It is widely acknowledged that genome-wide association studies (GWAS) of complex human disease fail to explain a large portion of heritability, primarily due to lack of statistical power-a problem that is exacerbated when seeking detection of interactions of multiple genomic loci. An untapped source of information that is already widely available, and that is expected to grow in coming years, is population samples. Such samples contain genetic marker data for additional individuals, but not their relevant phenotypes. In this article we develop a highly efficient testing framework based on a constrained maximum-likelihood estimate in a case-controlpopulation setting. We leverage the available population data and optional modeling assumptions, such as Hardy-Weinberg equilibrium (HWE) in the population and linkage equilibrium (LE) between distal loci, to substantially improve power of association and interaction tests. We demonstrate, via simulation and application to actual GWAS data sets, that our approach is substantially more powerful and robust than standard testing approaches that ignore or make naive use of the population sample. We report several novel and credible pairwise interactions, in bipolar disorder, coronary artery disease, Crohn's disease, and rheumatoid arthritis. G ENOME-WIDE association studies (GWAS) have implicated thousands of single-nucleotide polymorphisms (SNPs) in the human genome as associated with hundreds of phenotypes (Johnson and O'Donnell 2009). However, as many researchers have pointed out (Manolio et al. 2009;Eichler et al. 2010), the results from GWAS fail to explain the observed heritability of many phenotypes, including complex human diseases, whose genetic architectures remain largely unknown. One often-cited reason for this problem is that the high multiple-testing burden requires an exceedingly stringent statistical significance level. Furthermore, while most studies have employed univariate (locusby-locus) testing approaches, complex diseases are likely to be affected by interactions between loci (Eichler et al. 2010). Such interactions arise when there is a dependence of genotypic effects of one locus on genotypes at other loci (Cordell 2009).In the case of interactions, due to the overwhelming number of locus subsets, the multiple-testing problem becomes a serious computational and statistical challenge. Even when limiting exploration to pairwise SNP-SNP interactions, a modest study including 300,000 usable loci requires testing 45 billion SNP pairs, and associations must have P-values , 10 212 (the 0.05 Bonferroni-corrected significance level) to be declared statistically significant genome-wide. Recently, several authors have suggested sophisticated approximate and exhaustive methods for detecting pairwise interactions (Brinza et al. 2010;Liu et al. 2011;Prabhu and Pe'er 2012). These methods constitute a major step in dealing with the computational issue of carrying out the large number of tests, but their application to actual studies has led to surprisingly ...