Fig. 3.

A schematic of the general steps of the proposed algorithm to calculate the statistical significance of associations between variables (rows in Y) and their top r PCs ($V_r^T$). By independently permuting a small number (s) of variables and recalculating the PCs, we generate tractable “synthetic” null variables while preserving the overall systematic variation. Association statistics between the s synthetic null variables in $Y^{*}$ and $V_r^{*T}$ form the empirical null distribution, automatically taking account over-fitting intrinsic to testing for associations between a set of observed variables and their PCs