The gene score vectors before correction () calculated for the 1,000 DGI permuted data sets were used to quantify the correlation between six gene properties of potential confounding effects on and (Table 1). The permutations were also used to evaluate which of the correlated gene properties had a significant confounding effect on based on a step-wise multivariate linear regression model (Table S3). The resulting significant confounders were used in all gene set analyses presented in this study. To assess the performance of our regression-based correction of confounders on , Sidak's correction and a modified Sidak's correction, we compared the corrected gene p-values, to the corresponding gene p-values corrected with permutation analysis, for all genes g, using the actual DGI study. Permutation analysis was used as the gold standard for adjusting for confounders on SNP to gene scores as it generates gene-specific null distributions while maintaining the physical and genetic structure of SNPs across gene regions. This enables correcting for all possible confounding effects on gene association scores without requiring a priori knowledge of the confounders. The performance of our regression-based
