Comparison of measures of marker informativeness for ancestry and admixture mapping
 Lili Ding^{1},
 Howard Wiener^{2},
 Tilahun Abebe^{3},
 Mekbib Altaye^{1},
 Rodney CP Go^{2},
 Carolyn Kercsmar^{1},
 Greg Grabowski^{1},
 Lisa J Martin^{1},
 Gurjit K Khurana Hershey^{1},
 Ranajit Chakorborty^{4} and
 Tesfaye M Baye^{1}Email author
DOI: 10.1186/1471216412622
© Ding et al; licensee BioMed Central Ltd. 2011
Received: 13 July 2011
Accepted: 20 December 2011
Published: 20 December 2011
Abstract
Background
Admixture mapping is a powerful gene mapping approach for an admixed population formed from ancestral populations with different allele frequencies. The power of this method relies on the ability of ancestry informative markers (AIMs) to infer ancestry along the chromosomes of admixed individuals. In this study, more than one million SNPs from HapMap databases and simulated data have been interrogated in admixed populations using various measures of ancestry informativeness: Fisher Information Content (FIC), Shannon Information Content (SIC), F statistics (F_{ST}), Informativeness for Assignment Measure (I_{n}), and the Absolute Allele Frequency Differences (delta, δ). The objectives are to compare these measures of informativeness to select SNP markers for ancestry inference, and to determine the accuracy of AIM panels selected by each measure in estimating the contributions of the ancestors to the admixed population.
Results
F_{ST} and I_{n} had the highest Spearman correlation and the best agreement as measured by Kappa statistics based on deciles. Although the different measures of marker informativeness performed comparably well, analyses based on the top 1 to 10% ranked informative markers of simulated data showed that I_{n} was better in estimating ancestry for an admixed population.
Conclusions
Although millions of SNPs have been identified, only a small subset needs to be genotyped in order to accurately predict ancestry with a minimal error rate in a costeffective manner. In this article, we compared various methods for selecting ancestry informative SNPs using simulations as well as SNP genotype data from samples of admixed populations and showed that the I_{n} measure estimates ancestry proportion (in an admixed population) with lower bias and mean square error.
Background
Admixture is a common form of gene flow between populations. It refers to the process in which two or more genetically and phenotypically diverse populations with different allele frequencies mate and form a new, mixed or 'hybrid' population [1, 2]. A classic example of an admixed population in humans is the AfricanAmerican population. As a result of the genetic admixture, the AfricanAmerican population contains stretches of DNA as large as 2030 cM that resemble mosaics of chromosomal segments, or ancestry blocks [3]. These segments are derived from intermixing between European and African ancestry and have not had sufficient time to break up through recombination [4, 5]. As a result, in contrast to the million markers suggested to be necessary for genomewide association studies (GWAS) [6], modeling studies showed that between 2000 and 5000 welldistributed ancestry informative markers (AIMs) distinguishing parental origins are sufficient for wholegenome scanning under the admixture mapping strategy [7–9]. An ideal AIM should have one allele that is fixed (i.e., allele frequency of 1.0) in one ancestral population, and not present in the other [10]. However, in the context of human genetics, most alleles are shared among populations [11–13]. Hence, it is important to identify and choose most ancestry informative markers across populations [14]; the power of admixture mapping relies heavily on the ability of informative markers to infer ancestry along the chromosomes of admixed individuals.
Several measures of marker informativeness for ancestry have been developed to select the most ancestry informative markers (reviewed in Rosenberg et al., 2003 [10]) from an everincreasing wealth of genomic databases [15–18]. These measures include: Absolute Allele Frequency Differences (delta, δ), Shannon Information Content (SIC), Fisher Information Content (FIC), F statistics (F_{ST}), and the Informativeness for Assignment Measure (I_{n}). The cutoff value for δ is highly subjective and has steadily decreased over time from ≥ 0.5 [19] to 0.4 [20] to 0.3 [21, 22]. Cutoffs that have been used for other measures are F_{ST} ≥ 0.4 [21], FIC ≥ 2.0 [10], SIC ≥ 0.3 [23], and I_{n} ≥ 0.3 [10]. Informative measures such as δ can be used for only two ancestral populations at a time [10, 23]. On the other hand, F_{ST}, FIC, SIC, and I_{n} can be applied to select informative markers for admixed populations formed from two or more ancestral populations. For FIC and SIC indices, ancestral proportions in the admixed population need to be specified.
