data” section). Briefly, we used a coalescent-based simulator to simulate the Africa-Europe-Asia demographic history estimated by the 1000 Genomes Project [18], and added population bottlenecks and admixture occurring 12 generations ago.

Fig 1 shows true and estimated ancestry-specific effective population size. We see that important aspects of the effective population size trajectory are represented in the estimated trajectories, including the approximate timing of the population bottleneck, the approximate size of the ancestral population, and the effective size of the ancestry-specific fraction of the admixed population after the bottleneck. The bootstrap confidence intervals do not always cover the true value, but they provide an approximate measure of the precision of the estimates.

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Fig 1. Estimated ancestry-specific effective population size in simulated data.

Analysis of 500 simulated individuals from a three-way admixed population. Each column is one of the three simulated ancestries. The y-axes show ancestry-specific effective population size (N_e), plotted on a log scale. The x-axes show generations before present. The dashed lines show simulated effective population sizes. The solid black lines show estimated ancestry-specific effective population sizes, and the gray regions show 95% bootstrap confidence intervals.

https://doi.org/10.1371/journal.pgen.1007385.g001

In order to keep the simulation from becoming overly complex, the simulated population size decrease associated with migration occurs instantaneously, resulting in a sharp bottleneck in population size. The effective population size estimation procedure cannot fully capture this sharp change because it applies smoothing by fitting exponential growth curves to groups of 8 generations. In real life admixture events, such as that associated with the colonization of the Americas, we would not expect population changes to occur instantaneously. Rather, migration and population decline would have taken place over the course of several generations, resulting in smoother effective size trajectories.

We simulated two admixture scenarios that don’t include bottlenecks. One has the three populations merging without population size reductions, while the other has continuous migration from two of the populations into the third. We estimated ancestry specific effective population sizes and show the results for the merging scenario in S1 Fig and for the continuous migration scenario in S2 Fig. Because these scenarios don’t include large population size changes over short time intervals, the estimates closely match the underlying true values, and the bootstrap intervals mostly cover the true values.

We also simulated an admixture scenario with recent population structure, with or without biased sampling across the sub-populations, and show the results in S3 Fig. With unbiased sampling the results are similar to those for the main simulation without structure (Fig 1), while with biased sampling the ancestry-specific effective sizes are underestimated in the first few generations.

Ancestry-specific effective population size in the HCHS/SOL populations

We group individuals in HCHS/SOL into populations based on the reported country-of-origin of their grandparents. Individuals with missing grandparental origins or grandparents from different countries are omitted from the analysis. Fig 2 shows estimated ancestry-specific effective population sizes and overall effective population sizes for the HCHS/SOL populations for the past 100 generations. S1 Table shows average total length of detected IBD segments shared by unrelated pairs of samples, which is a summary of the data used to estimate the overall effective population size. Table 1 gives sample sizes for each population. When the overall sample size is low, the amount of data for estimating the overall effective population size is low and the estimates will have a high level of uncertainty. Similarly, when the average genome-wide ancestry proportion multiplied by the sample size is low, the amount of data for estimating the ancestry-specific effective population size is low. The bootstrap intervals give approximate measures of precision of the estimates. Estimates with wide intervals should be disregarded because the bootstrap intervals may not capture the full extent of the uncertainty in these estimates. The widest intervals occur when the sample size and/or average ancestry proportion is lowest. Wide pre-bottleneck confidence intervals are also seen for Puerto Rico, despite its high sample size, due to the extremely small bottleneck that occurred in this population. The small bottleneck means that many of the possible coalescences between haplotypes occurred in the post-bottleneck period, leaving few independent haplotypes to provide information about the pre-bottleneck period. In consequence, we recommend against drawing conclusions about the pre-bottleneck sizes of the populations ancestral to Puerto Ricans from this analysis.

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Fig 2. Estimated ancestry-specific effective population size in HCHS/SOL data.

