On Association Analysis of Rare Variants Under Population Substructure: An Approach for the Detection of Subjects That Can Cause Bias in the Analysis—T_opt: An Outlier Detection Method

Abstract

For the analysis of rare-variant data in population-based designs, we propose a method to detect study subjects that may create population substructure in the study sample. Our approach is computationally fast and simple, permitting applications to whole-genome sequencing studies. The method does not require the variants to be in linkage equilibrium and can be applied to all the genetic loci that are available in the study. For both rare and common variants, we assess the performance of our approach by its application to the 1000 Genome Project data, and in simulation studies. The results are compared to the commonly used outlier detection algorithm based on principal component analysis (PCA). The statistical power of both approaches to detect outliers are comparable in most of the scenarios, but the power of PCA to detect outliers is lower than the novel approach in the presence of linkage disequilibrium and for subpopulations that are genetically similar. The data analysis and the simulation studies suggest that the number of false-positive results appears to be different for the two approaches. Our approach maintains the type I error rate while the outlier detection approach based on PCA does not. Taking additionally into account the minimal computational requirements of our approach and the ability to incorporate all the marker information, the proposed method will have important application in sequencing studies and genome-wide association studies.

Introduction

Genetic association analysis has proven to be a powerful statistical tool for the identification of disease loci in the human genome [Consortium, 2007; McCarthy et al., 2008; Stranger et al., 2011]. Population-based association analysis is straight-forward and computationally fast, even at a whole-genome level. One of the main caveats of population-based association analysis, however, is that it can be susceptible to bias due to genetic confounding, i.e., population substructure.

This issue has been the focus of statistical research for some time. In designs of unrelated individuals, most genetic association tests take the form of a score test in which the numerator sums the contributions of the study subjects to the statistics and the denominator calculates the variance of the statistic, assuming independence of the study subjects. In the presence of mating among relatives or population substructure, the genotypes of the study subjects are no longer independent, leading to a potentially biased estimate for the variance of the test statistic. This can cause the test statistic to become anticonservative. Genomic control approach adjusts for the bias in the variance of the test statistic by estimating a variance inflation factor at a set of reference loci and scaling the variance of the test statistic accordingly [Devlin and Roeder, 1999; Reich and Goldstein, 2001]. With the arrival of Genome-wide genetic (GWAS) data, principal component analysis (PCA) gained popularity [Patterson et al., 2006; Price et al., 2006]. They infer population substructure and admixture based on the PCA of the variance-covariance matrix of genotyped markers [McVean, 2009;Novembre and Stephens, 2008]. Then, the principal components (PCs) are either used to identify genetically homogeneous subpopulations in the study [Luca et al., 2008] or to adjust the association for genetic confounding [Price et al., 2006].

For the association analysis of rare variants, the application of such approaches to avoid bias due to population substructure and admixture can be problematic. In PCA approach, the estimation of the variance/covariance matrix can become unstable for genetic loci with low minor allele frequencies, making the results of this approach less reliable. For example, the investigators usually select markers with allele frequencies greater than 10% before applying PCA [He et al., 2011; Sladek et al., 2007]. An alternative that could be considered here is to assess population substructure for loci with common alleles and apply the PC results to the rare-variant analysis, assuming that the population substructures for rare and common variants are the same. The transferability of population substructure between common and rare genetic loci is a hypothesis which has not been assessed thoroughly based on real data so far. The general applicability of this concept seems to be problematic in light of the age of the different variant types, i.e., common variants are genetically much older than rare variants [Mathieson and McVean, 2012]. Although rare-variant approaches rely mostly on permutation tests for the assessment of the significance, the concept of genomic control generally can be modified and applied to rare-variant analysis. However, it can give a reduced power [Price et al., 2010] and cannot be utilized to identify homogenous subpopulations.

Here, we propose a simple, computationally fast approach that allows the identification of genetic outliers to obtain a genetically homogeneous subpopulation in studies with sequence data, minimizing the impact of population substructure on rare-variants analysis. The approach is able to utilize the information on all available genetic loci and does not require the selection of a subset of markers that are in linkage equilibrium (LE). The test statistic is computed for each individual based on all the rare-variant information available. The power and the type I error of the approach are examined in simulation studies and by the applications to the 1000 Genome Project data. We compare the performance of our approach with the outlier detection algorithm based on PCA.

