  In neuroimaging, hundreds to hundreds of thousands of tests are performed
across a set of brain regions or all locations in an image. Recent studies have
shown that the most common family-wise error (FWE) controlling procedures in
imaging, which rely on classical mathematical inequalities or Gaussian random
field theory, yield FWE rates that are far from the nominal level. Depending on
the approach used, the FWER can be exceedingly small or grossly inflated. Given
the widespread use of neuroimaging as a tool for understanding neurological and
psychiatric disorders, it is imperative that reliable multiple testing
procedures are available. To our knowledge, only permutation joint testing
procedures have been shown to reliably control the FWER at the nominal level.
However, these procedures are computationally intensive due to the increasingly
available large sample sizes and dimensionality of the images, and analyses can
take days to complete. Here, we develop a parametric bootstrap joint testing
procedure. The parametric bootstrap procedure works directly with the test
statistics, which leads to much faster estimation of adjusted \emph{p}-values
than resampling-based procedures while reliably controlling the FWER in sample
sizes available in many neuroimaging studies. We demonstrate that the procedure
controls the FWER in finite samples using simulations, and present region- and
voxel-wise analyses to test for sex differences in developmental trajectories
of cerebral blood flow.
