  In this article we develop a method for performing post hoc inference of the
False Discovery Proportion (FDP) over multiple contrasts of interest in the
multivariate linear model. To do so we use the bootstrap to simulate from the
distribution of the null contrasts. We combine the bootstrap with the post hoc
inference bounds of Blanchard (2020) and prove that doing so provides
simultaneous asymptotic control of the FDP over all subsets of hypotheses. This
requires us to demonstrate consistency of the multivariate bootstrap in the
linear model, which we do via the Lindeberg Central Limit Theorem, providing a
simpler proof of this result than that of Eck (2018). We demonstrate, via
simulations, that our approach provides simultaneous control of the FDP over
all subsets and is typically more powerful than existing, state of the art,
parametric methods. We illustrate our approach on functional Magnetic Resonance
Imaging data from the Human Connectome project and on a transcriptomic dataset
of chronic obstructive pulmonary disease.