In spite of numerous studies with these measures of marker informativeness for ancestry, several questions are not systematically addressed, including how often are the same sets of SNPs selected by the different methods? To what degree do they overlap and share common sets of SNPs? How do AIM panels selected by these different methods perform in estimating ancestry population contributions under different proportion of ancestral population in an admixed population? With so many measures to choose from, it is very important to understand their common features as well as where they differ in terms of SNP selection. Answering these questions with a systematic study would help users in choosing appropriate measures in a costeffective manner. In absence of a comprehensive comparative study on the performance of the different marker informativeness measures in marker selection, researchers selected markers using only the measure of their personal choice. For example, the three major U.S. admixture mapping research groups led by David Reich, Michael Seldin and Mark Shriver in their recent independent admixture mapping panels for Latino populations used SIC, F_{ST} and δ [9, 24, 25], respectively. It is not clear which measurebased panel is the most informative for admixture mapping. In particular as more and more markers become available and as we study less differentiated populations, it is inevitable that prioritizing the most informative markers for ancestry inference or admixture mapping is critical. With the availability of common sets of SNPs from HapMap populations, we are given the opportunity to compare these methods directly. The objective of the present study is to compute and compare the commonly used measures of informativeness to select AIM panels for admixture mapping and structured association testing for admixed populations. To compare these methods, simulated as well as real data were used. In the simulated data, the ancestral populations and their contributions for each individual are known, allowing comparison of accuracy of the different measures of marker informativeness for ancestry using true and estimated individual ancestry values.
Results
SNP allele frequencies and comparisons of informative marker selection measures
Chromosome length and number of SNP markers across the genome in CEU and YRI population of the HapMap Phase III dataset
Number of SNPs  SNPs genotyped in both population^{a}  

Chr #  length  in each population  SNPs genotyped  and fulfilling the filtering criteria^{b}  
(Mb)  CEU  YRI  in both populations^{a}  Delta  F_{ST}  FIC  SIC  I_{n}  
1  246.6  111887  120349  103330  1663  1657  1653  1651  1654 
2  242.7  113613  122377  106053  1742  1751  1742  1740  1749 
3  199.3  94608  101070  88060  1464  1447  1448  1450  1450 
4  191.2  85403  92052  79856  1390  1394  1387  1398  1392 
5  180.6  87071  92350  81083  1310  1298  1314  1302  1304 
6  170.7  91415  95108  84536  1241  1235  1239  1231  1236 
7  158.7  75234  79231  69766  1139  1144  1152  1157  1148 
8  146.2  74443  79368  69177  1053  1055  1053  1048  1054 
9  140.2  63507  66437  58713  842  841  834  844  845 
10  135.3  72846  76787  67388  979  988  980  980  988 
11  134.3  69175  73942  64140  986  987  988  990  981 
12  132.3  67486  70909  62018  988  981  973  972  977 
13  96.2  51879  55392  48496  719  726  701  713  723 
14  88.2  44570  47354  41434  646  646  650  648  650 
15  82.0  40705  43837  37774  590  596  596  590  595 
16  88.7  42738  46366  39616  568  559  554  555  557 
17  78.6  36534  39075  33576  572  569  580  574  570 
18  76.1  40153  43467  37669  562  568  564  561  572 
19  63.6  25251  26798  23264  405  404  404  407  405 
20  62.4  35252  37476  32870  447  447  446  448  443 
21  37.1  19336  20556  18105  252  250  254  249  249 
22  35.1  19617  20595  17817  249  253  253  252  249 
Total  1362723  1450896  1264741  19807  19796  19765  19760  19791 
For CHB and JPT population, the distribution of the five measures of marker informativeness is show in Additional file 2, Figure S1 and summary statistics are shown in Additional file 3, Table S2. Almost all the means of the five measures were about 10fold less than those from the CEU and YRI population, indicating that CHB and JPT are much less differentiated than CEU and YRI and the ancestry estimation for an admixed population from CHB and JPT presents a much more difficult problem.
