  It was shown by LeCompte, Martin, and Oweans in 2010 that the existence of
mutually unbiased Hadamard matrices and the identity matrix, which coincide
with mutually unbiased bases, is equivalent to that of a $Q$-polynomial
association scheme of class four which is both $Q$-antipodal and $Q$-bipartite.
We prove that the existence of a set of mutually unbiased Bush-type Hadamard
matrices is equivalent to that of an association scheme of class five. As an
application of this equivalence, we obtain the upper bound of the number of
mutually unbiased Bush-type Hadamard matrices of order $4n^2$ to be $2n-1$.
This is in contrast to the fact that the upper bound of mutually unbiased
Hadamard matrices of order $4n^2$ is $2n^2$. We also discuss a relation of our
scheme to some fusion schemes which are $Q$-antipodal and $Q$-bipartite
$Q$-polynomial of class $4$.
