Enhancements to the ADMIXTURE algorithm for individual ancestry estimation

1Our program ADMIXTURE estimates individual ancestries by efficiently computing maximum likelihood estimates in a parametric model. The model posits that genotype nij for individual i at SNP j represents the number of type "1" alleles observed. Given K ancestral populations, the success probability in the binomial distribution nij ~ Bin(2, pij ) depends on the fraction qik of i's ancestry attributable to population k and on the frequency fkj of allele 1 in population k. ADMIXTURE maximizes the biconcave log-likelihood

2of the model using block relaxation. The alternating updates of the parameter matrices Q = (qik) and F = (fkj) both rely on sequential quadratic programming. Convergence is accelerated by applying a quasi-Newton extrapolation algorithm. Further details of our core algorithm are documented elsewhere. The performance of ADMIXTURE is compelling. An ADMIXTURE analysis is typically three to four orders of magnitude faster than a comparable STRUCTURE analysis.

3The advanced features of ADMIXTURE described here allow the user to automate the choice of the number of underlying populations K and to exploit known ancestral populations in a supervised learning mode. Our penalized estimation mechanism can provide many of the benefits of a Bayesian analysis at a fraction of the computation time. These features make ADMIXTURE a suitable replacement for STRUCTURE in most practical applications. Given the ever-increasing size of genotype datasets, the inherent speed of our optimization algorithm, coupled with the parallel-processing mode described here, may render ADMIXTURE the only viable model-based ancestry analysis tool for many users.

4The choice of the number of ancestral populations K can prove difficult when the underlying population genetics of a species is poorly understood. STRUCTURE provides a means of estimating the best value of K by computing the model evidence for each K from a range of choices. The model evidence is defined as

5where f represents the data likelihood and π represents a prior density on the parameters. STRUCTURE approximates the integral via Monte Carlo methods. Our optimization framework is not well suited to evaluating this integral. As an alternative, we employ cross-validation. In cross-validation, we aim to identify the best K value as judged by prediction of systematically withheld data points. A similar tactic is also employed by the haplotype analysis program fastPHASE and is inspired by Wold's method for cross-validating PCA models.

6Our v-fold cross-validation procedure partitions the non-missing genotypes into v roughly equally sized subsets (folds). At each of v iterations, the members of one of the folds are masked (temporarily marked as missing) to yield a new data matrix Analysis of the masked data matrix poses no new challenges. In computing the log-likelihood, score, and observed information matrix of, we simply ignore the entries (i, j) with missing values. Maximization of the log-likelihood readily yields new estimates and for the masked data. We then predict each masked value nij by. Prediction error is estimated by averaging the squares of the deviance residuals for the binomial model,

7across all masked entries over all folds. Minimizing this estimated prediction error on a grid of K values then suggests the most suitable K.

8ADMIXTURE's strategy of simultaneously estimating individual ancestry fractions Q and population allele frequencies F is ideal when nothing is known about the contributing ancestral populations. In many scenarios, however, these populations are known and several reference individuals from each population are available. Here it is of interest to estimate the potentially admixed ancestries of the remaining individuals. We term this supervised analysis, as the reference individuals furnish training samples in a supervised learning context. To perform supervised analysis in ADMIXTURE, an.ind file mapping individuals to populations must be provided, and the flag --supervised must be attached to the command line.

9Ancestry estimates can be estimated more accurately in supervised analysis because there is less uncertainty in allele frequencies. Interpretation of results is simplified, and run times are shorter owing to the reduced number of parameters to estimate. Both the number of iterations until convergence and the computational complexity per iteration decrease. However, we caution that supervised analysis is only suitable when the reference individuals can be assigned to ancestral populations with certainty and ancestral populations are fairly homogeneous. For exploratory analyses, unsupervised analysis is more appropriate and therefore remains the default in ADMIXTURE.

10As noted in our later comparison of supervised and unsupervised learning, datasets culled from closely related populations typed at a modest numbers of SNPs can pose substantial challenges in ancestry estimation. For instance, overfitting tends to yield ancestry estimates with inflated amounts of admixture. The Bayesian solution to this problem is to impose an informative prior to steer parameter estimates away from danger when data is sparse. Thus, STRUCTURE imposes Dirichlet prior distributions on ancestry parameters and estimates a hyperparameter α that controls the strength of the prior distributions.

11A suitable alternative in our optimization framework is to perform penalized estimation. Rather than maximizing the log-likelihood, we maximize an objective function consisting of the log-likelihood minus a penalty. The penalty is designed to discourage the undesirable biases in the estimated ancestry matrix just mentioned. The tuning constant λ controls the strength of the penalty. While it is tempting to consider the negated logarithm of the Dirichlet prior density appearing in STRUCTURE as a penalty, the Dirichlet(α,..., α) density is unbounded above in the parameter regime α < 1--arguably the most useful setting for the α parameter--and is therefore unusable in our optimization framework. A better alternative is the approximate ℓ0 penalty

12which encourages not only shrinkage but also aggressive parsimony. In particular, the approximate ℓ0 penalty drives small admixture coefficients to zero. Parsimony is desirable because it leads to more easily interpretable and probably more realistic parameter estimates. Estimation is performed by maximizing over its arguments. Increasing λ or the second tuning constant γ elevates the extent of shrinkage and parsimony in the resulting estimates and.

13Determination of the penalty tuning constants λ and γ is nontrivial. In our hands cross-validation has proved effective on simple simulated datasets. The tuning constants λ and γ are user-defined options, so users can explore different settings consistent with cross-validation or their own heuristics.