Methods

Introducing Test Statistics T₁ and T₂

Suppose in a genetic association study of unrelated individuals, genotype data are available at m bi-allelic loci for all the study subjects. We denote the number of the minor alleles at the ith marker locus by X_i for one subject. We define the genetic residual byΔ X_i = X_i − E (X_i) where E (X_i) is the expected number of the minor alleles at the ith locus in the study population. The genetic residual can be considered as the genetic deviation of the subject at ith locus from the study population. We define two genome-wide scores that measure the distance between a particular individual and the population across the genome. The scores are given by

$$S_1=\sum _{i=1}^m\mathrm{\Delta }{X}_i=\sum _{i=1}^m(X_i-E(X_i))$$

and

$$S_2=\sum _{i=1}^m|\mathrm{\Delta }{X}_i|=\sum _{i=1}^m|X_i-E(X_i)|.$$

Based on the scores, we can construct the score tests T₁ and T₂ which are given by

$$T_1=R_1^2=\frac{{(S_1-E(S_1))}^2}{\text{Var}(S_1)}$$

and

$$T_2=R_2^2=\frac{{(S_2-E(S_2))}^2}{\text{Var}(S_2)}.$$

The first score aggregates the residuals over all the marker loci for one subject. If, for the study population and the population where the outliers are from, there is preferentially a one-direction difference in the minor allele frequency (MAF), i.e., most of the markers have smaller MAF in one population than in the other population, then the test score S₁ will be more powerful in detecting the population outliers. This situation can occur due to the founder effects in one subpopulation [Reich et al., 2001; Roy-Gagnon et al., 2011], long-range haplotypes [Price et al., 2008], etc. However, if the differences in minor allele frequencies between two subpopulations do not follow this patterns, test statistic S₂ is generally better suited to identify genetically different subjects. In the supplementary Note III, we provide the theoretically justification for that. We will further outline these features of the score statistics S₁ and S₂ in the simulation section of this paper.

Under the assumption of Hardy-Weinberg equilibrium (HWE), the expected marker score can be calculated based on the minor allele frequency, i.e., E (X_i) = 2p_i, where p _i is the true minor allele frequency at ith marker locus. For large datasets, we can estimate the allele frequency p _i by the observed frequency of the minor allele in the actual data and the asymptotic distribution stays the same. Alternatively, the allele frequencies can be obtained from the corresponding reference populations. Assuming the absence of LD between the loci, the mean and variance of S₁ and S₂ can be derived analytically based on the allele frequencies, as shown in supplementary Note I. Then the test statistics are given by:

$$T_1=\frac{{[\sum _{i=1}^m(X_i-2p_i)]}^2}{\sum _{i=1}^m(2p_i(1-p_i))}$$ (1)

$$T_2=\frac{{[\sum _{i=1}^m|X_i-E(X_i)|-(2p_i(1-p_i))]}^2}{\sum _{i=1}^m(2p_i(1-p_i)(1-8p_i{(1-p_i)}^3))}$$ (2)

Then, under the null hypothesis of no population substructure, both test statistics T₁ and T₂ follow a χ² distribution with one degree of freedom asymptotically.

Adjusting T₁ and T₂ in the Presence of LD

For sequence data, the no LD assumption is not reasonable unless only a subset of loci that are in LE is selected. In the presence of LD, both standardized scores have to be adjusted accordingly. Since the variances of S₁ and S₂ do not depend on the actual genotype of the study subject and are constant across the subjects, ideally, we would need to adjust T₁ by

$$\frac{\text{Var}\left(\sum _{i=1}^m\mathrm{\Delta }{X}_i\right)}{\sum _{i=1}^m\text{Var}(\mathrm{\Delta }{X}_i)}$$