The y-axes show ancestry-specific effective population size (N_e), plotted on a log scale. The x-axes show generations before the present. The lines show estimated ancestry-specific effective population sizes, and the colored regions show 95% bootstrap confidence intervals.

https://doi.org/10.1371/journal.pgen.1007385.g002

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Table 1. HCHS/SOL populations.

https://doi.org/10.1371/journal.pgen.1007385.t001

The results for Puerto Rico show an apparent severe drop in overall effective population size in the most recent couple of generations. This is an artefact resulting from an excess of relatives in the Puerto Rican sample, particularly at the level of 2^nd to 3^rd cousins. Although we exclude IBD from relatives that are half-siblings or closer (including parent-offspring pairs and full siblings), it is not straightforward to exclude IBD from more distant relatives without the use of pedigree information, and the IBDNe program is not designed to allow for the removal of these more distant relatives. Inclusion of these relatives in the analysis results in an excess of long IBD segments which depresses the estimate of effective population size in the last few generations. This effect may also somewhat depress estimates of effective population size in the last few generations in the other HCHS/SOL populations, though clearly not to such an extreme as for the Puerto Rican sample.

Most of the populations and ancestries show a clear population bottleneck around 12 generations ago. Colonization began earlier, around 17 generations ago (approximately 500 years ago, assuming 30 years per generation), but occurred over the course of multiple generations. Overall bottleneck sizes vary considerably across the populations, ranging from 1,000 for Puerto Rico to 60,000 for Mexico. Growth in overall effective population size subsequent to the bottleneck is estimated to have been very high in all the populations, with estimated current effective sizes in the hundreds of thousands or millions (Table 2).

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Table 2. Maximum and minimum estimated effective population sizes (95% confidence intervals, in thousands).

https://doi.org/10.1371/journal.pgen.1007385.t002

When interpreting the drop in ancestry-specific effective population size at colonization, one must consider population structure. Populations are not closed units, since there is always migration between regions. When one considers the effective size of a “population”, one is considering the effective number of ancestors of the individuals in that population. For example, if the population is a village in a region with low migration, most of the parents and grandparents (corresponding to effective size at generations 1 and 2, respectively) will be derived from that village and the effective size will reflect the effective size of the village. However, when one looks back 20 generations, many of the ancestors may have come from nearby villages, and the effective size will reflect the effective size of the region containing those villages. Thus the effective size for the village may be lower in recent generations than in more distant generations, even if the census population size in the village and region has been stable. This effect was noted in a previous IBD-based analysis [15]. In the context of HCHS/SOL, the effective size 20 or so generations before admixture may reflect larger regional effective sizes while the immediate pre-bottleneck sizes reflect smaller sub-populations. Thus the changes in size between the maximum pre-admixture effective population size and the bottleneck effective population size (Fig 2 and Table 2) reflect these population structure effects as well as the effects of colonization.

Fig 3 shows selected population-ancestry combinations for which the precision is relatively high. African-American and European-American populations from Memphis are included for comparison. Considering each ancestry in turn, we see similarities and differences between populations in the estimated pre-admixture effective population sizes. In the African component, we see smaller estimated pre-admixture effective sizes for Cuba (150,000) and Mexico (100,000) than for the Dominican Republic (700,000), suggesting that the African ancestors of the former two populations came from smaller sub-populations of Africa than the African ancestors of the latter two populations. In the European component we see smaller estimated pre-admixture effective sizes for Cuba (200,000), Mexico (150,000), and Nicaragua (120,000) than for the Dominican Republic (400,000). In the American ancestral component, the estimated pre-admixture effective sizes are similar between Nicaragua (400,000), Ecuador (700,000), and Mexico (600,000).

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Fig 3. Ancestry-specific effective population size for selected populations.

The y-axes show ancestry-specific effective population size (N_e), plotted on a log scale. The x-axes show generations before present. The lines show estimated ancestry-specific effective population sizes, and the colored regions show 95% bootstrap confidence intervals. Each plot shows a different ancestral component. HCHS/SOL populations are included if the sample size multiplied by the average genome-wide ancestry proportion for the given ancestry in that population is at least 100. African ancestry for African American (AA) in Memphis, and European ancestry for European American (EA) in Memphis are included for comparison.

https://doi.org/10.1371/journal.pgen.1007385.g003

We can also look at the relative magnitude of the bottleneck population sizes to the pre-admixture sizes, bearing in mind the potential effects of population structure discussed above. For the African and European ancestral components, the drops in size are presumably mostly related to founder effects induced by migration. In contrast, for the American ancestral components, the drops in effective population size are likely due to the negative impacts of colonization including war and disease. Mexico had a relatively smaller estimated reduction in American-specific effective population size compared to the other populations (Fig 3 and Table 2).