14Very large datasets (millions of SNPs, thousands of individuals) can reduce even ADMIXTURE's efficient algorithms to a crawl. Since our original publication, we have tuned our core algorithm and improved its speed by a factor of two. We have also implemented a parallel execution mode that lets ADMIXTURE exploit multiple processors. This new option employs the OpenMP framework designed for simple parallelization using compiler #pragma directives. To perform analyses with, for example, four threads, the user need only add the flag -j4 to the command line. Hence

15$ admixture Data/hapmap3.bed 3 -j4

16analyzes the data file hapmap3.bed using 4 threads, assuming K = 3 ancestral populations. Analyses of our hapmap3 dataset with K = 3 were accelerated by 392% on a four processor machine.

17Figure 1 demonstrates the effectiveness of cross-validation on several datasets culled from HapMap 3. For these datasets, cross-validation was able to accurately identify the number of ancestral populations. While we have not performed extensive simulation studies, our experience has shown that the success of cross-validation depends in part on the degree of differentiation between the populations under study as quantified by Wright's fixation index FST. Very closely related populations cannot be accurately separated. We speculate that this phenomenon may have a theoretical connection to the "phase-change" phenomenon observed by Patterson et al.. For a dataset of fixed dimensions, they note that the FST value separating two populations must exceed a certain threshold before the population samples can be reliably distinguished in principal component analysis.

18To explore the benefits of supervised analysis, we generated a number of artificial datasets and evaluated the empirical precision of parameter estimates compared to the true Q and F. The ancestral allele frequencies F were first generated using the Balding-Nichols model for 10,000 markers in each of two populations differentiated by an FST value of.01 (comparable to the genetic distances observed between closely related populations within a continent) and with ancestral allele frequencies drawn uniformly from [0, 1]. Then, for each of 100 datasets, 400 individuals were simulated using ancestries fixed as follows: one hundred individuals with ancestry entirely from population 1, one hundred individuals from population 2, and the remaining two hundred with admixed ancestries spaced uniformly on a grid between population 1 and population 2. Supervised and unsupervised ADMIXTURE analyses performed on these datasets revealed several interesting patterns. First, supervised analysis more accurately recovered the underlying allele frequencies. On average the root-mean-squared error in estimating the vector f1 of reference allele frequencies for population 1 was.046 for unsupervised analysis but.040 for supervised. In general, it appears that errors in estimating F cause overestimation of the FST between the ancestral populations. Indeed, here the average FST estimate of.024 for unsupervised analysis fell to.019 for supervised analysis (true FST of.010).

19The flip-side of the systematic overestimation of the separation between populations is that ancestry fraction estimates suffer from bias. In particular, individuals will be ascribed a greater degree of admixture than they actually possess. Figure 2 illustrates this effect. Individuals with low qi1, reflecting a small degree of ancestry from population 1, have upward-biased estimates, while estimates for those with high qi1 exhibit a downward bias. The net effect is an apparent bias towards ancestry fractions of.5. Supervised analysis appears not to suffer from this bias.

20In our opinion the apparent bias in unsupervised ancestry estimates should not be cause for alarm. The bias becomes much less prominent for larger datasets or datasets where the ancestral populations are better differentiated. Performing the same simulation with an FST of.05, the bias in Q estimates is mitigated substantially, as seen in Figure 2b. A similar effect is apparent when we increase the number of markers J to 100,000 or more.

21Hence, it is evident that supervised analysis, when applicable, can yield more precise estimates that are less susceptible to the biases seen in unsupervised analysis. Another benefit of supervised analysis is that it runs considerably faster. For the 10 simulated datasets with 10,000 markers, supervised analysis took an average of 5.15 seconds, while unsupervised analysis averaged 27.5 seconds.

22The bias in ancestry estimates observed in Figure 2 is principally a problem for small datasets with closely related ancestral populations. Nevertheless, we designed our penalized estimation procedure partly to reduce this bias. To demonstrate the effectiveness of penalization, we explored penalized estimation in the context of the previous simulation of admixed individuals from two populations differentiated by FST =.01. Fixing γ =.1 and performing cross-validation on a single one of these simulated datasets for λ values spaced between 0 and 100, we identified λ = 5 as the value minimizing cross-validation error (Figure 3a). Comparing the ancestry estimates with those from maximum likelihood unsupervised and supervised analyses (Figure 3b) reveals that penalized estimation mitigates bias substantially.

23ADMIXTURE is a fully-featured, highly efficient, and easy-to-use tool for ancestry estimation from SNP datasets. The four enhancements described here promote great flexibility in both exploratory and focused studies of genetic ancestry. Cross-validation enables rational choice of the number of ancestral populations. Supervised analysis mode can yield more accurate ancestry estimates when the number and makeup of contributing populations are certain. Parallelizing the code reduces run times and allows more ambitious analyses involving more people and SNPs. Finally, penalizing weak evidence for admixture promotes model parsimony and yields ancestry fractions more in line with users' expectations.

24Project name: ADMIXTURE

25Project home page: http://www.genetics.ucla.edu/software/admixture; snapshot of software available as Additional File 1.

26Operating systems: Linux, Mac OS X

27Programming languages: C++

28Other requirements: None

29License: Binaries freely available; source code proprietary

30Any restrictions to use by non-academics: None

31The authors declare that they have no competing interests.

32DHA and KL devised the algorithms for penalized estimation, cross-validation, supervised analysis, and parallel execution. DHA implemented the software. DHA and KL designed the experiments, which DHA then executed and analyzed. DHA and KL composed the manuscript. The authors have approved the final manuscript.