  Let $F$ be a totally real field and let $S$ denote the geometric special
fiber of a Hilbert modular variety associated to $F$, at a prime unramified in
$F$. We show that the order of vanishing of the Hasse invariant on $S$ is equal
to the largest integer $m$ such that the smallest piece of the conjugate
filtration lies in the $m^{\text{th}}$ piece of the Hodge filtration. This
result is a direct analogue of Ogus' on families of Calabi-Yau varieties in
positive characteristic. We also show that the order of vanishing at a point is
the same as the codimension of the Ekedahl-Oort stratum containing it.
