Fig. 3

Illustration of federated GWAS algorithm. a 8 steps of null model fitting that is used in the GWAS protocol. First 4 steps utilize the matrix inversion (Fig. 2d) using the mask matrices \({H}_{1}\), \({H}_{2}\), and \({H}_{3}\) to calculate the encrypted inverse of the pooled covariance matrix of covariates, i.e., \({\left({X}^{\prime}WX\right)}^{-1}\). This matrix is also padded to the next power of 2 for usage later. In step 6, the weights are updated using a row-row multiplication, i.e., \({\alpha =\left({X}^{\prime}WX\right)}^{-1}\cdot \left({X}^{\prime}Wz\right)\). The parameter estimates for the current epoch is collectively decrypted and used in the next iteration. b 5 steps of p-value assignment, denoted by steps 9–13. Each site first calculates the components of the p-value statistics using local genotype and phenotype data. These are, \(T={G}_{1}^{\prime}\left({Y}_{1}-{\mu }_{\mathrm{0,1}}\right)\), \({G}_{1}^{\prime}{W}_{1}{G}_{1}\), \({G}_{1}^{\prime}{W}_{1}{X}_{1}\cdot {\left({X}^{\prime}WX\right)}^{-1}\), and \(G^{\prime}WX\). Next, each matrix is pooled among sites (Step 10) and the scale parameter is calculated, i.e., \(S={tr(G}^{\prime}WG)-{\langle {G}^{\prime}WX\cdot {\left({X}^{\prime}WX\right)}^{-1},{G}^{\prime}WX\rangle }_{r2r}\), where \(tr(A)\) denotes the trace of matrix \(A\). Next, each site generates a mask vector, denoted by \({N}_{1}\), and elementwise multiplies with both \(T\) and \(S\) vector with the same mask vector. The masked statistics are pooled among sites to calculate the final collectively masked statistics, which are collectively decrypted and used for assigning final p-values. c The meta-analysis steps. These steps start from the \(S\) and \(T\) statistics that GMMAT calculates. Each site performs the masking, pooling, and collective decryption followed by the p-value assignment step