  A path integral formulation of the Bianchi I models containing a massless
scalar field in loop quantum cosmology is constructed. Following the strategy
used in the homogenous and isotropic case, the calculation is extended to the
simplest non-isotropic models according to the $\bar{\mu}$ and
$\bar{\mu}^{\prime}$ scheme. It is proved from the path integral angle that the
quantum dynamic lacks the full invariance with respect to fiducial cell scaling
in the $\bar{\mu}$ scheme, but it does not in the $\bar{\mu}^{\prime}$ scheme.
The investigation affirms the equivalence of the canonical approach and the
path integral approach in loop quantum cosmology.
