When stratifying by population backgrounds, covariates such as PCs should still be used to correct for population stratification. Conventional linear or logistic regression with these covariates can be used for association testing as long as QC included exclusion of related individuals; mixed models or other alternatives with PC covariates may be applied in family-based samples stratified by ancestry (Walters et al., 2018). Computing these PCs separately within each ancestry subset instead of the full study ensures better control for residual structure specific to that subset (e.g., fine structure, genotyping/technical artifacts), but at the cost of potentially reduced control for stratification related to population structure shared across subsets (Patterson et al. 2006). For analyses of admixed or multi-ancestry cohorts, PCs may still be included in the regression but additional covariates may be required to control for stratification that is not linear in PCA space (Conomos et al., 2018; Heckerman et al., 2016; Zhang and Pan, 2015). For example, race and ethnicity are often correlated with socio-economic status and other environmental risk factors for disease. Self-reported ethnicity or other variables that capture
