Theoretical Formulation of Principal Components Analysis to Detect and Correct for Population Stratification

The Eigenstrat method, based on principal components analysis (PCA), is commonly used both to quantify population relationships in population genetics and to correct for population stratification in genome-wide association studies. However, it can be difficult to make appropriate inference about population relationships from the principal component (PC) scatter plot. Here, to better understand the working mechanism of the Eigenstrat method, we consider its theoretical or “population” formulation. The eigen-equation for samples from an arbitrary number () of populations is reduced to that of a matrix of dimension , the elements of which are determined by the variance-covariance matrix for the random vector of the allele frequencies. Solving the reduced eigen-equation is numerically trivial and yields eigenvectors that are the axes of variation required for differentiating the populations. Using the reduced eigen-equation, we investigate the within-population fluctuations around the axes of variation on the PC scatter plot for simulated datasets. Specifically, we show that there exists an asymptotically stable pattern of the PC plot for large sample size. Our results provide theoretical guidance for interpreting the pattern of PC plot in terms of population relationships. For applications in genetic association tests, we demonstrate that, as a method of correcting for population stratification, regressing out the theoretical PCs corresponding to the axes of variation is equivalent to simply removing the population mean of allele counts and works as well as or better than the Eigenstrat method.     

PCA-correlated SNPs for structure identification in worldwide human populations

Existing methods to ascertain small sets of markers for the identification of human population structure require prior knowledge of individual ancestry. Based on Principal Components Analysis (PCA), and recent results in theoretical computer science, we present a novel algorithm that, applied on genomewide data, selects small subsets of SNPs (PCA-correlated SNPs) to reproduce the structure found by PCA on the complete dataset, without use of ancestry information. Evaluating our method on a previously described dataset (10,805 SNPs, 11 populations), we demonstrate that a very small set of PCA-correlated SNPs can be effectively employed to assign individuals to particular continents or populations, using a simple clustering algorithm. We validate our methods on the HapMap populations and achieve perfect intercontinental differentiation with 14 PCA-correlated SNPs. The Chinese and Japanese populations can be easily differentiated using less than 100 PCA-correlated SNPs ascertained after evaluating 1.7 million SNPs from HapMap. We show that, in general, structure informative SNPs are not portable across geographic regions. However, we manage to identify a general set of 50 PCA-correlated SNPs that effectively assigns individuals to one of nine different populations. Compared to analysis with the measure of informativeness, our methods, although unsupervised, achieved similar results. We proceed to demonstrate that our algorithm can be effectively used for the analysis of admixed populations without having to trace the origin of individuals. Analyzing a Puerto Rican dataset (192 individuals, 7,257 SNPs), we show that PCA-correlated SNPs can be used to successfully predict structure and ancestry proportions. We subsequently validate these SNPs for structure identification in an independent Puerto Rican dataset. The algorithm that we introduce runs in seconds and can be easily applied on large genome-wide datasets, facilitating the identification of population substructure, stratification assessment in multi-stage whole-genome association studies, and the study of demographic history in human populations.     

Tracing sub-structure in the European American population with PCA-informative markers

Genetic structure in the European American population reflects waves of migration and recent gene flow among different populations. This complex structure can introduce bias in genetic association studies. Using Principal Components Analysis (PCA), we analyze the structure of two independent European American datasets (1,521 individuals-307,315 autosomal SNPs). Individual variation lies across a continuum with some individuals showing high degrees of admixture with non-European populations, as demonstrated through joint analysis with HapMap data. The CEPH Europeans only represent a small fraction of the variation encountered in the larger European American datasets we studied. We interpret the first eigenvector of this data as correlated with ancestry, and we apply an algorithm that we have previously described to select PCA-informative markers (PCAIMs) that can reproduce this structure. Importantly, we develop a novel method that can remove redundancy from the selected SNP panels and show that we can effectively remove correlated markers, thus increasing genotyping savings. Only 150-200 PCAIMs suffice to accurately predict fine structure in European American datasets, as identified by PCA. Simulating association studies, we couple our method with a PCA-based stratification correction tool and demonstrate that a small number of PCAIMs can efficiently remove false correlations with almost no loss in power. The structure informative SNPs that we propose are an important resource for genetic association studies of European Americans. Furthermore, our redundancy removal algorithm can be applied on sets of ancestry informative markers selected with any method in order to select the most uncorrelated SNPs, and significantly decreases genotyping costs.     