Correlation, concordance, and overlapping analysis
Spearman correlation
Concordance by deciles
Additional file 5, Figure S2 shows the scatter plot of allele frequencies of CEU and YRI, with different colors indicating which decile group a SNP fell into. It is interesting to see how the ten groups partitioned the SNPs into symmetric patterns from the topleft and bottomright corner representing the most informative SNPs to the center of the plot where the least informative SNPs resided. It is evident that δ showed a partition pattern different from other four measures due to the fact that δ depends only on the difference between allele frequencies of the two populations. F_{ST} and I_{n} exhibited very similar partition patterns, which is consistent with what we observed using the Spearman correlation coefficient, Mosaic plots, and Kappa statistics. It can also be recognized through the scatter plot that FIC favors the selection of markers that are closer to fixation in one of the populations.
Overlapping
Discrimination analysis and estimation of ancestral contribution
Discrimination analysis
Estimation of ancestral contribution in admixed populations with top AIMs
Additional file 7, Figure S4 shows an inferred population structure (STRUCTURE [26] and distruct[27]) for CEU, YRI, and ASW population with the top 200 AIMs selected by FIC. AIM panels by other measures gave the same population structure. Furthermore, Additional file 8, Figure S5 shows the estimates of ancestry contributions for the three populations as the number of AIMs increases. The top informative SNPs (100 to 200) obtained using each measure yielded similar estimates of ancestry contribution. With 200 AIMs, the estimate of ancestry in ASW was 78% YRI across all measures. However, for individuals from CEU and YRI population, where the estimated ancestry contribution from CEU and YRI, respectively, is expected to be or close to 1, AIMs selected by I_{n} performed slightly better than those selected by other measures of informativeness.
Estimation of ancestral contribution in admixed populations with random subsets of top AIM panels
Overall, the AIM panels chosen by I_{n} performed the best, giving the lowest bias and RMSE, whereas those by FIC performed the worst across the real dataset and the simulated datasets. For the real dataset (ASW), the combined method by using the average ranking of the five measures outperformed all the five measures (Additional file 10, Table S4). However, for the simulated datasets, the average ranking and minimal ranking methods were outperformed by I_{n} and F_{ST}, but their performance were either better or very similar to the most commonly used method Delta, FIC and SIC (Additional file 11, Table S5 and Additional file 12, Table S6).
Discussion
Admixture mapping is a powerful gene mapping approach [5, 28]. However, the power of this method relies on the ability of ancestry informative markers (AIMs) to infer ancestry along the chromosomes of previously separated but recently admixed individuals. In a recent paper from ASHG, Royal et al. (2010) [29] outlined the challenges, opportunities and implications of genetic ancestry inference. Over 40 companies provide genetic ancestry testing to the public. However, these companies differ in their approaches, the types of ancestry markers used and tests they offer. The promise of utilizing genetic ancestry information to advance medical genomics depends on our ability to correctly and precisely infer/measure ancestry using informative markers.
Several methods have been proposed to measure ancestry informativeness of markers and to choose a panel of markers to be genotyped while maintaining the power of detecting ancestral chromosome segments in each genomic location. The choice of which of these measures to use should depend on the efficiency of each measure in selecting most ancestry informative markers. However, there is no consensus as to which criteria to use to select markers for ancestry inference or admixture mapping, and the performance of these methods has not been carefully evaluated and compared. The rule of thumb is to select markers with large allele frequency differences between ancestry populations. However, the number of markers required for population assignment will depend on the populations under consideration, their respective level of genetic differentiation and the desired stringency of assignment [30]. For instance, in humans, the level of genetic variation between populations is only 5%10% whereas genetic variation within dogs is about 27% [31]. As a result, the number and types of markers required for individual assignment and discrimination amongst populations is different between populations/species under consideration [30]. Previous studies selected markers based on different datasets and marker types using only one of the methods at a time, and there has not yet been a formal comparison of the performance of these methods. Therefore there is a need to compare all methods using the same data sets and evaluate their efficiencies and accuracies in estimating ancestral proportions for admixed populations.
