  The dependence of polarization fraction $p$ on total intensity $I$ in
polarized submillimeter emission measurements is typically parameterized as
$p\propto I^{-\alpha}$ $(\alpha \leq 1)$, and used to infer dust grain
alignment efficiency in star-forming regions, with an index $\alpha=1$
indicating near-total lack of alignment of grains with the magnetic field. In
this work we demonstrate that the non-Gaussian noise characteristics of
polarization fraction may produce apparent measurements of $\alpha \sim 1$ even
in data with significant signal-to-noise in Stokes $Q$, $U$ and $I$ emission,
and so with robust measurements of polarization angle. We present a simple
model demonstrating this behavior, and propose a criterion by which
well-characterized measurements of polarization fraction may be identified. We
demonstrate that where our model is applicable, $\alpha$ can be recovered by
fitting the $p-I$ relationship with the mean of the Rice distribution, without
statistical debiasing of polarization fraction. We apply our model to JCMT
BISTRO Survey POL-2 850$\mu$m observations of three clumps in the Ophiuchus
Molecular Cloud, finding that in the externally-illuminated Oph A region,
$\alpha\approx 0.34$, while in the more isolated Oph B and C, despite their
differing star formation histories, $\alpha \sim 0.6-0.7$. Our results thus
suggest that dust grain alignment in dense gas is more strongly influenced by
incident interstellar radiation field than by star formation history. We
further find that grains may remain aligned with the magnetic field at
significantly higher gas densities than has previously been believed, thus
allowing investigation of magnetic field properties within star-forming clumps
and cores.