Molecular Reclassification of Crohn’s Disease: A Cautionary Note on Population Stratification

Complex human diseases commonly differ in their phenotypic characteristics, e.g., Crohn’s disease (CD) patients are heterogeneous with regard to disease location and disease extent. The genetic susceptibility to Crohn’s disease is widely acknowledged and has been demonstrated by identification of over 100 CD associated genetic loci. However, relating CD subphenotypes to disease susceptible loci has proven to be a difficult task. In this paper we discuss the use of cluster analysis on genetic markers to identify genetic-based subgroups while taking into account possible confounding by population stratification. We show that it is highly relevant to consider the confounding nature of population stratification in order to avoid that detected clusters are strongly related to population groups instead of disease-specific groups. Therefore, we explain the use of principal components to correct for population stratification while clustering affected individuals into genetic-based subgroups. The principal components are obtained using 30 ancestry informative markers (AIM), and the first two PCs are determined to discriminate between continental origins of the affected individuals. Genotypes on 51 CD associated single nucleotide polymorphisms (SNPs) are used to perform latent class analysis, hierarchical and Partitioning Around Medoids (PAM) cluster analysis within a sample of affected individuals with and without the use of principal components to adjust for population stratification. It is seen that without correction for population stratification clusters seem to be influenced by population stratification while with correction clusters are unrelated to continental origin of individuals.     

Number of SNPS loci needed to detect population structure

The study of the association of polymorphic genetic markers with common diseases is one of the most powerful tools in modern genetics. Interest in single nucleotide polymorphisms (SNPs) has steadily grown over the last decade. SNPs are currently the most developed markers in the human genome because they have a number of advantages over other marker types. One of the critical problems responsible for 'spurious' association findings in case-control studies is population stratification. There are many statistical approaches developed for detecting population heterogeneity. However the power to detect population structure by known methods is highly dependent on the number of loci utilised. We performed an analysis of SNPs data available in the public domain from The Single Nucleotide Consortia Ltd. (TSCL). Three populations, Afro-American, Asian and Caucasian, were compared. Estimation of the minimum number of SNPs loci necessary for detection of the population structure was performed. Two clustering approaches, distance-based and model-based, were compared. The model-based approach was superior when compared with the distance-based method. We found more than 65 random SNPs loci are required for identifying distinct geographically separated populations. Increasing the number of markers to over 100 raises the probability of correct assignment of a particular individual to an origin group to over 90%, even with conventional clustering methods.     

A PCA-based method for ancestral informative markers selection in structured populations

Identification of population structure can help trace population histories and identify disease genes. Structured association (SA) is a commonly used approach for population structure identification and association mapping. A major issue with SA is that its performance greatly depends on the informativeness and the numbers of ancestral informative markers (AIMs). Present major AIM selection methods mostly require prior individual ancestry information, which is usually not available or uncertain in practice. To address this potential weakness, we herein develop a novel approach for AIM selection based on principle component analysis (PCA), which does not require prior ancestry information of study subjects. Our simulation and real genetic data analysis results suggest that, with equivalent AIMs, PCA-based selected AIMs can significantly increase the accuracy of inferred individual ancestries compared with traditionally randomly selected AIMs. Our method can easily be applied to whole genome data to select a set of highly informative AIMs in population structure, which can then be used to identify potential population structure and correct possible statistical biases caused by population stratification.     

Inference of population structure using genetic markers and a Bayesian model averaging approach for clustering.

The analysis of the structure of populations on the basis of genetic data is essential in population genetics. It is used, for instance, to study the evolution of species or to correct for population stratification in association studies. These genetic data, normally based on DNA polymorphisms, may contain irrelevant information that biases the inference of population structure. In this paper we adapt a recently proposed algorithm, named multistart EMA, to be used in the inference of population structure. This algorithm is able to deal with irrelevant information when obtaining the (probabilistic) population partition. Additionally, we present a maker selection test able to obtain the most relevant markers to retrieve that population partition. The proposed algorithm is compared with the widely used STRUCTURE software on the basis of the F(ST) metric and the log-likelihood score. It is shown that the proposed algorithm improves the obtention of the population structure. Moreover, information about relevant markers obtained by the multi-start EMA can be used to improve the results obtained by other methods, correct for population stratification or even also reduce the economical cost of sequencing new samples. The software presented in this paper is available online at http://www.sc.ehu.es/ccwbayes/members/guzman.     