In this study, we applied five different analytic tools to evaluate the concordance of selected informative SNPs using the same dataset. Our investigation using 500 top ranked markers for each measure and accounting for the physical distance between consecutive AIMs to be at least 100 kb, showed the following overlap between the different measures: δ vs F_{ST} (n = 479), δ vs FIC (n = 220), δ vs SIC (n = 319), δ vs I_{n} (n = 424), F_{ST} vs FIC (n = 230), F_{ST} vs SIC (n = 329), F_{ST} vs I_{n} (n = 445), FIC vs SIC (n = 395), FIC vs I_{n} (n = 258), and SIC vs I_{n} (n = 354) (Additional file 13, Table S7). On average, the overlap of each measure with the other four was 361, 371, 276, 349, and 370 for δ, F_{ST}, FIC, SIC, and I_{n}, respectively. FIC had the least overlap with other measures. However, based on current cutoff values used for each measure, the δ measure included a number of loci that were not selected by the remaining four methods. F_{ST}, FIC, and I_{n} gave relatively smaller and similar AIM panels, whereas SIC gave a very small panel of AIMs (Additional file 14, Figure S7). Analyses based on deciles showed that sets of SNPs at the highest or lowest SNP information content selected for admixture mapping were highly similar across the different measures of informativeness. Towards the middle of the informativeness scales, the agreement among the sets of SNPs selected by different methods to discriminate between populations decreased (Figure 3, 4, and 5). Furthermore, FIC and SIC were more likely to pick the same set of SNPs, δ, F_{ST}, and I_{n} were more likely to pick the same set of SNPs, and FIC was more likely to choose SNPs that were not chosen by the other measure.
Analytically, the FIC and SIC measures require predefined ancestral proportions in an admixed population, whereas F_{ST}, δ, and I_{n} do not. We ran sensitivity analysis to study the impact of ancestral proportion in choosing informative markers using arbitrary values using CEU and YRI population. Compared with FIC, SIC was less sensitive to the proportion of ancestry contribution in the selection of AIMs (Additional file 15, Table S8 and Additional file 16, Table S9). Proportion of ancestry had virtually no effect on the selection of top 1% AIMs for SIC. For two proportions of ancestry contribution within a distance of 0.1, FIC selected 56%71% common sets of AIMs and SIC selected 75%100% common sets of AIMs. Therefore, it is important, when using FIC, to have a good a priori estimate of proportion of ancestry contribution.
There are some limitations for some of these measures, for example, FIC favors selection of markers that are closer to fixation in one parental population and may not be appropriate to assess the level of informative markers when ancestral populations are more than two [32]. Compared with other methods such as F_{ST}, δ is easy to calculate and independent of mutation and model assumptions, however, δ has a major limitation of being only useful for admixed populations from two parental populations and it doesn't account for multiallelic situations at a locus. F_{ST} may not be appropriate to assess the level of genetic information in SNP markers when the number of populations is > 2, as the method could result in the selection of SNP markers which are specific for a single most genetically distinct population. The selected SNP markers that were specific for only the most distinct population are expected to have low heterozygosity. Genetic markers with high expected heterozygosity are informative and therefore useful in individual assignment analysis [33]. Although most researchers traditionally focus on global axes of variation in a dataset, substantial information about population ancestry exists locally across chromosomes. Adjustment of global ancestry between study subjects may lead to false positives when chromosomal (local) population ancestry is an important confounding factor [34]. In a recent chromosomebased study by Baye (2011) [35], finescale substructure was detectable beyond the broad population level classifications that previously have been explored using genomewide average estimates. The study of population ancestry in terms of local ancestry has broader practical relevance because genetic diversity is directly related to recombination rate (meiosis), which differs among regions of the genome, and genes are not randomly distributed along chromosomes. The current analytical approach using genomewide average estimates will control for confounding due to global ancestry but will not control for confounding due to the local ancestry effect because the global ancestry information is obtained from all markers across the genome and may not accurately reflect local ancestry variation. It is becoming increasingly important to recognize local ancestry variation, especially when populations have been recently admixed [35]. Future studies should focus on the applications of these measures to important genomic regions.
