Calculation of Tajima’s D and other neutrality test statistics from low depth nextgeneration sequencing data
 Thorfinn Sand Korneliussen^{1}Email author,
 Ida Moltke^{2, 3},
 Anders Albrechtsen^{3} and
 Rasmus Nielsen^{4}
https://doi.org/10.1186/1471210514289
© Korneliussen et al.; licensee BioMed Central Ltd. 2013
Received: 6 May 2013
Accepted: 25 September 2013
Published: 2 October 2013
Abstract
Background
A number of different statistics are used for detecting natural selection using DNA sequencing data, including statistics that are summaries of the frequency spectrum, such as Tajima’s D. These statistics are now often being applied in the analysis of Next Generation Sequencing (NGS) data. However, estimates of frequency spectra from NGS data are strongly affected by low sequencing coverage; the inherent technology dependent variation in sequencing depth causes systematic differences in the value of the statistic among genomic regions.
Results
We have developed an approach that accommodates the uncertainty of the data when calculating site frequency based neutrality test statistics. A salient feature of this approach is that it implicitly solves the problems of varying sequencing depth, missing data and avoids the need to infer variable sites for the analysis and thereby avoids ascertainment problems introduced by a SNP discovery process.
Conclusion
Using an empirical Bayes approach for fast computations, we show that this method produces results for lowcoverage NGS data comparable to those achieved when the genotypes are known without uncertainty. We also validate the method in an analysis of data from the 1000 genomes project. The method is implemented in a fast framework which enables researchers to perform these neutrality tests on a genomewide scale.
Keywords
Background
In the past decade there has been considerable interest in detecting natural selection in humans and other organisms [14] from DNA sequence data. An often used approach for detecting selection is to use a neutrality test statistic based on allele frequencies, with Tajima’s D being the most famous. Such frequency based tests have been used to identify a number of genes, that have undergone selection: lactase [5], ABO blood group [6], the HLA immune complex [7], and the calcium pathway trpv6 [8].
In recent years NextGeneration Sequencing (NGS) has revolutionized the field of genetics in general and population genetics in particular. The underlying technology behind these different highthroughput sequencers are different [9, 10], but common is the unprecedented level of data produced, allowing population geneticists access to full genome data for large population samples at an affordable price.
Applying the neutrality tests directly to the genotypes obtained from SNP chips can lead to biased results because of the way the SNPs were selected. Even though work has been done to correct for this [11], corrections are not possible without knowledge of the selection criteria used to ascertain SNPs for the specific chip. NGS data do not suffer from such ascertainment biases. However, with low and medium depth (<20×), genotype calling from NGS data can lead to other biases [12]. The biases will depend on the method used for genotype calling. Early genotype calling methods were based on simple criteria, such as calling a heterozygous site, if the proportion of nonreference alleles is between 20% and 80% [13, 14]. Most commonly used genotype callers today are probabilistic, i.e. they are based on genotype likelihoods (GLs) [15]. A GL is the probability of the observed sequencing data given a genotype. GLs are obtained from the raw NGS read data, typically using information such as position in the read, quality score, and possibly type of mutation [16, 17]. A standard approach for data analysis is to call genotypes, and then estimate neutrality statistics. Most genotype callers are relatively conservative and only call a site to be heterozygous if there is substantial evidence that it actually is heterozygous. Such methods will tend to underestimate the allele frequency of the minor allele. Less conservative methods will tend to produce too many rare variants. In either case, an unbiased estimate of the allele frequency cannot be obtained for all classes of allele frequencies [1, 18, 19]. In some studies, an attempt to alleviate this problem has made by only calling a genotype when there is substantial statistical evidence supporting the genotype call. Such approaches will generate a considerable amount of missing data which leads to biases if not adequately dealt with [20]. The effect of these errors on downstream analyses can be severe. For most data produced, the strongest bias has been an excess of singletons (derived alleles segregating in a single copy in the sample). If just a single individual in a sample is miscalled to be heterozygous in a monomorphic site, the site will appear as polymorphic singleton site. The effects of genotype calling for downstream population genetic analyses have been extensively illustrated using simulations by [12]. Other approaches have been suggested: Achaz, 2008 [21] proposed modified statistics that excludes the singletons to address the problem of high error rate in sequencing data. Liu et al., 2010 [22] does not use the quality scores but uses an error rate for each nucleotide for all samples and [23] which uses a fixed error rate. This is contrasted by the methods of [19, 24] that can use quality scores for SNP discovery and incorporating the quality scores into the parameter estimation directly.