Comparison of population-based association study methods correcting for population stratification

Population stratification can cause spurious associations in population-based association studies. Several statistical methods have been proposed to reduce the impact of population stratification on population-based association studies. We simulated a set of stratified populations based on the real haplotype data from the HapMap ENCODE project, and compared the relative power, type I error rates, accuracy and positive prediction value of four prevailing population-based association study methods: traditional case-control tests, structured association (SA), genomic control (GC) and principal components analysis (PCA) under various population stratification levels. Additionally, we evaluated the effects of sample sizes and frequencies of disease susceptible allele on the performance of the four analytical methods in the presence of population stratification. We found that the performance of PCA was very stable under various scenarios. Our comparison results suggest that SA and PCA have comparable performance, if sufficient ancestral informative markers are used in SA analysis. GC appeared to be strongly conservative in significantly stratified populations. It may be better to apply GC in the stratified populations with low stratification level. Our study intends to provide a practical guideline for researchers to select proper study methods and make appropriate inference of the results in population-based association studies.     

Family-Based Bivariate Association Tests for Quantitative Traits

The availability of a large number of dense SNPs, high-throughput genotyping and computation methods promotes the application of family-based association tests. While most of the current family-based analyses focus only on individual traits, joint analyses of correlated traits can extract more information and potentially improve the statistical power. However, current TDT-based methods are low-powered. Here, we develop a method for tests of association for bivariate quantitative traits in families. In particular, we correct for population stratification by the use of an integration of principal component analysis and TDT. A score test statistic in the variance-components model is proposed. Extensive simulation studies indicate that the proposed method not only outperforms approaches limited to individual traits when pleiotropic effect is present, but also surpasses the power of two popular bivariate association tests termed FBAT-GEE and FBAT-PC, respectively, while correcting for population stratification. When applied to the GAW16 datasets, the proposed method successfully identifies at the genome-wide level the two SNPs that present pleiotropic effects to HDL and TG traits.     

Case-control association studies in mixed populations: correcting using genomic control

OBJECTIVE: Case-control association studies in mixed populations can result in spurious disease-marker associations if subpopulation disease prevalence and marker frequencies both differ. Genomic control (GC) uses neutral loci to correct for spurious association (due to population stratification), but how well this works remains undetermined. METHODS: We simulated and mixed populations with different disease and marker frequencies but without marker-disease association. We generated case-control datasets, calculated the chi2 for disease association with each marker, and applied two GC procedures, dividing by the mean chi2 or median-chi2/0.456. RESULTS: Corrections became conservative (false positive rate [FPR] <5%) with increasing subpopulation prevalence and marker differences. The mean correction resulted in FPRs close to 5% at average subpopulation allele frequency differences <0.26, but inclusion of just a few markers with large frequency differences resulted in conservative FPRs. FPRs from the median correction were mostly conservative but became anticonservative when a few markers with large frequency differences were included. CONCLUSION: GC can both lead to a notable loss of power to detect a true association (conservative) in many circumstances or may fail to eliminate the spurious associations (anticonservative). The mean correction factor is useful in certain situations to correct population stratification, but it is difficult to know when those situations exist.     

Fast Principal Component Analysis of Large-Scale Genome-Wide Data

Principal component analysis (PCA) is routinely used to analyze genome-wide single-nucleotide polymorphism (SNP) data, for detecting population structure and potential outliers. However, the size of SNP datasets has increased immensely in recent years and PCA of large datasets has become a time consuming task. We have developed flashpca, a highly efficient PCA implementation based on randomized algorithms, which delivers identical accuracy in extracting the top principal components compared with existing tools, in substantially less time. We demonstrate the utility of flashpca on both HapMap3 and on a large Immunochip dataset. For the latter, flashpca performed PCA of 15,000 individuals up to 125 times faster than existing tools, with identical results, and PCA of 150,000 individuals using flashpca completed in 4 hours. The increasing size of SNP datasets will make tools such as flashpca essential as traditional approaches will not adequately scale. This approach will also help to scale other applications that leverage PCA or eigen-decomposition to substantially larger datasets.     

Group normalization for genomic data

Data normalization is a crucial preliminary step in analyzing genomic datasets. The goal of normalization is to remove global variation to make readings across different experiments comparable. In addition, most genomic loci have non-uniform sensitivity to any given assay because of variation in local sequence properties. In microarray experiments, this non-uniform sensitivity is due to different DNA hybridization and cross-hybridization efficiencies, known as the probe effect. In this paper we introduce a new scheme, called Group Normalization (GN), to remove both global and local biases in one integrated step, whereby we determine the normalized probe signal by finding a set of reference probes with similar responses. Compared to conventional normalization methods such as Quantile normalization and physically motivated probe effect models, our proposed method is general in the sense that it does not require the assumption that the underlying signal distribution be identical for the treatment and control, and is flexible enough to correct for nonlinear and higher order probe effects. The Group Normalization algorithm is computationally efficient and easy to implement. We also describe a variant of the Group Normalization algorithm, called Cross Normalization, which efficiently amplifies biologically relevant differences between any two genomic datasets.     