In interpreting these results, it is important to recognize that the sampled individuals were residents of four major cities in the United States. Thus the results apply to those particular urban populations, and not necessarily to the entire countries-of-origin represented. If the populations in the US are derived from regional subsets of the countries-of-origin, the estimated effective sizes will be smaller than would be found for the countries as a whole if one had samples of individuals drawn randomly from those countries. This would be expected to have a significant influence on the estimated effective size of the most recent generations, and less influence on more distant generations due to mixing within the population over time.

Ancestry-specific effective population sizes in the Health ABC populations

In order to further investigate the demographic history of US populations we analyzed data from the Health ABC study, which is comprised of samples from Memphis and Pittsburg (Table 3). As in the HCHS/SOL populations, the Health ABC populations show significant growth in the most recent generations, as expected. The overall bottleneck effective sizes for the Memphis populations and for the Pittsburgh African-American population are 130,000–190,000 which is more than twice as large as those for any of the HCHS/SOL populations. The pre-admixture African-specific effective population sizes for the Memphis and Pittsburg populations are 1–2 million, and thus are higher than those for HCHS/SOL populations (Fig 3). The pre-admixture European-specific effective sizes for the African-American populations and for the Memphis European-American population are around 1 million, and thus are also higher than those for most of the HCHS/SOL populations.

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Table 3. Health ABC populations.

https://doi.org/10.1371/journal.pgen.1007385.t003

The estimated ancestry-specific effective population sizes for the African-American populations in these two cities are similar to each other (Fig 4). The similarity of the estimated demographic histories of the Memphis and Pittsburgh African-American populations suggests significant historical mixing within the larger African ancestry population that encompasses these cities, so that the two populations have a shared demographic history. In particular, the similarity of the estimated demographic histories of the European ancestry component is consistent with previous analysis of genetic data from African Americans [19], which indicates that most of the European admixture in US African American populations occurred in the southern US prior to the Great Migration movement of African Americans from the South to elsewhere in the US.

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Fig 4. Estimated effective population size in two US cities.

The y-axes show ancestry-specific effective population size (N_e), plotted on a log scale. The x-axes show generations before present. The solid lines show estimated effective population sizes, and the colored regions show 95% bootstrap confidence intervals. Overall effective sizes are shown for the African American (AA) and European American (EA) populations, as well as ancestry-specific effective sizes for African and European ancestry in the African-American populations.

https://doi.org/10.1371/journal.pgen.1007385.g004

In contrast, the demographic histories of the European-American populations in Memphis and Pittsburgh differ, both before and after the founding bottleneck. Prior to the founding bottleneck, the effective population size of the Memphis European-American population was similar to that of the European component of the two African-American populations, suggesting that the European ancestors of these populations were drawn from the same European source, which again is consistent with the European admixture in the African-American populations having occurred in the South. Memphis European-Americans have a higher estimated current effective size than Pittsburgh European-Americans, which contrasts with a previous report of more long segments of IBD between European Americans in the South than in the Northeast [19]. Urban areas often differ from the general population in having more immigrants, both domestic and foreign. During the period 1910–1950, Memphis grew rapidly, tripling in size from 130 thousand to 400 thousand, while Pittsburgh’s population only increased slowly, from 530 thousand to 680 thousand [20]. Since 1950, Memphis’s population has continued to grow, while Pittsburgh’s population has declined. Thus, it is likely that Memphis’s European American Health ABC population has more diverse geographical origins on average than Pittsburgh’s European American Health ABC population, leading to the larger current effective population size.