Many factors impact the accuracy of the estimation of ancestry contributions, which include but are not limited to sample size, the panel of AIMs used, the number of AIMs used, and the underlying distribution of ancestry contribution of the individuals in the sample. The use of a phased HapMap dataset allowed us to simulate individuals that share common founding populations. Moreover, the ancestry proportion for each individual is known, allowing for the comparison of true and estimated individual admixture values, thus enabling the comparison of different methods by estimation accuracy. Our findings indicate that, the different measures of marker informativeness [δ, F_{ST}, FIC, SIC, and I_{n}] performed well and as few as the top 20 ranked informative markers were adequate for accurate classification of ancestral populations. This is in agreement with the commonly made claim in the literature on marker selection for population assignment that 'classification accuracy can be substantially improved if only a subset of loci is used in the assignment test' [36]. For instance, Lao et al., (2006) [37] found that 10 SNP markers from a 10 K SNP array contained enough genetic information to differentiate individuals from Africa, Europe, Asia and America and no further gain in power of assignment was achieved by including more SNP markers. Indeed, it is generally considered that uninformative markers (i.e., monomorphic loci) may add variability and noise to the results and compromise the power of population genetic studies [38].
Although the marker selection methods explored in this study agreed to a large extent in identifying the most informative SNPs, there were differences in their performance in ancestry estimation. The simulation study revealed that I_{n} was the best in selecting the set of AIMs giving the smallest bias and mean square error in ancestry estimation. Analysis based on random subsets of top 1% to 10% ranked AIMs indicated that, compared to other methods, AIM panels selected by I_{n} behaved consistently and reasonably well for both the ASW population and simulated admixed populations. These results illustrate that effective exploration of all these methods can help to not only identify the most informative markers but also produce an optimal minimum set of markers that can accurately and efficiently differentiate among populations.
We suggest that the different measures may provide unique insights into a marker's informativeness under different scenarios, including varying ancestral proportion and when more than two ancestral populations are present. To identify all potentially informative SNPs, results from all measures could be considered. For example, the union of the top 500 SNPs for all five measures could be considered as the best AIMs panel. Researchers need to be aware of the differences between the various methods for evaluating ancestry informativeness of SNP markers. Furthermore, as we attempted in our simulation studies using either average rank or minimal rank of all five measures, combined information from more than one method may provide a reliable means, although may not be the best, in selecting markers for ancestry inference. Further research on this topic may shed light on how to best integrate different measures to obtain a set of AIMs most effective for the populations under consideration. We believe that the information that a set of markers provides for assigning or discriminating individuals to their source populations or different relationships must be critically evaluated before investing millions of dollars on an admixture or ancestry related project. We anticipate identification of more complex patterns of ancestry will require explorations of these and newer methods yet to be defined, to identify an optimal set of markers to use, however this should become increasingly feasible as genotyping costs decrease and available data grow on different populations. This in turn will allow the development of higher resolution of genogeographic and ethnic maps and help investigators designing genetic association studies in stratified homogeneous groups.
Conclusions
Although millions of SNP markers with varying levels of information content for ancestry inference have been identified, only small subsets of highly informative markers need to be genotyped in order to accurately predict ancestry with a minimal error rate in a costeffective manner. In this article, we compared various methods for selecting informative SNPs and showed that the I_{n} measure estimated ancestry proportion (in an admixed population) with lower bias and mean square error. In summary, we showed the utility of several measures of informativeness using simulations and real SNP genotype data from samples of admixed populations. The use of several available methods to prioritize informative markers for ancestry inference can reduce genotyping costs and avoid false positive genotypephenotype associations while retaining most of the power found in much larger sets of published AIM panels.
Methods
Five measures of ancestry informativeness
Notation
Consider populations i = 1, 2,..., K with K ≥ 2 and a locus with N ≥ 2 alleles. Let p_{ ij }denote the frequency of allele j, j = 1, 2, ..., N, in population i. Let p_{ j }denote the average frequency of allele j over the K populations, i.e., ${p}_{j}={\sum}_{i=1}^{K}{p}_{ij}\u2215K$. Consider an admixed population with two parental populations, the frequency of allele j at a locus in the admixed population p_{ Aj }is a linear combination of allele frequencies in the ancestral populations, and can be written as p_{ Aj }= m_{1}p_{1j}+ m_{2}p_{2j}, where m_{ i }is the proportion of contribution of the i^{th} ancestral population, and m_{1} + m_{2} = 1.
Absolute allele frequency difference (delta, δ)
A marker with δ = 1 provides perfect information regarding ancestry whereas a marker with δ = 0 carries no information. It has been shown that δ by itself only provides limited information regarding a marker's informativeness for ancestry [10]. The sum of the allele frequencies in the two parental populations, or equivalently the value of the smaller of the two frequencies, can provide additional information independent of δ.
