GWASs may utilize either case-control cohorts to test for associations with diseases or population cohorts to identify associations with quantitative traits. In both cases, it is assumed that the cohorts consist of unrelated individuals that share the same population background, although this may not hold in practice for cohorts used in many current GWASs. The presence of related individuals within a study sample results in sample structure, a term that encompasses population stratification and hidden relatedness. Population stratification refers to the inclusion of individuals from different populations within the same study sample. Hidden relatedness refers to the presence of unknown genetic relationships between individuals within the study sample^{1,2}. The effects of sample structure present in cohorts used for genetic association studies have been well documented and identified as a cause for some spurious associations^{3,4}.

Although limiting study samples entirely to unrelated individuals may be difficult or impossible, genotype data provides valuable information on the sample structure that can inform genetic association analysis. For example, the STRUCTURE software^{5} uses genotype data to partition the sample into subpopulations within which there is no sample structure and subsequently carries out association tests within the identified subpopulations. To eliminate the effects of hidden relatedness, one can estimate the proportion of genes identical by descent (IBD) between any pair of individuals in the sample and exclude from the analysis those individuals that appear closely related^{1,6}. Population stratification and hidden relatedness, however, constitute just two extreme manifestations of sample structure, and methods are needed to correct for other forms of sample structure. In the genomic control approach^{7,8}, which has been widely adopted, the distribution of test statistics from the single-marker analysis is used to estimate the inflation factor, *λ*, with which the test statistics are subsequently rescaled, constraining the risk of false positives. The EIGENSTRAT software^{9,10} uses principal components analysis (PCA) to detect and describe sample structure and has been widely used in GWASs. Some principal components may represent broad differences across individuals within a given data set, effectively capturing a few major axes of population structure, but it is unclear how to interpret the rest of the principal components as surrogates of sample structure^{11,12}. Currently, association studies typically use a combination of these strategies, first identifying close relatives to remove them from analysis, then correcting for broad sample structure using principal components or spatial information and finally correcting for the residual inflation with genomic control^{6,13,14}.

If we knew the complete genealogy of the population, we could, in principle, apply a variance component method to model the effects of the genetic relationships on the phenotypes; this approach would be similar in spirit to the classical polygenic model^{15} directly applied to association mapping^{16}. The variance component would capture the complex mixture of both population stratification and hidden relatedness that directly results from the genealogy and would correct for these relationships during the mapping. Although the exact genetic relationships between individuals in the samples are unknown, we could take advantage of the high-density genotype information to empirically estimate the level of relatedness between reportedly unrelated individuals.

We report here an approach for correcting for sample structure within GWASs, based on a linear mixed model (also sometimes referred to as a mixed linear model) with an empirically estimated relatedness matrix to model the correlation between phenotypes of sample subjects. Similar variance component approaches have been used successfully in animal models^{17-19}. However, applying even an efficient implementation of a variance component approach, such as EMMA (ref. ^{19}), is computationally intractable for data sets consisting of thousands of individuals, owing to the heavy computational burden in the estimation of variance parameters. Capitalizing on the characteristics of complex traits in humans, we make a few simplifying assumptions that allow us to markedly increase the speed of computations, making our approach readily applicable to GWASs with tens of thousands of individuals assayed at hundreds of thousands of SNPs. For most genetic association studies in humans, because the effect of any given locus on the trait is very small^{20}, we need to estimate the variance parameters only once for each data set, and we can globally apply them to each marker. Our computational improvements reduce the running time for the analysis of a typical GWAS data set using a variance component model from years to hours. The advantage of the variance component approach is that the empirical relatedness matrix encodes a wide range of sample structures, including both hidden relatedness and population stratification. Principal component–based methods, in contrast, by estimating major axes of the pairwise genetic similarity matrix, capture some, but not all, of the sample structure, as we show below.

We evaluate our method using two human GWAS data sets, from the 1966 Northern Finland Birth Cohort (NFBC66)^{13,21} and the Wellcome Trust Case Control Consortium (WTCCC)^{6}. The NFBC66 is based on a founder population, which is expected to minimize genetic heterogeneity, increasing the chances of mapping genes underlying traits of interest^{22}. This is an ideal sample to evaluate our method because a detailed study^{23} of this data set has revealed the presence of substantial population structure that could influence the results of genetic association studies. In addition, we apply our method to the case-control studies for seven common complex diseases conducted by the WTCCC^{6}. In both data sets, our method consistently outperforms both genomic control and principal component analysis. We term our method EMMA eXpedited (EMMAX) because it builds on the previous approach EMMA (ref. ^{19}) and markedly reduces the computational cost.