This adjustment is only reasonable when the effect of LD is a linear inflation of T₁, which is the case here. This is due to the fact that the denominator of the test statistic is the same across all the subjects as it aggregates over all subjects and does not depend on the each subject’s genotype. Thus, we can use one value to correct for the inflation under LD. To estimate the inflation factor, since the calculation of the correlations of the residuals across the genome requires a great amount of computation time, the genomic inflation factor for each test statistic can be estimated based on the distribution of the test statistic across the study subjects. For test statistic T₁, we estimate the genomic inflation λ₁ by

$${\mathrm{\lambda ̂}}_1=\frac{\text{Median of}{T}_1\text{across all subjects}}{0.455}$$ (3)

where 0.455 is the 50th percentile of a ${\chi }_{(1)}^2$ distribution. Similarly for T₂, we estimate the genomic inflation factor λ₂ by

$${\mathrm{\lambda ̂}}_2=\frac{\text{Median of}{T}_2\text{across all subjects}}{0.455}$$ (4)

In the presence of LD, we can adjust T₁ using the subject inflation factor λ₁ by

$$\frac{1}{{\lambda }_1}{T}_1=\frac{{(S_1-E(S_1))}^2}{{\lambda }_1\sum _{i=1}^m\text{Var}(\mathrm{\Delta }{X}_i)}\sim {\chi }_{(1)}^2$$ (5)

The adjusted test statistic T₂ is derived in the same way. Under the null-hypothesis that the subject is from the study population, the test statistics T₁ and T₂ have an asymptotic χ²-distribution with one degree of freedom.

The Optimal Test and Its Asymptotic Distribution

Since, prior to the calculation of the test statistic, we do not have any knowledge whether test statistic T₁ or T₂ is more suitable for the analyzed study subject, we define the genome-wide test statistic to detect genetic outliers in rare-variant data as:

$$T_{\text{opt}}=\text{max}(T_1,T_2)$$ (6)

We already know that assuming no LD between the markers, and under the null hypothesis that the subject under study is from the given population, the standardized test statistics T₁ and T₂ follow a ${\chi }_{(1)}^2$ distribution asymptotically. To derive the asymptotic distribution of T_opt, we need to incorporate the correlation between the test statistics T₁ and T₂. In the absence of LD between the genetic loci, an estimator of the correlation between R₁ and R₂ based on the allele frequencies of the loci can be easily derived (supplementary Note II). As an alternative approach or in the presence of LD, the correlation between R₁ and R₂ can also be estimated by the empirical correlation between the statistics R₁ and R₂ in the study (supplementary Note II). Given the estimate for the correlation/covariance of R₁ and R₂, the asymptotic distribution of T_opt can be obtained under the null hypothesis, by simulating from a bivariate normal distribution with the estimated correlation. In supplementary Note II, we outline the derivation of the asymptotic distribution for T_opt in more details.

Results

We examined the performance of the proposed test statistic T_opt by its applications to the third version of 1000 Genome Project data, and in simulation studies with sequencing and GWAS data. In all applications and simulation scenarios, the approach was compared to the outlier detection algorithm based on PCA. For this comparison, we selected the smartpca implementation of PCA in the package EIGENSOFT version 3.0 [Price et al., 2006].

Applications to 1000 Genome Project Data

The 1000 Genome Project [The 1000 Genomes Project Consortium, 2010] data provide a unique framework to validate our approach based on real data. We applied the novel test to the third release of the variant call set based on both low coverage and exome whole-genome sequence data from the 1000 Genome Project [The 1000 Genomes Project Consortium, 2010]. The release contains the genotype calls of 1,092 samples from 14 different populations. We combined three pairs of populations to investigate the power, type I error, and family-wise error rate (FWER) of the test. The three pairs are Han Chinese in Beijing, China (CHB) and Japanese in Tokyo, Japan (JPT) (F_st = 0.007; supplementary information of Altshuler et al. [2010]), Tuscany in Italy (TSI) and Finnish from Finland (FIN; F_st = 0.020) [Nelis et al., 2009], Yoruba in Ibadan, Nigeria (YRI) and Luhya in Webuye, Kenya (LWK) (F_st = 0.008; supplementary information of Altshuler et al. [2010]). Since these populations can be considered to be genetically homogeneous they are an ideal validation tool for methodology to detect population substructure. The general idea is to create datasets that consist of one population, and include one additional subject that is not part of the population.