A striking difference between the two demographic histories is that the estimated bottleneck size in Pittsburg European-Americans is significantly smaller than that in Memphis, and the timing of the bottleneck appears to be earlier (> 20 generations ago versus around 10 generations ago), which is earlier than the timing of European migration to North America. Also the decrease in population size approaching the bottleneck is very gradual, which suggests that the ancestors came from sub-populations that had quite slow rates of mixing with the broader European ancestral population. One possible explanation that could fit both of these characteristics is that many of the ancestors of the European-American population in Pittsburgh may have been members of groups that formed relatively small separated populations within Europe prior to their migration to the US. An example of such an ancestral group that fits the historical record would be the Anabaptists (including Mennonites and Amish), who separated from other European populations around 500 years ago and migrated to the US in large numbers to escape religious persecution. Many of these Anabaptists settled in Pennsylvania [21]. Although this is one possible explanation, it is not the only possibility, and our data do not address the question of origins.

Simulated data

We used msprime [26] to simulate three continental populations (Africa, Europe and Asia) with recent admixture. Our pre-admixture model is based on a published model inferred from the 1000 Genomes project data [18]. The initial population size (in Africa) was 7310, which increased 5920 generations ago to 14474. The out of Africa event occurred 2040 generations ago, with the out-of-Africa population size being 1861, and the migration rate between Africa and out-of-Africa being 1.5 × 10⁻⁴ (migrant proportion of population, per generation). The Asian and European populations split from the out-of-Africa population 920 generations ago, with sizes of 1032 for Europe and 554 for Asia, and growth rates of 3.8 × 10⁻³ for Europe and 4.8 × 10⁻³ for Asia (the African population size remained constant at 14474). Migration rates post-split were 2.5 × 10⁻⁵ between Africa and Europe, 7.8 × 10⁻⁶ between Africa and Asia, and 3.11 × 10⁻⁵ between Europe and Asia [18].

We create an admixed population with admixture occurring 12 generations ago. The admixed population had an initial size of 30,000 and grew at a rate of 5% per generation, with 1/6 of the population of African ancestry, 1/3 European, and 1/2 Asian.

We simulated sequence data with a mutation rate of 1.25 × 10⁻⁸ per basepair per meiosis, and a constant recombination rate of 1 × 10⁻⁸ per basepair per meiosis (i.e., 1 cM = 1 Mb). We simulated 500 individuals from the simulated admixed population, and 100 reference individuals from the three ancestral populations (Africa, Europe, and Asia). We simulated 30 chromosomes each of length 100 Mb. Our simulation code can be found in S1 Source Code.

After simulating the data, we removed all variants with minor allele frequency less than 5%, and 70% of the remaining variants. After removing these variants, 926,159 variants remained for analysis across the 30 simulated chromosomes, so that the data are similar to those on a 1M SNP array. We then added genotype error: 0.1% of genotypes were randomly chosen to have a random allele changed.

We followed the same analysis pipeline as for the real data, using RFMix [8] to infer local ancestry, Refined IBD [17] with a gap filling procedure to infer segments of identity by descent of length 2 cM and longer, and IBDNe [14] to estimate ancestry-specific effective population sizes. Further details are given below. The analysis pipeline can be found at http://faculty.washington.edu/sguy/asibdne/.

In addition, for the simulated data we needed to obtain the ground-truth effective population sizes for comparison with the estimated values. It is not straightforward to determine the populations’ effective sizes directly from the simulation parameters. So we simulated coalescence trees from the same simulation model as in the main simulation, and we utilized the known coalescence times and ancestral origins to determine the ground-truth ancestry-specific effective sizes. Specifically, we simulated 5000 replicate trees, each with 10,000 haplotypes sampled from the current-day admixed population. For each sampled haplotype, we determined its ancestral origin by tracing its path back through the tree to the first pre-admixture coalescence node. Considering a given ancestral population and a given generation (g < 100) before present, in simulated tree i we determined the number of branches, b_i, that are ancestral to haplotypes derived from the given ancestral population. Any pair of branches could coalesce, so the number of opportunities for coalescence is b_i choose 2, or b_i(b_i − 1)/2. We also determined the actual number of coalescences occurring in this generation for this ancestry population, c_i. Combining results across the simulated trees, we obtained the conditional coalescence rate . The large number of simulated trees (5000) and large number of sampled haplotypes per tree (10,000) result in an estimated that is close to the true value q_g. We then obtained the ancestry-specific effective size at generation g from the relationship N_g = 1/(2q_g).