We will here show, using both simulated and real human data that, that Tajima’s D calculated using genotypes called from NGS data can lead to severely biased results. The level of bias depends on the sequencing depths and error rates, but disturbingly it also depends on whether or not the data set is neutral or not. This in effect leads to lower power to detect selection.
We argue, that the solution to this problem is to avoid genotype calling, but instead incorporate uncertainty in the genotypes through direct analyses of the genotype likelihood (GL). This approach also solves the issue of missing data, and the problem of inferring which sites are variable. We propose two methods for estimating the neutrality tests statistics from low depth NGS data. In both methods the estimates are based on GLs instead of called genotypes. The first approach is based on obtaining a Maximum Likelihood (ML) estimate of the sample allele frequency spectrum. This approach takes all the uncertainty of the NGS data into account, and provides estimates of the neutrality tests statistics from the estimated sample frequency spectrum. However, it is too slow for genomewide window based scans. The second approach is computationally feasible for entire genomes of any magnitude. It uses an empirical Bayes approach, that also uses an ML estimate of the site frequency spectrum. This estimate is used as a prior for calculating site specific posterior probabilities for the sample frequency spectrum. We show, using both simulated and real low depth NGS data, that the test statistics estimated using the empirical Bayes approach are at most weakly biased and provide a computationally attractive alternative to genotype calling methods.
Methods
Estimators of θand neutrality tests
In a set of n aligned DNA sequences, the frequency spectrum (SFS) is defined as the vector η = (η_{ k })_{k = 0,…,n} where η_{ k } is the number of sites with k derived alleles. A derived allele is a new mutation and the ancestral allele is the nucleotide occurring in the site before mutation. Notice that these population genetic models assume an infinite sites model, and there is therefore an unambiguous definition of ancestral and derived. In real data, outgroup information is used to estimate which allele is ancestral and which allele is derived. Using this notation, the number of segregating, i.e. polymorphic, sites is then given by $S={\displaystyle {\sum}_{i=1}^{n1}{\eta}_{\mathrm{i}}}$. The most commonly applied estimators of θ=4 N μ, (N is the effective population size and μ the mutation rate) are linear functions of the frequency spectrum. These estimators take the general form: $\widehat{\theta}={\displaystyle \sum _{i=0}^{n}{\alpha}_{i}{\eta}_{i}}$, where α_{ i } are different vectors of weights used in constructing the estimators. Some of the common estimators include:

${\widehat{\theta}}_{\mathrm{w}}{=\mathrm{a}}_{1}^{\u20121}{\displaystyle \sum _{\mathrm{i}=1}^{n1}{\eta}_{\mathrm{i}},}\phantom{\rule{0.25em}{0ex}}$ with ${a}_{1}={\displaystyle \sum _{i=1}^{n1}i}.$[25]

${\widehat{\theta}}_{\pi}={\left(\begin{array}{c}\hfill \mathrm{n}\hfill \\ \hfill 2\hfill \end{array}\right)}^{1}{\displaystyle \sum _{\mathrm{i}=1}^{n1}i\left(ni\right){\eta}_{\mathrm{i}}}.$[26]

${\widehat{\theta}}_{\mathit{FL}}={\eta}_{1}.$[27]

${\widehat{\theta}}_{H}={\left(\begin{array}{c}\hfill \mathrm{n}\hfill \\ \hfill 2\hfill \end{array}\right)}^{1}{\displaystyle \sum _{\mathrm{i}=1}^{n1}{i}^{2}{\eta}_{\mathrm{i}}}.$[28]
For more background on these estimators, see for example [29]. In brief, notice that the estimators differ in how they weight polymorphisms with different allele frequencies. The ${\widehat{\theta}}_{\mathrm{w}}$, and ${\widehat{\theta}}_{\pi}$ estimators are symmetric in that they assign equal weight to ancestral and derived alleles of the same frequency. However, ${\widehat{\theta}}_{\pi}$ assigns more weight to alleles segregating at intermediate frequencies while ${\widehat{\theta}}_{\mathrm{w}}$ weights all categories equal. In contrast, ${\widehat{\theta}}_{\mathit{FL}}$ only assigns positive weight to derived singletons and ${\widehat{\theta}}_{H}$ assigns more weight to derived alleles of high frequency. The first two estimators can be calculated without knowledge of the ancestral by using the folded SFS, ${\eta}^{*}={\left({\eta}_{k}^{*}\right)}_{k=0,\dots ,n/2},{\eta}_{k}^{*}={\eta}_{k}+{\eta}_{nk}$, assuming n is even. The latter two estimators require knowledge regarding ancestral states, i.e. they use the unfolded SFS (η).