We focused only on the single-nucleotide polymorphisms (SNPs) calls, thus any information on the short Indels or large deletions was ignored. Quality control process has been implemented (autosomal SNPs with call rate > 98%, HWE P-value > 0.000001 that are not in the long-range LD regions [Price et al., 2008], and unrelated subjects with call rate > 98%), we are left with approximately 11M variants for the combined datasets CHB and JPT, and FIN and TSI, and with approximately 19M variants for the combined datasets LWK and YRI. To apply PCA, the three combined datasets, CHB and JPT, FIN and TSI, and LWK and YRI, have been pruned to include SNPs with MAF > 10% and with pairwise r² < 0.05 in each 50 SNPs window with a step size of five SNPs. This pruned dataset for CHB and JPT includes about 92K SNPs, similar for FIN and TSI. The pruned dataset for LWK and YRI includes about 150K SNPs. To compare with the new test, PCA was also applied to the variants with MAF≤ 5% without any LD pruning. There are about 6M–7M such variants for the combined datasets of CHB and JPT, and of FIN and TSI, and there are about 13M such variants (MAF ≤ 5%) for the combined datasets of LWK and YRI.

In each application/replicate, we assess whether the two methods correctly identify the subject that is not part of the population as an outlier. A subject is rejected as an outlier if its test statistic T_opt is greater than the value corresponding to the significance level 0.05/n where n is the number of subjects in the dataset. The type I error is the average percentage of incorrectly rejected subjects among the combined datasets for each scenario. The family-wise error rate (FWER) is the percentage of times that there is at least one incorrectly rejected subjects in the datasets. The methods were applied first to all the common SNPs, which, for PCA, are the pruned SNPs with minor allele frequency >10% and which, for our approach, are all the available SNPs, including rare SNPs and SNPs in the long-range LD regions [Price et al., 2008]. Then, we applied the two approaches to the rare SNPs (minor allele frequency < 5%).

The power, type I error, and FWER estimates are shown in Table 1. For this table, we used the default values recommended in the package, i.e., 10 for the number of PCs used for determining outliers, and 6 for the number of standard deviations of which the subject must deviate in any of the top 10 PCs to be removed as an outlier. We also used the default maximum number of outlier removal iterations, which is 5 in the process. The table shows that PCA cannot detect the outlier using the pruned SNP set with MAF > 10% due to the small number of SNPs included in the pruned data and the closeness of the two populations. PCA has a good power to detect the outlier using SNPs with MAF ≤ 5%. However, the outlier detection algorithm based on PCA does not control for the type I error or the FWER, which would result in the unnecessary removal of samples. The new statistic T_opt has a good power to detect the outliers, especially for the more distant pairs, TSI and FIN, and LWK and YRI. The type I error and the FWER are mostly controlled well except the case where the JPT population is combined with one CHB subject. In this scenario, there are three JPT subjects that are detected as outliers in most of the combined datasets due to the small genetic difference between JPT and CHB (F_st = 0.007; supplementary information of Altshuler et al. [2010]), and two examples are shown in Figure 1. We can see that in the examples, T_opt is able to detect the outlier, but it also detected some CHB subjects as the outliers, whereas the outlier detection algorithm based on PCA applying to SNPs with MAF < 5% could not identify the CHB outlier, but rejects the JPT subject at index 34 (NA18978).

Table 1.

The estimated family-wise error rate (FWER), the average type I error (TI) and the power of T_opt and the outlier detection process based on PCA when they were applied to the combined 1,000 genome datasets

Estimates	Pop Outlier	CHB JPT	JPT CHB	TSI FIN	FIN TSI	LWK YRI	YRI LWK
PCA (MAF > 10%)	FWER	0.00	0.00	0.00	0.00	1.00	0.00
	TI	0.00	0.00	0.00	0.00	0.0125	0.00
	POWER	0.00	0.00	0.151	0.00	0.0349	0.00
PCA (MAF < 5%)	FWER	1.00	1.00	0.151	0.990	1.00	1.00
	TI	0.144	0.0935	0.00351	0.0765	0.0489	0.0234
	POWER	0.843	0.443	0.957	0.888	0.570	1.00
Topt (MAF < 5%)	FWER	0.00	1.00	0.00	0.00	0.00	0.00
	TI	0.00	0.0415	0.00	0.00	0.00	0.00
	POWER	0.146	0.495	1.00	1.00	0.988	1.00
Topt (all SNPs)	FWER	0.0225	1.00	0.882	0.00	0.00	0.00
	TI	0.00	0.0365	0.000869	0.00	0.00	0.00
	POWER	0.0562	0.0928	0.720	0.969	0.00	0.861

Figure 1.

