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Abstract This paper proposes a novel statistical method to address population structure in genome-wide association studies while controlling the false discovery rate, which overcomes some limitations of existing approaches. Our solution accounts for linkage disequilibrium and diverse ancestries by combining conditional testing via knockoffs with hidden Markov models from state-of-the-art phasing methods. Furthermore, we account for familial relatedness by describing the joint distribution of haplotypes sharing long identical-by-descent segments with a generalized hidden Markov model. Extensive simulations affirm the validity of this method, while applications to UK Biobank phenotypes yield many more discoveries compared to BOLT-LMM, most of which are confirmed by the Japan Biobank and FinnGen data.

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