  Metacalibration is a recently introduced method to accurately measure weak
gravitational lensing shear using only the available imaging data, without need
for prior information about galaxy properties or calibration from simulations.
The method involves distorting the image with a small known shear, and
calculating the response of a shear estimator to that applied shear. The method
was shown to be accurate in moderate sized simulations with galaxy images that
had relatively high signal-to-noise ratios, and without significant selection
effects. In this work we introduce a formalism to correct for both shear
response and selection biases. We also observe that, for images with relatively
low signal-to-noise ratios, the correlated noise that arises during the
metacalibration process results in significant bias, for which we develop a
simple empirical correction. To test this formalism, we created large image
simulations based on both parametric models and real galaxy images, including
tests with realistic point-spread functions. We varied the point-spread
function ellipticity at the five percent level. In each simulation we applied a
small, few percent shear to the galaxy images. We introduced additional
challenges that arise in real data, such as detection thresholds, stellar
contamination, and missing data. We applied cuts on the measured galaxy
properties to induce significant selection effects. Using our formalism, we
recovered the input shear with an accuracy better than a part in a thousand in
all cases.