PC plots and T_opt plots for two randomly selected datasets with JPT subjects and one CHB subject. The outlier (one CHB subject) in both examples has an index of 90. Only SNPs with MAF < 5% were considered. The indices of some of the subjects are shown on the left of their points in the plots.

Note that there are a few surprises here. One is that the asymmetry in the power for the dataset of LWK with one YRI sample as the outlier and the dataset of YRI with one LWK as the outlier. This can be explained by the larger genetic variation of the LWK population than the YRI population. However, we observe that, using SNPs with MAF ≤ 5%, we have a much better power to detect the YRI outliers included in the LWK samples than using all the SNPs. This may be due to the fact that a lot of variants that contribute in distinguishing the two populations are rare since the separation of the two populations are relatively recent. Adding more common variants only adds noise to the difference and it overwhelms the information given by the rare variants. This suggests the importance of using SNPs with small MAF to distinguish genetically close populations. This is also true for the other combined datasets.

In practice, researchers would examine the PC plots manually to determine the outliers rather than using the default parameters and thresholds. However, due to the large number of simulated datasets we have, we are unable to do that for each combined datasets. To maximize the performance of the outlier detection algorithm based on PCA, we examined the effects of different parameters used in the outlier removal process based on PCA. Table 2 shows the effects of including different numbers of PCs on the power, type I error, and FWER to detect the outlier. The number of PCs to be used can be determined using the Tracy-Widom statistic in practice [Patterson et al., 2006]. There is one outlier included in each combined dataset and only SNPs with MAF < 5% are considered. We observe a decrease in the type I error rate as the number of PCs decreases, but it is still above the significance level (0.05/n where n is the number of subjects in the dataset). Also, we observe a decrease in the power to detect outliers and a constant FWER of 1.0 as the number of PCs decreases in most of the cases. Similarly, as the number of standard deviations used to determine the outlier decreases, the power increases but the type I error rate also increases. We also changed the number of iterations to 1 rather than using the default value 5, and we observe lower type I error in most of the cases, but the power is lower too (supplementary Table S1). Therefore, changing the parameters used in the outlier detection process based on PCA does not improve its overall performance in terms of both FWER and power comparing to T_opt as T_opt performs better in both aspects.

Table 2.

The estimated family-wise error rate (FWER), the average type I error (TI), and the power of the outlier detection process based on PCA when different numbers of principal components are used to determine the outliers. There is one outlier included in each combined dataset and only SNPs with MAF < 5% are used

Pop	Outlier	Number of PC	2	4	10	20
CHB	JPT	FWER	1.00	1.00	1.00	1.00
		TI	0.010	0.064	0.144	0.147
		Power	0.169	0.427	0.843	0.888
JPT	CHB	FWER	1.00	1.00	1.00	1.00
		TI	0.044	0.057	0.0935	0.120
		Power	0.289	0.340	0.443	0.567
TSI	FIN	FWER	0.118	0.118	0.151	0.151
		TI	0.0024	0.00318	0.00351	0.00351
		Power	0.785	0.946	0.957	0.957
FIN	TSI	FWER	0.418	0.929	0.990	1.00
		TI	0.0045	0.0106	0.0765	0.0856
		Power	0.398	0.827	0.888	0.929
LWK	YRI	FWER	1.00	1.00	1.00	1.00
		TI	0.038	0.0489	0.0489	0.0511
		Power	0.00	0.00	0.570	0.733
YRI	LWK	WER	1.00	1.00	1.00	1.00
		TI	0.0233	0.0233	0.0234	0.0234
		Power	1.00	1.00	1.00	1.00