Estimating IBD segments in heterogeneous data

In our previous development of IBDNe [14] we used IBDseq [27] to detect IBD segments because we were analyzing homogenous data and because assignment of an IBD segment to a particular haplotype was not necessary. However, IBDseq overestimates IBD segment lengths in heterogeneous data such as the admixed data analyzed in this study, because it does not account for linkage disequilibrium induced by population structure. The haplotype-based IBD detection method Refined IBD [17] is robust to genetic heterogeneity because the requirement that two haplotypes are identical over an extended region is very strong, and because haplotype frequencies estimated using appropriate approaches such as the Beagle model account for linkage disequilibrium. Further, Refined IBD assigns IBD segments to individual haplotypes which is necessary for determining the local ancestry within the IBD segment, whereas IBDseq does not assign phase to the IBD segments. We used the Beagle 4.1 version of Refined IBD with default settings unless otherwise noted, and we applied a 2 cM IBD segment length threshold.

In order to successfully apply IBDNe, we need an IBD detection method that has high power to detect IBD segments with length greater than a threshold (2 cM in this study), and that estimates the lengths of these segments accurately. Genotype errors and haplotype phase errors can result in gaps in the estimated IBD segments when using a haplotype-based approach such as Refined IBD. A single long IBD segment may be reported as two shorter segments with a gap between them. This leads to underestimation of IBD segment lengths. We developed a software tool merge-ibd-segments.jar (available from the Beagle Refined IBD website http://faculty.washington.edu/browning/refined-ibd.html) to fill these gaps between reported segments. In this study we filled a gap between two detected IBD segments if the length of the gap was less than 0.6 cM and there was at most one pair of genotypes inconsistent with IBD (e.g. opposite homozygotes) in the gap. This strategy produces estimates of IBD segment length that are reasonably accurate for segments > 2 cM length, even in admixed populations (S4 Fig).

Estimation of local ancestry

We used RFMix [8] version 1.5.4 to estimate local ancestry in each data set, simulated and real. For the Health ABC data, we used 112 CEU (CEPH from Utah) and 147 YRI (Yoruba from Ibadan Nigeria) samples from Hapmap3 [28] as reference samples. For the HCHS/SOL data, we used reference data from 195 West Africans, 63 Amerindians, and 527 Europeans obtained from the 1000 Genomes project and the Human Genome Diversity Project as previously described [13].

We used the “PopPhased” option and “-n 5” argument as recommended in the RFMix documentation. RFMix requires phased input data, and we used the phased genotypes generated by Beagle in the Refined IBD analysis. RFMix performs some rephasing of the data while inferring local ancestry. In order to match the local ancestry haplotypes to the IBD segment haplotypes, we adjusted the phasing of the RFMix Viterbi local ancestry calls to match the genotype phasing used in the IBD segment detection. Moving along the chromosome for an individual, we checked whether the phase of heterozygous genotypes matched between the RFMix rephasing and the Beagle phasing for the individual. We inferred a switch in the RFMix phasing relative to the Beagle phasing whenever the relative phasing of consecutive heterozygous genotypes changed. These switches were then applied to the RFMix Viterbi local ancestry calls for the individual in order to make the phasing of the ancestry calls consistent with the phasing of the IBD segments. The local ancestry proportion for an IBD segment was then determined from the phase-adjusted local ancestry calls for the corresponding haplotypes, using the mean called local ancestry from the two haplotypes. For example, if 41% of the length of the IBD segment had called local ancestry 1 for the first haplotype, and 43% of the length of the IBD segment had called local ancestry 1 for the second haplotype, then we considered the IBD segment to have 42% ancestry 1. We investigated the concordance between the called local ancestry proportions for pairs of IBD haplotypes in the HCHS data. Considering one ancestry at a time, we record the proportion of that ancestry for each IBD haplotype, and calculate the correlation of these proportions between the pairs of IBD haplotypes. For African ancestry, we obtain a correlation of 0.980; for European ancestry, 0.982; for Native American ancestry, 0.987.

