Principal Component Analysis (PCA)

PCA transforms a high-dimensional genotype matrix (individuals × SNPs) into a smaller set of uncorrelated principal components (PCs) that capture the directions of maximum genetic variance. The first PC typically separates individuals along the most divergent ancestral axis, while subsequent PCs resolve finer-scale population structure. This is achieved through eigenvalue decomposition of the genetic relationship matrix (GRM) or singular value decomposition (SVD) of the standardized genotype matrix, producing eigenvectors that serve as continuous ancestry covariates.

In GWAS and PRS analyses, population stratification creates spurious genotype-phenotype associations when both allele frequencies and disease prevalence differ systematically between subpopulations. Including the top 10-20 principal components as fixed-effect covariates in a linear or logistic regression model effectively adjusts for this confounding. This approach assumes that genetic ancestry acts as a continuous variable, with PCs capturing the latent population structure that would otherwise inflate the genomic inflation factor (λ) and produce false-positive associations.

Each principal component represents a linear combination of SNP genotypes weighted by their contribution to population differentiation. Key interpretive properties include:

• PC1 often separates continental-level ancestry groups (e.g., African vs. European populations)
• PC2 typically resolves within-continent clines (e.g., North-South European gradients)
• Eigenvector loadings identify the specific SNPs driving each ancestry axis, which frequently correspond to high-FST markers under differential selection
• PCs are orthogonal by construction, ensuring each axis captures independent ancestry information

Study samples can be projected onto principal components computed from an external reference panel of known ancestry (e.g., 1000 Genomes Project) rather than performing de novo PCA on the study cohort alone. This projection method:

• Prevents the PCs from being influenced by case-control imbalance or family structure within the study
• Enables direct ancestry labeling by comparing projected coordinates to reference population clusters
• Requires the same set of LD-pruned, common SNPs to be available in both datasets
• Uses the reference panel's SNP loadings to compute projected PC scores for new individuals

PCA is sensitive to linkage disequilibrium (LD) and genotyping artifacts that can distort the covariance structure. Standard preprocessing includes:

• LD pruning to retain a set of approximately 50,000-100,000 approximately independent SNPs (r² < 0.2 in sliding windows)
• Removal of high-LD regions (e.g., MHC locus on chromosome 6, inversions on chromosomes 8 and 17)
• Exclusion of SNPs with minor allele frequency < 5% to reduce noise from rare variants
• Filtering out related individuals (pi-hat > 0.2) to prevent family structure from dominating the top PCs

The Tracy-Widom distribution provides a formal statistical test for determining how many principal components represent true population structure versus random noise. The test statistic compares observed eigenvalues against the theoretical null distribution expected under no population stratification. This enables objective selection of the number of PCs to include as covariates, avoiding both under-correction (residual confounding) and over-correction (loss of statistical power from unnecessary degrees of freedom). Software implementations include EIGENSOFT's twstats and the AssocTests R package.