Accounting for probe-level noise in principal component analysis of microarray data

MOTIVATION: Principal Component Analysis (PCA) is one of the most popular dimensionality reduction techniques for the analysis of high-dimensional datasets. However, in its standard form, it does not take into account any error measures associated with the data points beyond a standard spherical noise. This indiscriminate nature provides one of its main weaknesses when applied to biological data with inherently large variability, such as expression levels measured with microarrays. Methods now exist for extracting credibility intervals from the probe-level analysis of cDNA and oligonucleotide microarray experiments. These credibility intervals are gene and experiment specific, and can be propagated through an appropriate probabilistic downstream analysis. RESULTS: We propose a new model-based approach to PCA that takes into account the variances associated with each gene in each experiment. We develop an efficient EM-algorithm to estimate the parameters of our new model. The model provides significantly better results than standard PCA, while remaining computationally reasonable. We show how the model can be used to 'denoise' a microarray dataset leading to improved expression profiles and tighter clustering across profiles. The probabilistic nature of the model means that the correct number of principal components is automatically obtained.     

A robust statistical method for association-based eQTL analysis

Background

It has been well established that theoretical kernel for recently surging genome-wide association study (GWAS) is statistical inference of linkage disequilibrium (LD) between a tested genetic marker and a putative locus affecting a disease trait. However, LD analysis is vulnerable to several confounding factors of which population stratification is the most prominent. Whilst many methods have been proposed to correct for the influence either through predicting the structure parameters or correcting inflation in the test statistic due to the stratification, these may not be feasible or may impose further statistical problems in practical implementation.

Methodology

We propose here a novel statistical method to control spurious LD in GWAS from population structure by incorporating a control marker into testing for significance of genetic association of a polymorphic marker with phenotypic variation of a complex trait. The method avoids the need of structure prediction which may be infeasible or inadequate in practice and accounts properly for a varying effect of population stratification on different regions of the genome under study. Utility and statistical properties of the new method were tested through an intensive computer simulation study and an association-based genome-wide mapping of expression quantitative trait loci in genetically divergent human populations.

Results/Conclusions

The analyses show that the new method confers an improved statistical power for detecting genuine genetic association in subpopulations and an effective control of spurious associations stemmed from population structure when compared with other two popularly implemented methods in the literature of GWAS.     

SHIPS: Spectral Hierarchical Clustering for the Inference of Population Structure in Genetic Studies

Inferring the structure of populations has many applications for genetic research. In addition to providing information for evolutionary studies, it can be used to account for the bias induced by population stratification in association studies. To this end, many algorithms have been proposed to cluster individuals into genetically homogeneous sub-populations. The parametric algorithms, such as Structure, are very popular but their underlying complexity and their high computational cost led to the development of faster parametric alternatives such as Admixture. Alternatives to these methods are the non-parametric approaches. Among this category, AWclust has proven efficient but fails to properly identify population structure for complex datasets. We present in this article a new clustering algorithm called Spectral Hierarchical clustering for the Inference of Population Structure (SHIPS), based on a divisive hierarchical clustering strategy, allowing a progressive investigation of population structure. This method takes genetic data as input to cluster individuals into homogeneous sub-populations and with the use of the gap statistic estimates the optimal number of such sub-populations. SHIPS was applied to a set of simulated discrete and admixed datasets and to real SNP datasets, that are data from the HapMap and Pan-Asian SNP consortium. The programs Structure, Admixture, AWclust and PCAclust were also investigated in a comparison study. SHIPS and the parametric approach Structure were the most accurate when applied to simulated datasets both in terms of individual assignments and estimation of the correct number of clusters. The analysis of the results on the real datasets highlighted that the clusterings of SHIPS were the more consistent with the population labels or those produced by the Admixture program. The performances of SHIPS when applied to SNP data, along with its relatively low computational cost and its ease of use make this method a promising solution to infer fine-scale genetic patterns.     