F statistics (F_{ST})
Here, j = 1 or 2 is the reference allele. Values of F_{ST} can range from 0 to 1. A high F_{ST} value implies a considerable degree of differentiation between populations. F_{ST} is a pairwise population measure of differentiation or relatedness (genetic distance measure between the two populations) based on genetic polymorphism data such as SNPs and was recently utilized as a criterion for selecting markers for ancestry estimation [40].
Fisher Information Content (FIC)
Here, ${\widehat{p}}_{Aj}={p}_{2j}+{\widehat{m}}_{1}{\delta}_{j}$ is the expected frequency of the j^{th} allele in the admixed population or individual, δ_{ j }= p_{1j} p_{2j}is the allele frequency difference, ${\widehat{m}}_{1}$ is the maximum likelihood estimate of the contribution from ancestral population one. FIC measure allows selection of markers that are particularly informative in an admixed population in which the contribution of one parental population is substantially greater than that of the other parental population. It favors selection of markers that are closer to fixation in the parental population with the greater contribution.
Shannon Information Content (SIC)
Informativeness for assignment (I_{n}) measure
This formula is a generalization to more than two populations. From a likelihood perspective, it gives the expected logarithm of the likelihood ratio that an allele is assigned to one of the populations compared with a hypothetical 'average' population whose allele frequencies equal the mean allele frequency across the K populations. Its value is smaller when all alleles have similar frequencies in all populations.
Data
HapMap phase III dataset
We downloaded the complete HapMap phase III genotype data (http://www.hapmap.org, release #3, May 2010) available for Yoruban population in Ibadan, Nigeria [YRI], Caucasian population from the United States with northern and western European ancestry [CEU], and the African American population from Southwest USA [ASW]. HapMap is a public resource created by the International HapMap Project to catalogue genetic variants (SNPs) that are common in human populations. The HapMap phase III release #3 contains genotypes from 147 unrelated individuals (parents) from YRI population, 113 unrelated parents from CEU population, and 87 individuals from ASW population. For the purpose of the present study, YRI and CEU populations are assumed to be the ancestors of ASW population. Two criteria were used to filter the SNPs included in the final analysis: 1) the SNP should be shared by both YRI and CEU populations, i.e., SNPs for which allele frequencies were available in both YRI and CEU populations, and 2) SNPs with missing frequency for over 10% of the samples were excluded. Furthermore, to avoid the possibility of choosing two redundant SNPs that are in strong LD (linkage disequilibrium), for each measure, we calculated the informativeness on all shared SNPs, then filtered them for the most informative ones such that the physical distance between consecutive selected SNPs must be at least 100 kb.
Simulated dataset
To compare marker informativeness measures in the estimation of ancestry population contribution, we simulated two artificially admixed datasets from the phased HapMap III dataset (with known allele frequencies). The first one is an admixed population from two parental populations with relatively high divergence: 113 unrelated individuals in CEU population and 113 unrelated individuals in YRI population. The second one is an admixed population from less differentiated ancestral populations: 84 unrelated individuals of Han Chinese in Beijing, China [CHB] and 86 unrelated individuals of Japanese in Tokyo, Japan [JPT]. The simulations were run using simuPOP [32, 33] for 10 generations. During the simulation, we tracked the true ancestry contributions for each individual and calculated average ancestry contributions for each of the two admixed populations.
Statistical analysis
Python (http://www.python.org) scripts were written to retrieve and preprocess SNP and frequency data. Five measures of marker ancestry informativeness were calculated for shared SNPs between CEU and YRI population and for shared SNPs between CHB and JPT population, with YRI or JPT contribution fixed as 80% for the calculation of FIC and SIC. Sensitivity analysis of different YRI contributions on the selection of AIMs was performed for FIC and SIC. For each data set, the number of alleles per locus (SNP) was coded to a string of numbers to obtain a full design matrix of alleles where the cells give the number of copies of each major allele for each individual (zero, one, or two). R (R Foundation for Statistical Computing, 2010), SAS software (SAS 9.1.3, SAS Institute Inc.), and JMP Genomics (JMP^{®} Genomics, v.5, SAS Institute Inc.) programs were used to analyze the various measures of informativeness for ancestry.