Ancestry-specific effective population size estimation

An IBD segment may span a change in ancestry if the most recent common ancestor lived more recently than the commencement of admixture. The total length of an IBD segment provides information about the time to the most recent common ancestor, so one cannot simply cut the IBD segment into smaller ancestry-homogeneous segments. In order to estimate effective size, it is necessary to consider both the length of detected ancestry-specific IBD (which gives information about coalescence probabilities) and the length of the overall IBD segment (which gives information about the coalescence time). Specifically, the IBDNe program assigns each IBD segment fractionally to various coalescence times (measured in discrete numbers of generations) depending on the IBD segment length and the current estimates of effective population size history, and then uses the total (sum of fractional counts across IBD segments) for each generation to re-estimate the effective population size of that generation. For ancestry-specific effective size, we only want to consider pairs of haplotypes of the particular ancestry, that is those parts of IBD segments that are of that ancestry, but we need to use the full IBD segment lengths to estimate the coalescence times. One way to achieve this would be to weight the IBD segments by their proportion of the given ancestry. Implementing this approach would add complexity to the IBDNe program so instead we randomly assign the IBD segment to an ancestry based on the ancestry proportions of the segment. For example, if the segment has 80% of its length called as ancestry 1 and 20% as ancestry 2, we assign it to ancestry 1 with probability 80% and to ancestry 2 with probability 20%. This approach has the same average input as the fractional assignment approach, and results in an unbiased estimate of the observed total length of IBD corresponding to a particular ancestry.

Ancestry-adjusted number of pairs of sampled haplotypes

The expected total length of IBD attributable to generation g, E_g, is proportional to the number of pairs of haplotypes that can generate that IBD. Two haplotypes can only be IBD with respect to a given ancestry at genomic positions where both haplotypes have that ancestry. We assume the local ancestry of individuals i and j is independent, which is approximately true for all but close relatives. If individuals i and j have proportion p_i and p_j of the given ancestry, respectively, then the expected proportion of genome over which a random haplotype drawn from individual i and a random haplotype drawn from individual j both have that ancestry is p_ip_j. If there are n individuals, then the ancestry-adjusted number of pairs of haplotypes is .

Removal of relative pairs

Estimation of effective population size assumes a random population sample. Non-random ascertainment can affect the estimated effective size for the first few generations before present. In particular, an excess of close relatives in a data set will result in an excess of very large IBD segments, and hence a downward-biased estimate of the most recent effective size. By default, the IBDNe program searches for close relatives (half-sibs and closer) by looking for high genome-wide levels of IBD sharing and removes such pairs from consideration [14]. In the ancestry-specific case, the total proportion of the genome that could be IBD for the specific ancestry will tend to be low, and hence IBDNe won’t identify the pairs. We therefore identify pairs of close relatives using all IBD segments, and manually remove pairs with over 1200 cM of IBD genome-wide from the ancestry-specific analysis by removing the corresponding IBD segments and removing the pairs from the calculation of the ancestry-adjusted numbers of pairs of sampled haplotypes.

Bootstrap confidence intervals

We obtain confidence intervals for the estimated effective population size trajectories by bootstrap resampling of chromosomes. The bootstrapping is performed by the IBDNe program. Each bootstrap replicate resamples from the chromosomes with replacement to obtain the same number of chromosomes (22 for autosomal human data) as in the original data. The program estimates the effective population size trajectory for each bootstrap replicate in the same manner as for the original data. We use the default number of bootstrap replicates (80) and show the 2.5^th and 97.5^th percentiles. These intervals provide an approximate measure of the precision of the effective population size estimates.

Improvements to the IBDNe algorithm

In the course of this project, we made some changes to the IBDNe software to simplify the algorithm and reduce computation time.

The original version of the IBDNe program performs 50 runs of an iterated method of moments algorithm and reports the harmonic mean of the population size at each generation from the 50 runs. Each run starts with a random initial effective population size at each generation before the present (a trajectory) and updates this estimate at each iteration for 50 iterations. In each iteration, the algorithm fits a piecewise exponential growth curve, with a new growth rate every 8 generations. For each run, the first change in growth rate is randomly chosen to occur between 5 and 12 (inclusive) generations before the present [14].