where the variance usually is the variance expected under a standard neutral model without recombination. For example, Tajima’s D is given by setting ${\widehat{\theta}}_{1}={\widehat{\theta}}_{\pi}$ and ${\widehat{\theta}}_{2}={\widehat{\theta}}_{\mathrm{w}}$. Tajima’s D is often used for detecting selective sweeps, i.e. the effect of an advantageous mutation going to fixation in the population. After fixation, of an advantageous allele, there is an excess of rare alleles in the population. This will cause Tajima’s D to become negative as the expectation of ${\widehat{\theta}}_{\pi}$ will be smaller than the expectation of ${\widehat{\theta}}_{\mathrm{w}}$, when the SFS contains relatively more rare alleles than expected under a standard neutral model.
Genotype likelihoods
The GL can be calculated by assuming independence of the reads covering a position. Some methods take into account the position within the read and does recalibration which take into account cycle dependencies [17]. The implementation in [17] however assumes that all reads have the same length. The widely used SAMtools uses a model derived from the Maq program [15, 31]. These GL methods are all single sample based in contrast to the method of [32], which estimates type specific errors for multi samples jointly. In this paper we base our simulations on a simplified method of [16].
Model
Full maximum likelihood
Several recent papers have suggested methods for estimating the SFS from NGS data [31, 3335]. One approach that we will explore for calculating neutrality statistics (T) for a region, is to estimate the SFS, η = (η_{ k })_{k = 0,…,n}, using one of these methods, in this case the maximum likelihood method by [35] (${\widehat{\eta}}_{\mathit{ML}}$), and then calculate T using estimated values of η instead of the (unknown) true values. As (${\widehat{\eta}}_{\mathit{ML}}$) has previously been shown to perform well under simple conditions, we would expect this approach for calculating the frequency spectrum to perform well under similar conditions. We note that the uncertainty regarding genotypes and the effect of sequencing and mapping errors are incorporated in this approach through the calculation of GLs as described in the section above, but we will otherwise refer readers to [35] for information regarding algorithmic details. Neutrality tests are normally performed using sliding windows in a genome. Maximum likelihood estimation of the SFS for all windows may be computationally challenging. For this reason we also explore other approaches below.
Empirical Bayes
Calling genotypes
Notice that we here, for notational convenience, code genotypes as elements of {0, 1, 2} instead of pairs of nucleotides that each are elements of {A, C, T, G}. The first method chooses the genotype with the highest GL. The second procedure (a type of maximum a posteriori probability estimate) calls genotypes by first estimating the sample allele frequency f from a larger group of individuals, and then uses this estimate of f as a prior on the 3 genotypes using the assumption of HardyWeinberg equilibrium (HWE).
Simply calling all genotypes would result in too many false heterozygotes due to the sequencing errors, resulting in too many variable sites and an inflation of the singleton category of the SFS. Therefore, we also need to select an inclusion criterion for the sites we deem variable, i.e. sites in which we allow the existence of heterozygotes (SNP calling). To do so we first perform a likelihood ratio test of the null hypothesis H_{0}: f = 0, combining the data from all individuals [32]. We do so assuming that all sites are diallelic, and use the approach described in the supplementary of [36] to identify the two most likely alleles for a given site. Throughout we use the maximum likelihood estimator of f by [32].
Simulations
To assess the performance of our estimators we simulate genomic data under a standard neutral model and under models that deviate from the neutral model using msms [37]. We set the population size, mutation rate and the recombination rate to realistic values for humans, N = 10, 000, μ = 2.35 · 10^{− 8}, r = 10^{− 8}[38, 39] and use an infinite sites model. As msms only prints out variable sites with binary coding, we insert invariable sites in the sequences and convert from binary coding to nucleotides by sampling randomly with equal probability from all four nucleotides. We then collapse the simulated haplotypes into genotypes. For the nonneutral scenarios we simulate strong positive selection under an additive model with a selection coefficient of 0.1.
where M is the sequencing depth, b_{ i } the observed base in read i, e is the probability of error and G = {A_{1},A_{2}}. In our simulations, we assume an equal error rate for all bases and for all sites. The sequencing depth is sampled from a Poisson distribution with mean equal to the specified mean depth. The simulations then proceed by first simulating G using msms and then simulating D in accordance with the formula given above. The latter stage is achieved by sampling M nucleotides from G, and then introducing errors independently in each of them with probability e. For each site GLs are then calculated according to the GATK model given above.