Correlation, concordance, and overlapping analysis
To assess the level of similarity of the estimates of genetic information contained in each SNP marker across the five measures of marker informativeness, we used three statistical procedures: Spearman correlation coefficient, Cohen's Kappa statistics, and overlapping frequency analysis of top n ranked AIMs by different measures. Although the three approaches share some common information, each provides unique and complementary views of the behavior of the five measures. Spearman correlation coefficient is a global measure of statistical dependence and provides a general sense regarding pairwise monotonic relationship of the five measures of marker informativeness. The Cohen's Kappa coefficient based on deciles quantifies agreement between two measures and the corresponding mosaic plot exhibits overlapping structures, i.e., the distribution of markers according to one measure of informativeness relative to another. Finally, the overlapping frequency analysis demonstrates how often the same set of SNPs is selected by two or more different measures or which measures tend to select the same set of SNPs.
Spearman correlation coefficient [44] is a measure of correlation based on the ranks of the data values. It is a nonparametric alternative to Pearson's correlation coefficient and do not require the knowledge of the distribution of the data. The formula for Spearman correlation coefficient is $\frac{{\sum}_{i}\left(\left({X}_{i}\overline{X}\right)\left({Y}_{i}\overline{Y}\right)\right)}{\sqrt{{\sum}_{i}{\left({X}_{i}\overline{X}\right)}^{2}{\sum}_{i}{\left({Y}_{i}\overline{Y}\right)}^{2}}}$, where, X_{i} and Y_{i} are the ranks of observed data values, $\overline{X}$ is the mean of X_{i}'s, and $\overline{Y}$ is the mean of Y_{i}'s. In case of ties, the averaged ranks are used. Spearman correlation coefficient takes values between 1 and +1. A +1 or 1 indicates that the two measures are in a perfectly monotonically increasing or decreasing relationship, respectively, and a 0 means no relationship.
To show the distribution of markers according to one measure of informativeness relative to another, we further analyzed the data by grouping and rating SNP markers using deciles, producing mosaic plots and calculating Cohen's Kappa coefficients. Deciles are the nine values of a variable dividing its distribution into ten groups with equal frequencies. For each measure, based on its deciles we created a new categorical variable with values 1, 2..., and 10, indicating to which group a SNP belongs. We then used the new categorical variables to build mosaic plots and to examine the relationship between measures of marker informativeness. The mosaic plots show, for example, how the top 10% SNPs from one measure of informativeness distribute relative to another measure of informativeness. To assess the concordance of decilebased ratings of the informativeness of AIMs between measures, we computed the Cohen's kappa coefficient, a commonly used index to quantify agreement between two measurements [45]. It takes into account the concordance by chance and is calculated as κ = [Pr(a)  Pr(e)]/[1 Pr(e)], where Pr(a) is the observed agreement percentage, and Pr(e) is the chance agreement percentage. The larger the kappa coefficient, the better the concordance is between two measurements. Kappa takes values between 0 and 1. κ = 1 indicates a perfect agreement while κ = 0 indicates no agreement other than what would be expected by chance.
To answer the question of how often the same set of SNPs is selected by the different methods or which methods tend to select the same set of SNPs, we studied the overlap pattern of the top n AIMs selected by different measures of informativeness. Each SNP was assigned a 5digit binary vector, where each digit represents a measure. From the first to the last these correspond to δ, F_{ST}, FIC, SIC, and I_{n}, respectively. A 1 in the digit indicates that the SNP is selected by the corresponding measure as one of the top n AIMs. For example, a binary vector 11001 represents the SNP is selected by δ, F_{ST}, and I_{n} as one of the top n AIMs, but not by FIC and SIC. For a specific n, the frequency of the different combinations of the 5digit numbers (such as 11001 and 00110) shows how often the different methods select the same set of SNPs. The higher the frequencies, the higher the chance that the same set of SNPs are selected by the methods corresponding to the 1's in the 5digit vector.