We further investigated the performance of the test by introducing more than one outliers into the dataset. We randomly selected 5 or 10 outliers from the outlier population and they were combined with the corresponding study population to assess the performance of the approaches. There are 1,000 such randomly generated datasets in all the scenarios except for evaluating the performance of PCA on variants with MAF ≤ 5%. We observe that even with more outliers included in the dataset, our method performs generally better than PCA, especially in the combined datasets of TSI and FIN, and LWK and YRI. PCA continues to have a large type I error rate and FWER in all the scenarios. We also observe again that the performance of the novel test is much better using the SNPs with MAF≤ 5%, than using all the SNPs. The results are shown in supplementary Tables S2 and S3.

Note that as the proportion of outliers included in the dataset continues to increase, the power of our test decreases. This is because that as more outliers are included in the dataset, the estimated MAF and the expected number of alleles obtained from the data are biased toward the outlier population. Then the test statistics would be biased and would not follow the same distribution as under the null hypothesis Thus, the novel method is mainly used to detect a relatively small set of outliers. The effect of the proportion of the outliers included in the dataset on the test statistics also depends on the genetic distance of the populations.

Simulations

In addition to the analyses on the 1000 Genome Project data, we performed simulation studies under the alternative hypothesis to examine whether the proposed test T_opt has sufficient power to detect outliers. We assessed the power of the approach to detect genetic outliers based on both rare variants and common variants. In our simulations, the Balding-Nichols model [Balding and Nichols, 1995] was applied to generate the allele frequencies of the two subpopulations: $p_{i1},p_{i2}\sim \text{Beta}(\frac{1-F}{F}{p}_i,\frac{1-F}{F}(1-p_i))$, where p_i1, p_i2 are the allele frequencies of marker locus i for the two subpopulations; F is the F_st, the genetic distance between the two subpopulations [Holsinger and Weir, 2009]; and the parameter p_i is the background allele frequency for marker locus i. In the simulation studies for the rare variants, the background allele frequencies p_i were generated from Wright’s distribution [Wright, 1949] using Metropolis-Hastings algorithm: f (p) = cp^{β_s −1}(1 − p)^β_n−1e^σ(1−p), where the scaled mutation rates are elected to be β_s = 0.001, β_n = β_s/3, the selection rate σ = 12, and c is a normalizing constant. Wright’s distribution is expected to simulate the MAF spectrum of the human genome, where most of the MAFs generated are smaller than 5% [Pritchard, 2001]. To generate common variant data, we generated the background allele frequencies p_i from the Uniform distribution Unif (0, 0.5).

Under the models defined by these parameters, we draw two sets of allele frequencies for the two subpopulations in each trial. In analogy to the 1000 Genome Project analysis, one study subject was generated from the first subpopulation, while the remaining study subjects in the dataset were generated from the second subpopulation.

Power

Based on 1,000 replicates, we estimated the power of the test under each scenario for both common variant and rare-variant data. The power is estimated by the percentage of trials in which the outlier is detected using T_opt, where the test statistic is adjusted for multiple comparisons, i.e., study subjects, using the Bonferroni correction. The results are shown in Table 3 and supplementary Table S4. Table 3 suggests that the test T_opt has sufficient power in most of the scenarios for rare-variant data, especially for datasets with 1 million markers, as shown in Figure 2. This is expected since, as the number of markers increases, there is more information about the genetic structure of the population, and it is easier for the test to capture any small difference between the outlier and the rest of the subjects in the dataset. The genetic distance of the two subpopulations, F_st, is varied over a wide range, and we observe a decrease in the power of the test as F_st decreases, as we would expect. Also, we find that the percentage of markers with smaller MAF in the first population than in the second population also influences the power of the test, especially when the two populations are genetically close to each other. However, after assessing the power for the percentage between 50% and 75% (data not shown), we found that, as long as the percentage is above 50%, i.e., there is LD in the sample, we have a good power to detect the outlier under the scenarios we considered.

Table 3.