It should be clarified that we do not actually simulate reads, but sample bases for every site independently. Hence there is a dependency in sequencing depth between adjacent sites in real data that we do not model.
HapMap 2 and 1000genomes
For evaluating the performance of the estimators on real data we used 15 unrelated CEU samples from HapMap phase 2 data [41], sequenced by the 1000 genomes project [42] using Illumina sequencing. Based on the mapped reads we used ANGSD http://www.popgen.dk/angsd to align the 15 mapped samples and calculate the genotype likelihoods using the GATK error model. Ancestral states for all sites were obtained from the multiz46way dataset http://hgdownload.cse.ucsc.edu/goldenPath/hg19/multiz46way/ available from the UCSC browser.
Results
The effect of genotype calling for low or medium coverage data
The effect of SNP calling criteria when calling genotypes
In the following sections, we therefore compare our methods to results using several different cutoffs for genotype calling.
Estimating Tajima’s Dusing ML estimates of the SFS
The effect of sample size and sequencing depth
Mean estimate of theta estimators for 100x1MB regions for 25 samples
Selection  No selection  

Method  θ _{ w }  θ _{ π }  θ _{ FL }  θ _{ w }  θ _{ π }  θ _{ FL } 
Known  649.8  567.1  745.1  930.0  923.5  939.0 
ML  648.6  567.9  734.7  932.4  923.7  968.3 
GChwe 10^{6}  341.7  372.1  72.2  541.3  640.3  99.9 
GCmLike 10^{6}  341.7  497.1  32.9  541.3  768.7  44.8 
GChwe 10^{3}  556.1  434.7  583.5  784.8  712.4  680.6 
GCmLike 10^{3}  556.1  598.9  360.4  784.8  897.9  410.1 
GChwe 10^{5}  1011.0  594.8  653.7  1244.4  874.1  752.8 
GCmLike 10^{5}  1011.0  831.8  420.4  1244.4  1131.6  471.8 
Application to data simulated with a selective sweep
Effect of the prior
We observe the same trend for other neutrality tests statistics that can be seen in Additional file 4: Figure S4 and Additional file 5: Figure S5. However, the other investigated neutrality test statistics, Fu & Li’s D and F, are not estimated with the same accuracy as Tajima’s D. For both selection and neutral data sets, we seem to underestimate the statistics with the EB approach and neutral prior (Additional file 4: Figure S4 and Additional file 5: Figure S5). From figure (S4,S5) we also see that we have problems estimating the true value in the region of the targeted locus using the 50 kb ML approach. The difference is perhaps caused by the fact that Fu & Li’s statistics are based on a single category of the frequency spectrum, whereas Tajima’s D is based on all categories.
Power to detect a selective sweep
Often the goal of the investigator is not to estimate the neutrality test statistics in an unbiased manner but instead the goal is to identify regions with extreme values. To investigate our ability to discriminate between regions with selection and neutral regions, we show receiver operating characteristic (ROC) for the different approaches. These are shown in Additional file 6: Figure S6, for different depth, error rates and number of individuals. When the depth is high, all methods perform almost as well as when the genotypes are known without error. But when the depth decreases or the error rates increases, only the full maximum likelihood approach and the EB approach have similar power as the known genotypes.
As expected, the simulations show that our methods have improved power to discriminate between regions evolving neutrally and under positive selection as more samples are added (Additional file 6: Figure S6). The genotype calling methods perform worse when the error rate is increased and the depth is decreased, especially at low depth with a low pvalue cutoff, while the ML method for all scenarios performs almost as well as if the true genotypes where known (Additional file 7: Figure S7).
Variable depth and variable error rates from NGS data
So far, we have used a simple simulation model assuming sequencing depths follow a Poisson distribution and a fixed error rate. To examine the robustness of our conclusions to these assumptions, we made an additional set of simulations using the observed distribution of quality scores and sequencing depths tabulated from BAM files from the 1000 Genomes project (Additional file 8: Figure S8). As before we simulate 100 1 MB regions with and without selection for 25 individuals, and apply our two genotype calling methods and the EB method to the simulated data (Additional file 9: Figure S9). For the genotype calling approaches we only included sites that were likely to be polymorphic with a pvalue <10^{6}. We observe results that are highly compatible with the previous results. The EB method is approximately unbiased in both scenarios and have similarly small mean squared error (6e4, 5e4). In contrast, the genotype calling methods are biased in both scenarios, and we notice again that the bias depends on whether or not the data is simulated under a neutral or under a selection model. We observe that the MSE for the genotype calling methods are orders of magnitude larger than the EB method, and that the MSE for the GCmLike model is more than twice as high under the selection scenario as under the neutral scenario, whereas we observe the opposite trend for GChwe method.
