To present the local false discovery rate based on our GPA model, we begin with the simplest case: single GWAS without annotation data. In this case, there are only two groups: null and non-null. The false discovery rate is defined as the probability that the j-th SNP belongs to the null group given its p-value, i.e., (10)For joint analysis of two GWAS data sets, we are interested in the local false discovery rate of the j-th SNP, if it is claimed to be associated with the first phenotype, the second one, and both, i.e., (11)Similarly, when annotation data are available, the false discovery rates can be calculated as (12)Then, we use the direct posterior probability approach [51] to control global false discovery rates. More details for the estimation of false discovery rates are provided in Section 2 in Text S1.
