A basic requirement for on-off intermittency to occur is that the system po
ssesses an invariant subspace. We address how on-off intermittency manifest
s itself when a perturbation destroys the invariant subspace. In particular
, we distinguish between situations where the threshold for measuring the o
n-off intermittency in numerical or physical experiments is much larger tha
n or is comparable to the size of the perturbation. Our principal result is
that, as the perturbation parameter increases from zero, a metamorphosis i
n on-off intermittency occurs in the sense that scaling laws associated wit
h physically measurable quantities change abruptly. A geometric analysis, a
random-walk model, and numerical computations support the result.