The revised IBDNe program performs 8 runs of the iterated method of moments algorithm and uses a fixed initial effective size trajectory for each run (a constant effective population size of 100). As before the algorithm fits a piecewise exponential growth curve, with a new growth rate every 8 generations. The first change in growth rate still occurs between generations 5 and 12 (inclusive), and each of the 8 runs uses a different first time of change in growth rate. As before the program averages results from runs of the algorithm using a harmonic mean. We increased the default number of iterations per run from 50 to 1000, as we found continuing improvements in fit up to this point. These changes are implemented in the revised release of the IBDNe program (version 1.1). The program can be freely downloaded from http://faculty.washington.edu/browning/ibdne.html.

To allow ancestry-specific estimation of effective population size, we further modified IBDNe to include a parameter that specifies the ancestry-adjusted number of pairs of sampled haplotypes (see Ancestry-adjusted number of pairs of sampled haplotypes). If this parameter is missing, IBDNe will assume that overall (rather than ancestry-specific) effective size is being estimated, and determine the number of sampled haplotype pairs directly from the data. Thus one can give IBDNe ancestry-specific IBD segments and an ancestry-adjusted number of pairs of sampled haplotypes (both of which are obtained using the procedures described above) in order to obtain ancestry-specific estimates of effective population size.

HCHS/SOL data

HCHS/SOL is a study of 16,415 self-identified Hispanic/Latino individuals aged 18–74 (mean 41), with baseline examination in 2008–2011. The individuals were sampled by household in four US field centers (Chicago, IL; Miami, FL, Bronx, NY; San Diego, CA). The individuals were genotyped on an Illumina Omni 2.5M array with additional custom content, and the genotype data and phenotype data for 12,437 individuals are posted on dbGaP (accession numbers phs000880.v1.p1 and phs000810.v1.p1). In our IBDNe analysis, we excluded data from individuals who were not included in the dbGaP posting.

Segments of IBD were called with Beagle version 4 (r1203) [17] using genotyped SNPs with minor allele frequency > 0.02. Local ancestry calls were made previously with RFMix [13]. In analysis of effective population size, we considered only individuals whose four grandparents all had the same country of origin. Countries with fewer than 120 sampled individuals were excluded. Sample sizes by country can be found in Table 1. Due to the household-based sampling design, the data include many close relatives. We excluded IBD from close relative pairs as described above. We used the HapMap recombination map [29] for analyses with RFMix, for analyses with Beagle, and to determine IBD segment lengths.

Health ABC data

Individuals from the Healthy Aging and Body Composition (Health ABC) study were genotyped for the CIDR Visceral Adiposity Study. Genotype data were obtained for around 600 self-identified black and 800 self-identified white individuals (here referred to as African American and European American, respectively) from each of Memphis and Pittsburgh (Table 2). All individuals were 70–79 years old at recruitment in 1997–1998. Participants were identified from a random sample of white Medicare beneficiaries and all age-eligible community-dwelling black residents in designated zip code areas surrounding the field centers in Pittsburgh, Pennsylvania, and Memphis, Tennessee [30]. Genotype data from an Illumina Human1M-Duo BeadChip array were downloaded from dbGaP (study accession phs000169.v1.p1). We removed SNPs with call rate <99% in either the African American or the European American samples, Hardy-Weinberg equilibrium p-value < 10⁻⁶ in either the African American or the European American samples, or minor allele frequency < 1% in either the African American or the European American samples.

We called segments of IBD using Beagle 4.1. We used HapMap phase 3 CEU and YRI samples (Utah residents with ancestry from northern and western Europe and Yoruba in Ibadan, Nigeria respectively) [28] as reference panels for the local ancestry calling, utilizing SNPs genotyped in both the HapMap phase 3 and Health ABC data. We phased the reference HapMap data using the phased Health ABC data from our Beagle Refined IBD analysis as a phasing reference panel.

We used the HapMap recombination map [29] for analyses with RFMix, for analyses with Beagle, and to determine IBD segment lengths. The Health ABC data did not contain any close relatives.