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Population substructure can lead to confounding in tests for genetic association, and failure to adjust properly can result in spurious findings. Here we address this issue of confounding by considering the impact of global ancestry (average ancestry across the genome) and local ancestry (ancestry at a specific chromosomal location) on regression parameters and relative power in ancestry-adjusted and -unadjusted models. We examine theoretical expectations under different scenarios for population substructure; applying different regression models, verifying and generalizing using simulations, and exploring the findings in real-world admixed populations. We show that admixture does not lead to confounding when the trait locus is tested directly in a single admixed population. However, if there is more complex population structure or a marker locus in linkage disequilibrium (LD) with the trait locus is tested, both global and local ancestry can be confounders. Additionally, we show the genotype parameters of adjusted and unadjusted models all provide tests for LD between the marker and trait locus, but in different contexts. The local ancestry adjusted model tests for LD in the ancestral populations, while tests using the unadjusted and the global ancestry adjusted models depend on LD in the admixed population(s), which may be enriched due to different ancestral allele frequencies. Practically, this implies that global-ancestry adjustment should be used for screening, but local-ancestry adjustment may better inform fine mapping and provide better effect estimates at trait loci., (© 2017 WILEY PERIODICALS, INC.)