The standard TWAS Z-score is a test against the null of no SNP-trait association; that is ZTWAS = WZ/(W Σs,s Wt )1/2 is well calibrated (i.e. has a 0 mean and unit variance) only under the null model of Z ~ N(0,Σs,s). In the alternate model where Z is drawn from a non-zero mean distribution59,60, ZTWAS has a distribution that depends both on Z as well as the weights W. To quantify the impact of the weights on ZTWAS regardless of whether Z is null or non-null we conduct permutations conditional on the observed Z vector. For each gene, the expression labels were randomly shuffled and the summary-based TWAS analysis trained on the resulting expression to compute a permuted new null for ZTWAS. Testing against this permuted null distribution is equivalent to testing for an expression-trait association (or genetic correlation between expression and trait, see below) conditional on the observed GWAS statistics at the locus (which may not be drawn from the null of no association). The permutation test empirically computes this distribution of ZTWAS values conditional on the observed
