In a linear regression analysis of multiple SNPs, the least-squares estimates of the joint effects of one set of SNPs conditional on another set of SNPs (b2 | b1) are (15)b^2∣b^1=(X2′X2)−1X2′y−(X2′X2)−1X2′X1(X1′X1)−1X1′y(16)var(b^2∣b^1)=σC2(X2′X2)−1−σC2(X2′X2)−1X2′X1(X1′X1)−1X1′X2(X2′X2)−1 where σC2 is the residual variance in the conditional analysis and all the other variables and parameters are defined as above, with the subscripts 1 and 2 indicating the two SNP sets. We can perform a multi-SNP conditional analysis using summary data from single-SNP analyses and individual-level genotype data of the sample without accessing the phenotype data by (17)b^2∣b^1=(X2′X2)−1D2β^2−(X2′X2)−1X2′X1(X1′X1)−1D1β^1(18)σ^C2=y′y−b^1′D1β^1−(b^2∣b^1)′D2β^2(n−N1−N2) where N1 and N2 are the number of SNPs in the two sets. If there is only one SNP to be tested in the conditional analysis (N2 = 1), then equations (17) and (18) simplify to (19)b^2∣b^1=β^2−(X2′X2)−1X2′X1(X1′X1)−1D1β^1(20)var(b^2∣b^1)=σ^C2[D2−X2′X1(X1′X1)−1X1′X2]∕D22 where b̂2, β^2 and D2 are scalars.
