We have defined above that Dj=∑inxij2 and, as xij is not available in this case, we thus take Dj = 2pj (1 – pj)n, assuming HWE and can show this in matrix form: (11)B=D1∕2DW−1∕2W′WDW−1∕2D1∕2 Therefore, we can approximate a joint analysis of multiple SNPs as (12)b~=B−1Dβ^and var(b~)=σJ2B−1 where b̃ = {b̃j} is an N × 1 vector of approximate estimates of joint SNP effects. If a SNP is uncorrelated with all other SNPs in the model, then the estimate of the effect size from the joint analysis will be identical to that from the meta-analysis. In a genetically homogenous population of large effective size, the expected value of LD correlation between two SNPs on different chromosomes or a large distance apart is approximately zero, and the observed LD correlations between such pairs of SNPs in a sample are just a result of random sampling. We show with empirical data that the observed LD correlation between SNPs more than 10 Mb apart is consistent with what we would expect by chance (Supplementary Fig. 7). We use the expected values (zeros) in
