  Abundance matching (AM) is a popular technique for assigning galaxy mass or
luminosity to haloes produced in $N$-body simulations. The method works by
matching the cumulative number functions of the galaxy and halo properties, and
is therefore sensitive both to the precise definitions of those properties and
to the selection criteria used to define the samples. Further dependence
follows when AM parameters are calibrated with galaxy clustering, which is
known to depend strongly on the manner in which galaxies are selected. In this
paper we introduce a new parametrisation for AM and derive the best-fit AM
parameters as a function of various properties of the selection of the galaxy
sample and of the photometric definition, including S\'ersic vs Petrosian
magnitudes, stellar masses vs $r$-band magnitudes and optical (SDSS) vs HI
(ALFALFA) selection. In each case we calculate the models' goodness-of-fit to
measurements of the projected two-point galaxy correlation function. In the
optically-selected samples we find strong evidence that the scatter in the
galaxy-halo connection increases towards the faint end, and that AM performs
better with luminosity than stellar mass. The AM parameters of optically- and
HI-selected galaxies are mutually exclusive, with the latter suggesting the
importance of properties beyond halo mass. We provide best-fit parameters for
the AM galaxy-halo connection as a function of each of our input choices,
extending the domain of validity of the model while reducing potential
systematic error in its use.