Power of T_opt for rare-variant data

Fst	Percentage	500 × 10k	1,000 × 10k	500 × 100k	1,000 × 100k	500 × 1M	1,000 × 1M
0.20	100	0.898	0.905	0.991	0.990	0.998	1.000
	75	0.879	0.867	0.990	0.995	0.997	0.998
	50	0.914	0.870	0.988	0.986	1.000	1.000
0.15	100	0.904	0.900	0.983	0.987	1.000	0.998
	75	0.854	0.857	0.989	0.954	0.999	0.998
	50	0.880	0.856	0.987	0.987	1.000	1.000
0.10	100	0.894	0.887	0.988	0.984	1.000	1.000
	75	0.859	0.833	0.981	0.982	0.998	0.999
	50	0.835	0.796	0.981	0.986	1.000	0.999
0.05	100	0.878	0.875	0.980	0.983	0.997	0.996
	75	0.807	0.777	0.973	0.974	0.998	0.997
	50	0.388	0.354	0.967	0.970	0.998	0.996
0.01	100	0.828	0.825	0.973	0.979	0.999	0.999
	75	0.453	0.401	0.968	0.963	0.995	0.997
	50	0.003	0.006	0.031	0.025	0.863	0.839
0.005	100	0.757	0.748	0.987	0.977	0.997	0.999
	75	0.183	0.119	0.947	0.950	0.993	0.997
	50	0.005	0.003	0.002	0.004	0.149	0.118

The number of subjects in the datasets is either 500 or 1,000. The number of SNPs included is 10,000, 100,000, or 1 million. The first column refers to the genetic distance of the two subpopulations in the dataset. The second column shows the percentage of the markers with a smaller MAF in the first subpopulation than in the second subpopulation.

Figure 2.

The power of T_opt for rare-variant data with 1,000 subjects at F _st = 0.1 and 0.005 with the three percentage levels as a function of the number of SNPs included in the dataset, which are 10k, 100k, and 1 million.

For common variant data, the power of the test becomes very small when the percentage of markers with smaller MAF in one population than in the other population is approximately 50%, even for large number of SNPs. However, as the percentage increases, the power increases rapidly. This is due to the small power offered by the score S₂ since S₁ does not have much power under the 50% scenario. It is important to note that, in a real dataset, we would not expect the percentage to be exactly 50% if all available genetic loci are included in the calculation of the test statistic and there is LD between the loci.

We also compared our approach and the PCA approach for rare-variant data, as shown in Table 4. We observe that both the proposed test statistic and the outlier detection algorithm based on PCA have sufficient statistical power in most scenarios. However, when there is a systematic difference in allele frequencies between the two populations, the PCA approach does not perform well. In practice, this effect on PCA can be minimized by the removal of long-range LD regions and LD-pruning for common variant analysis, but would be unavoidable for sequence data. Note that there are only 10k SNPs included in the simulation due to the computational cost of PCA, and the power of T_opt increases dramatically as the number of SNPs included increases, whereas the results of the PCA analysis would be biased since LD would be introduced as the number of SNPs included increases.

Table 4.

Power of T_opt and the outlier detection process based on PCA for rare-variant data

		500 subjects × 10k		1,000 subjects × 10k
Fst	Percentage	PCA	Topt	PCA	Topt
0.2	100	0.05	0.93	0.00	0.90
	75	0.90	0.84	0.90	0.87
	50	0.93	0.87	0.93	0.87
0.15	100	0.06	0.92	0.05	0.87
	75	0.86	0.88	0.90	0.79
	50	0.94	0.90	0.96	0.92
0.10	100	0.03	0.96	0.05	0.88
	75	0.65	0.84	0.67	0.81
	50	0.92	0.80	0.94	0.86
0.05	100	0.03	0.89	0.02	0.88
	75	0.20	0.80	0.38	0.77
	50	0.90	0.33	0.94	0.28
0.01	100	0.04	0.84	0.04	0.78
	75	0.05	0.37	0.04	0.37
	50	0.37	0.00	0.48	0.00
0.005	100	0.00	0.73	0.00	0.73
	75	0.03	0.15	0.07	0.12
	50	0.10	0.00	0.19	0.01

The ancestral MAFs are generated from Wright’Õs distribution for the datasets. The number of SNPs included is 10,000.

