  The Sun has been found to be depleted in refractory (rock-forming) elements
relative to nearby solar analogs, suggesting a potential indicator of planet
formation. Given the small amplitude of the depletion, previous analyses have
primarily relied on high signal-to-noise stellar spectra and a strictly
differential approach to determine elemental abundances. We present an
alternative, likelihood-based approach that can be applied to much larger
samples of stars with lower precision abundance determinations. We utilize
measurements of about 1700 solar analogs from the Apache Point Observatory
Galactic Evolution Experiment (APOGEE-2) and the stellar parameter and chemical
abundance pipeline (ASPCAP DR16). By developing a hierarchical mixture model
for the data, we place constraints on the statistical properties of the
elemental abundances, including correlations with condensation temperature and
the fraction of stars with refractory element depletions. We find evidence for
two distinct populations: a depleted population of stars that makes up the
majority of solar analogs including the Sun, and a not-depleted population that
makes up between 10-30% of our sample. We find correlations with condensation
temperature generally in agreement with higher precision surveys of a smaller
sample of stars. Such trends, if robustly linked to the formation of planetary
systems, provide a means to connect stellar chemical abundance patterns to
planetary systems over large samples of Milky Way stars.