Discriminant analysis
To compare the discrimination power of the five measures of informativeness and assess how many markers are needed for accurate ancestral CEU vs. YRI population and CHB vs. JPT population membership assignment, discriminant analysis was performed using the top 1, 2, ..., and up to 150 ranked AIMs. Discriminant analysis [46] is a method of projecting highdimensional data onto a lowerdimensional space in a way that data points from different classes are wellseparated. The projection is given by $y={\sum}_{i=1}^{p}{w}_{i}{x}_{i}$, where x = (x_{1}, x_{2}, ..., x_{ p }) is a p dimensional data point (individuals with SNP information), and y is the projection of x onto w = (w_{1}, w_{2}, ..., w_{ p }), or a linear combination of x_{ i }with weights w_{ i }. The weights are chosen such that the projections of the data points (individuals) in the same class (CEU or YRI population) are close to each other while those of the data points from different classes are far from each other. Linear discriminant can be derived using a measure of generalized squared distance. An optimal linear classifier then can be found by minimizing classification error (probability of misclassification). The classifier can take into account of prior probabilities of the classes, which, in our analysis, were specified as proportional to the sample sizes in each class. A data point is classified into the class for which the posterior probability of the observation belonging to this class is the largest among all classes. Crossvalidation is used to obtain prediction accuracy. The analysis was carried out using PROC DISCRIM in SAS (SAS 9.1.3, SAS Institute Inc.). We also examined the number of AIMs needed to achieve 90% or 95% classification accuracy.
Estimation of ancestry contribution in admixed ASW population
We estimated ancestry contribution for the admixed ASW population using up to 200 top ranked AIMs by different measures. We also estimated ancestry contribution using 100 sets of randomly selected 20 SNPs from the top 1%, 2%, 5%, and 10% ranked AIM panels. The analysis was performed using the software PSMIX [47]. This analysis allowed us to compare the consistency in the estimation of ancestry contribution when the number of informative markers in the pool increases or decreases. The rationale for conducting this analysis is that AIM panels generated by different measures are more similar when only the top AIMs are considered, which makes it difficult to compare their performances using only the top AIMs. By using random subsets of AIMs from the top AIM panels of various sizes, we expect to select less informative markers and the estimate of the ancestry contribution is expected to become less accurate (or more biased) with more variability. More importantly, we will be able to determine if there is clear separation in the performance of different measures based on the information content taken from similar (1% to 10%) pools of markers as determined by each method.
We constructed two new methods of ranking marker informativeness for ancestry by combining the information from all the five measures. For each marker, we assigned a ranking or score based on either the average ranking (AVE) or the minimum ranking (MIN) of the five measures. We didn't use the raw values from the five measures because they have different scales; thus, any score computed by weighted average of the raw values needs to be preceded by standardization of the raw values, which is beyond the scope of this paper.
Estimation of ancestry contribution in simulated admixed population
To validate the ancestral estimates of the five measures, the same set of analyses in the previous section were conducted for the two simulated admixed populations. In the simulated admixed populations, the ancestry proportion for each individual is known, so is the mean ancestry proportion across individuals in the same population. Estimation accuracy by different measures was compared at two different levels. At the population level, the estimate of the mean ancestry contribution across individuals was compared with the true value and bias was calculated for the five measures. At the individual level, individual true and estimated admixture values were compared, and root mean square error (RMSE) was used as a summary measure of precision in the estimation of individual ancestry proportion. RMSE is defined as ${\left[\frac{1}{M}\sum _{i=1}^{M}{\left({q}_{i}{\widehat{q}}_{i}\right)}^{2}\right]}^{1\u22152}$, where M is the number of individuals in the sample, and q_{ i }and ${\widehat{q}}_{i}$ represent the true and estimated individual ancestries, respectively. We also plotted individual estimated contributions vs. true contributions.
Web resources
HapMap: http://www.hapmap.org
PYTHON: http://www.python.org
Abbreviations
 AIMs:

ancestry informative markers
 GWAS:

genomewide association studies
 YRI:

Yoruban population in Ibadan, Nigeria
 CEU:

Caucasian population from the United States with northern and western European ancestry
 ASW:

African American population from Southwest USA
 CHB:

Chinese population from Beijing, China
 JPT:

Japanese population from Tokyo, Japan.
Declarations
Acknowledgements
This work was supported by National Institutes of Health grant 1K01HL103165 (TMB) and MH066181 (RCPG).
Authors’ Affiliations
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