Type I Error

Using the same set of simulated data, the type I error is estimated as the average percentage of subjects who are incorrectly rejected. Part of the results for the rare variants is shown in Figure 3, and the complete results are in supplementary Table S5. In the scenarios considered, the nominal type I error is 0.05/n, where n is the number of subjects included in the dataset, to maintain the FWER at 0.05 level. Thus, the nominal type I error is 0.0001 for 500 subjects and 0.00005 for 1,000 subjects. From the results, we observe that for rare SNPs, the type I error for 10,000 SNPs is inflated, but for datasets with a large number of SNPs, the type I error rate is acceptable. For common variants, the type I error is well maintained in all the scenarios (data not shown).

Figure 3.

The type I error rate of T_opt for rare-variant data with 1,000 subjects at six different F_st values with the percentage level 75% as a function of the number of SNPs included in the dataset, which are 10k, 100k, and 1 million.

The same pattern is observed for the FWER as for the type I error rate. FWER is estimated as the number of trials among 1,000 trials such that at least one subject is wrongly rejected. For common variant data, the FWER is well below 0.05 for all the scenarios. For rare-variant data, we do see an inflation in the FWER as the type I error rate when the number of SNPs included in the dataset is small. However, in real dataset, e.g., whole-exome sequencing data, GWAS, etc., we expect that a sufficient number of loci is available to guarantee that the FWER is maintained.

As a last comparison, we assessed the performance of both approaches under the null hypothesis. As shown in Table 5, for rare variants generated from Wright’s distribution, PCA has much larger FWER compared to T_opt. In almost all the trails, PCA rejected at least one subject incorrectly, whereas T_opt has been shown above that the FWER is acceptable when the number of SNPs included is sufficiently large. For common variants, the FWER of both approaches is well maintained with 500 or 1,000 subjects included in the datasets.

Table 5.

FWER of T_opt and the outlier detection process based on PCA for rare-variant data

	500 × 10k		1,000 × 10k		500 × 100k		1,000 × 100k
FWER dist	PCA	Topt	PCA	Topt	PCA	Topt	PCA	Topt
Wright	0.912	0.106	0.978	0.123	0.928	0.066	0.989	0.060

Discussion

The large-scale applications of next-generation sequencing technology to association studies require the development of robust and powerful analysis approaches. While substantial progress has been made in terms of the development of association tests for rare variants [Ionita-Laza et al., 2011; Li and Leal, 2008; Madsen and Browning, 2009; Mukhopadhyay et al., 2010; Neale et al., 2011], there is yet no standard statistical approach that addresses the issues of population substructure for sequence data.

Recently, a permutation procedure is proposed by Epstein et al. [2012] to address the problem in association tests of rare variations. It provides the option for the rare-variant association tests that cannot correct for confounding to adjust for covariates. However, it is subject to the same problem as other rare-variant association tests that may be adjusted for ancestry due the fact that the ancestry covariates obtained using PCA may not be accurate as the type I error of the association tests after adjusting for ancestry using PCA has been shown to be still inflated under certain scenarios [Mathieson and McVean, 2012]. Here, we try to approach the problem from a different direction, by obtaining a homogeneous subpopulation to remove confounding and avoid the hassle of estimating the ancestry covariates for rare variants. In this communication, we proposed a method that can detect study subject that introduce population substructure in the sample, potentially confounding the association analysis. Our approach is computationally fast and simple, i.e., the method is computed based on all available genetic loci, making LD estimation and pruning unnecessary. This is especially useful for data from Exome chips or disease-specific fine-mapping chips, in which case the pruning of the SNPs would not be desirable, and thus PCA would give biased results. The approach works well for both rare and common variants. We illustrated this by the applications to the 1000 Genome Project data, and in our simulation studies.

While these are the advantages over the standard PCA analysis for rare-variant data, our approach assumes that most of the subjects in the dataset are from the same population and there is only a small proportion of outliers included. Also, our method does not assess the pairwise similarity of study subjects, e.g., PC plots. This restricts our approach to the role of an outlier detection tool. Unlike PCs, an integration of the test statistic into a regression model as an adjustment for population substructure is problematic for this reason. Additional research on this topic is required.