A permutation-free family-wise error rate for the moderated top-gene scan under gene correlation

A differential-expression scan reports the genes with the largest moderated t-statistics, so controlling the family-wise error rate means controlling the null distribution of the maximum statistic over genes. Under gene correlation this is widely believed to require permutation, because correlation both changes the effective multiplicity and corrupts the empirical-Bayes variance prior that underlies the moderated t-statistic. This paper decomposes that liberality by an error-budget ablation and shows that, within the simulated model class, it reduces principally to an inflation of the empirical-Bayes prior degrees of freedom: substituting the true prior returns the family-wise error to the small-sample baseline that is already present for independent genes, so the dependence imposes no separate barrier once the prior is correct. The inflation has a simple mechanism: correlation deflates the cross-gene spread of the log sample variances, and because the prior degrees of freedom is a decreasing function of that spread, the prior is over-estimated and the moderated maximum turns liberal. The mechanism is reversed by dividing the observed spread by one minus the mean gene correlation squared, which is estimated by a spectral U-statistic that needs no tuning constant and whose target trace is unbiased under any correlation structure. The resulting cutoff holds the family-wise error near the small-sample baseline across a range of correlation structures at retained power and without permutation, and an observable instability index flags when severe co-expression should defer to permutation. On a pre-specified real-data panel the correction changes gene-level verdicts only in small-sample, strongly co-expressed sets, and where a permutation gold standard is informative the analytic and permutation calls agree only about half the time; the method is therefore an analytic safeguard for that corner rather than a routine replacement for permutation.