The dependence of the polarization fraction p on total intensity I in polarized submillimeter emission measurements is typically parameterized as p ∝ I ？α (α ≤ 1) and used to infer dust grain alignment efficiency in star-forming regions, with an index α = 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 the polarization fraction may produce apparent measurements of α ~ 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 the polarization fraction may be identified. We demonstrate that where our model is applicable, α can be recovered by fitting the p？I relationship with the mean of the Rice distribution without statistical debiasing of the polarization fraction. We apply our model to JCMT BISTRO Survey POL-2 850 μm observations of three clumps in the Ophiuchus molecular cloud, finding that in the externally illuminated Oph A region, α ？ 0.34, while in the more isolated Oph B and C, despite their differing star formation histories, α ~ 0.6？0.7. Our results thus suggest that dust grain alignment in dense gas is more strongly influenced by the 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.