Now we extend the above model to handle multiple GWAS data sets. To keep the notation uncluttered, we present the model for the case of two GWAS data sets. Suppose we have p-values from two GWAS: (6)Let be the matrix collecting all the p-values, where denotes the p-value of the j-th SNP in the k-th GWAS. Similarly, we introduce latent variables indicating the association between the j-th SNP and the two phenotypes: means the j-th SNP is associated with neither of them, means it is only associated with the first one, means it is only associated with the second one, and means it is associated with both. The two-groups model (4) is extended to the following “four-groups model”: (7)where . When the genetic bases of the two phenotypes are independent of each other (i.e., no pleiotropy), then we have by expectation. Therefore, the difference between and can be used to characterize pleiotropy. Statistical inference on pleiotropy is given in Section “Hypothesis testing of annotation enrichment and pleiotropy”.