Detailed analysis of Japanese population substructure with a focus on the southwest islands of Japan

Uncovering population structure is important for properly conducting association studies and for examining the demographic history of a population. Here, we examined the Japanese population substructure using data from the Japan Multi-Institutional Collaborative Cohort (J-MICC), which covers all but the northern region of Japan. Using 222 autosomal loci from 4502 subjects, we investigated population substructure by estimating F(ST) among populations, testing population differentiation, and performing principal component analysis (PCA) and correspondence analysis (CA). All analyses revealed a low but significant differentiation between the Amami Islanders and the mainland Japanese population. Furthermore, we examined the genetic differentiation between the mainland population, Amami Islanders and Okinawa Islanders using six loci included in both the Pan-Asian SNP (PASNP) consortium data and the J-MICC data. This analysis revealed that the Amami and Okinawa Islanders were differentiated from the mainland population. In conclusion, we revealed a low but significant level of genetic differentiation between the mainland population and populations in or to the south of the Amami Islands, although genetic variation between both populations might be clinal. Therefore, the possibility of population stratification must be considered when enrolling the islander population of this area, such as in the J-MICC study.     

European American stratification in ovarian cancer case control data: the utility of genome-wide data for inferring ancestry

We investigated the ability of several principal components analysis (PCA)-based strategies to detect and control for population stratification using data from a multi-center study of epithelial ovarian cancer among women of European-American ethnicity. These include a correction based on an ancestry informative markers (AIMs) panel designed to capture European ancestral variation and corrections utilizing un-thinned genome-wide SNP data; case-control samples were drawn from four geographically distinct North-American sites. The AIMs-only and genome-wide first principal components (PC1) both corresponded to the previously described North or Northwest-Southeast axis of European variation. We found that the genome-wide PCA captured this primary dimension of variation more precisely and identified additional axes of genome-wide variation of relevance to epithelial ovarian cancer. Associations evident between the genome-wide PCs and study site corroborate North American immigration history and suggest that undiscovered dimensions of variation lie within Northern Europe. The structure captured by the genome-wide PCA was also found within control individuals and did not reflect the case-control variation present in the data. The genome-wide PCA highlighted three regions of local LD, corresponding to the lactase (LCT) gene on chromosome 2, the human leukocyte antigen system (HLA) on chromosome 6 and to a common inversion polymorphism on chromosome 8. These features did not compromise the efficacy of PCs from this analysis for ancestry control. This study concludes that although AIMs panels are a cost-effective way of capturing population structure, genome-wide data should preferably be used when available.     

Advanced Intercross Lines, an Experimental Population for Fine Genetic Mapping

An advanced intercrossed line (AIL) is an experimental population that can provide more accurate estimates of quantitative trait loci (QTL) map location than conventional mapping populations. An AIL is produced by randomly and sequentially intercrossing a population that initially originated from a cross between two inbred lines or some variant thereof. This provides increasing probability of recombination between any two loci. Consequently, the genetic length of the entire genome is stretched, providing increased mapping resolution. In this way, for example, with the same population size and QTL effect, a 95% confidence interval of QTL map location of 20 cM in the F(2) is reduced fivefold after eight additional random mating generations (F(10)). Simulation results showed that to obtain the anticipated reduction in the confidence interval, breeding population size of the AIL in all generations should comprise an effective number of >/=100 individuals. It is proposed that AILs derived from crosses between known inbred lines may be a useful resource for fine genetic mapping.     

Properties of structured association approaches to detecting population stratification

OBJECTIVE: To examine the properties of the structured association approach for the detection and correction of population stratification. METHOD: A method is developed, within a latent class analysis framework, similar to the methods proposed by Satten et al. (2001) and Pritchard et al. (2000). A series of simulations illustrate the relative impact of number and type of loci, sample size and population structure. RESULTS: The ability to detect stratification and assign individuals to population strata is determined for a number of different scenarios. CONCLUSION: The results underline the importance of careful marker selection.     

A semiparametric test to detect associations between quantitative traits and candidate genes in structured populations

MOTIVATION: Although population-based association mapping may be subject to the bias caused by population stratification, alternative methods that are robust to population stratification such as family-based linkage analysis have lower mapping resolution. Recently, various statistical methods robust to population stratification were proposed for association studies, using unrelated individuals to identify associations between candidate genes and traits of interest. The association between a candidate gene and a quantitative trait is often evaluated via a regression model with inferred population structure variables as covariates, where the residual distribution is customarily assumed to be from a symmetric and unimodal parametric family, such as a Gaussian, although this may be inappropriate for the analysis of many real-life datasets. RESULTS: In this article, we proposed a new structured association (SA) test. Our method corrects for continuous population stratification by first deriving population structure and kinship matrices through a set of random genetic markers and then modeling the relationship between trait values, genotypic scores at a candidate marker and genetic background variables through a semiparametric model, where the error distribution is modeled as a mixture of Polya trees centered around a normal family of distributions. We compared our model to the existing SA tests in terms of model fit, type I error rate, power, precision and accuracy by application to a real dataset as well as simulated datasets.