Application to real data
Discussion
We have developed two methods to perform the neutrality tests on NGS data that take the uncertainty of the genotypes into account. Both methods perform better than using called genotypes and in most instances they are approximately unbiased. The full ML method is computationally slow when applied in sliding windows at a genomewide scale, which is why we also present a fast empirical Bayes method with a prior that is estimated from the data itself, for example the entire genome, or a reasonable subset of the genome. This makes the method computationally feasible for full genomic data of any magnitude and any windows size.
There is not a single obvious way to identify SNP sites and call genotypes from NGS data. Here we have tried different approaches with different cutoffs. Regardless of method and chosen cutoff they all show a large bias in some or all simulated scenarios. This is evident from the raw theta estimates (Table 1), and the actual test statistics (Figure 1). The level of bias varies between the different scenarios, not only for different depths and error rates, but it also depends on whether or not the data set is neutral or affected by selection (Figure 2). The results from this study suggest/confirms it is not unproblematic to perform neutrality tests on genotypes called from low or medium coverage NGS data.
In contrast, both the ML and the EB approach give fairly accurate estimates for all the examined measures. When applying a neutral genomewide prior for our analysis, we observed only small deviations from the true values of Tajima’s D even for very low depth data. When applying the EB approach we did observe a small bias in the regions under selection when the prior was estimated from regions without selection (see Figure 7). However, the bias is always much smaller than the bias of the GC approach. Even though the EB approach can give small biases it can still have an advantage over the full ML approach. When estimating the neutrality test statistics for small windows, we often obtained a few extreme outliers with positive values of Tajima’s D for the ML approach (see Figure 6). Since the EB method uses the entire 1 Mb region as prior we do not see a similar problem with extreme positive outliers and the variance of the estimates is smaller overall. If the goal is to identify regions under selection the smaller variance of the EB approach will give fewer regions with extreme values. This is because regions with little data will increase the variance for the full ML approach while they will give results closer to the prior for the EB approach. This problem could also be circumvented by using a sliding window approach with window sizes determined by using a fixed number of SNPs.
We applied the EB method to data from 15 individuals from the 1000 genomes project, and observed trends similar to previous published results for the LCT region in Europeans. For SNP chip data from the same region, we observed a large overrepresentation of positive Tajima’s D values, presumably due to ascertainment biases introduced by the SNPdiscovery and selection process from the SNPchip itself. Similar positive values were not observed using the EB method based on the NGS data. For both the GC and EB methods, we observe negative values around the LCT gene, however the estimates are not very extreme for the GC approach. The EB approach is the only approach for which LCT has the most extreme Tajima’s D value.
The main advantage of the approaches presented here, is that they, in expectation, have at most a very weak dependence on sequencing depth. This facilitates the use of genomewide scans on genomes with varying sequencing depth without introducing biases. The computational framework suggested here, based on the EB approach, provides a robust and computationally fast method for scanning a genome for regions with outlying or extreme frequency spectrum. Such a method should be of great use in years to come when analyzing population genomic data from a variety of different species.
Conclusion
In this paper we show through simulations that estimating neutrality test statistics using called genotypes can lead to highly biased result. The bias is dependent not only on sequencing depth and error rate but interestingly the bias also depends on whether or not the region is under selection or not. We have developed an empirical Bayes method that can calculate the test statistic fast and efficiently. This method circumvents the problem of SNP discovery, genotype calling and missing data, which is a fundamental problem of NGS data. This is done by working with genotype likelihoods, which contains all relevant information about the uncertainty of the data. Using this approach leads to approximately unbiased estimates in most instances.
Availability
All methods discussed in this paper are freely available as part of the Analyses of Next generation Sequencing Data (ANGSD) package (http://www.popgen.dk/angsd).
Declarations
Acknowledgements
Work for this manuscript was financed through Danish National Research Foundation, (DNRF94) Villum Foundation and Danish Council for Independent Research.
Authors’ Affiliations
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