Algorithm 2.

Procedure with P-values of gene-level statistics computed via permutation

For each study k, compute the gene-level statistics: same as Step I of Algorithm 1. Meta-analysis: Within study k, randomly permute Ykss M times, and calculate the permuted statistics ${({\mathbf{U}}_{\mathit{kgm}},{\mathbf{V}}_{\mathit{kgm}})}_{g=1}^G$ for 1 ≤ m ≤ M and 1 ≤ k ≤ K. For each study k, randomly choose one pair of permuted statistics from the M pairs obtained in Step 1, and denote the selected pair by ${({\mathbf{U}}_{kg}^{(1)},{\mathbf{V}}_{kg}^{(1)})}_{g=1}^G$. In the RE model, calculate the corresponding ${\mathbf{B}}_g^{(1)}$. Repeat this process N times to obtain ${[{({\mathbf{U}}_{kg}^{(n)},{\mathbf{V}}_{kg}^{(n)})}_{k=1}^K]}_{g=1}^G$ for the FE model or ${[{({\mathbf{U}}_{kg}^{(n)},{\mathbf{V}}_{kg}^{(n)})}_{k=1}^K,{\mathbf{B}}_g^{(n)}]}_{g=1}^G$ for the RE model, where n = 1, …, N. Note that we usually choose N ≫ M for computational efficiency, but require N ≪ MK to remove the effect of repeatedly using the same M permutations for each study (e.g., for K = 5, set M = 50, and N = 1000). Compute Qg for the original data and ${[Q_g^{(n)}]}_{n=1}^N$ for the N sets of K permuted studies, based on the FE model or RE model. Estimate P-values of by the permutation for 1 ≤ g ≤ G: $p(Q_g)={\sum }_{n=1}^{N}T\left(Q_g\le Q_g^{(n)})\right)/N$. Set enrichment analysis: same as Step III of Algorithm 1.