  Clusters of galaxies are powerful probes with which to study cosmology and
astrophysics. However, for many applications an accurate measurement of a
cluster's mass is essential. A systematic underestimate of hydrostatic masses
from X-ray observations (the so-called hydrostatic bias) may be responsible for
tension between the results of different cosmological measurements. We compare
X-ray hydrostatic masses with masses estimated using the caustic method (based
on galaxy velocities) in order to explore the systematic uncertainties of both
methods and place new constraints on the level of hydrostatic bias. Hydrostatic
and caustic mass profiles were determined independently for a sample of 44
clusters based on Chandra observations of clusters from the Hectospec Cluster
Survey. This is the largest systematic comparison of its kind. Masses were
compared at a standardised radius ($R_{500}$) using a model that includes
possible bias and scatter in both mass estimates. The systematics affecting
both mass determination methods were explored in detail. The hydrostatic masses
were found to be systematically higher than caustic masses on average, and we
found evidence that the caustic method increasingly underestimates the mass
when fewer galaxies are used to measure the caustics. We limit our analysis to
the 14 clusters with the best-sampled caustics where this bias is minimised
($\ge210$ galaxies), and find that the average ratio of hydrostatic to caustic
mass at $R_{500}$ is $M_X/M_C=1.12^{+0.11}_{-0.10}$. We interpret this result
as a constraint on the level of hydrostatic bias, favouring small or zero
levels of hydrostatic bias (less than $20\%$ at the $3\sigma$ level). However,
we find systematic uncertainties associated with both mass estimation methods
remain at the $10-15\%$ level, which would permit significantly larger levels
of hydrostatic bias.
