In a GWAS or meta-analysis, however, each SNP is usually tested for association separately based on a single-SNP model (3)y=xjβj+e where xj is the jth column of X and βj is the marginal effect of SNP j. The marginal effects of multiple SNPs estimated from a single SNP–based genome scan can be written in matrix form as (4)β^=D−1X′yand var(β^)=σM2D−1 where β = {βj} is an N × 1 vector of marginal SNP effects, D = {Dj} is the diagonal matrix of X′X with Dj=∑inxij2 and σM2 is the residual variance in the single-SNP analyses. The marginal SNP effects do not take the LD correlations between SNPs into account compared with the joint SNP effects. There are two issues involved in such single-SNP analyses for SNPs a short distance from each other: (i) if the increasing (or risk) alleles of two SNPs is negatively correlated, the effects of both SNPs will be attenuated; therefore, the single-SNP analysis is underpowered, and one SNP or both SNPs may be undetected and (ii) if both SNPs reach genome-wide significance, it is difficult to determine their degree of dependency by interrogating the LD afterwards.
