  We present a uniform analysis of the stellar initial mass function (IMF) from
integrated light spectroscopy of 15 compact stellar systems (11 globular
clusters in M31 and 4 ultra compact dwarfs in the Virgo cluster, UCDs) and two
brightest Coma cluster galaxies (BCGs), covering a wide range of metallicities
($-$1.7 $<$ [Fe/H] $<$ 0.01) and velocity dispersions (7.4 km~s$^{-1}$ $<
\sigma <$ 275 km~s$^{-1}$). The S/N $\sim 100$ \AA$^{-1}$ Keck LRIS spectra are
fitted over the range $4000<\lambda/\mbox{\AA}<10,000$ with flexible,
full-spectrum stellar population synthesis models. We use the models to fit
simultaneously for ages, metallicities, and individual elemental abundances of
the population, allowing us to decouple abundance variations from variations in
IMF slope. We show that compact stellar systems do not follow the same trends
with physical parameters that have been found for early-type galaxies. Most
globular clusters in our sample have an IMF consistent with that of the Milky
Way, over a wide range of [Fe/H] and [Mg/Fe]. There is more diversity among the
UCDs, with some showing evidence for a bottom-heavy IMF, but with no clear
correlation with metallicity, abundance, or velocity dispersion. The two Coma
BCGs have similar velocity dispersion and metallicity, but we find the IMF of
NGC~4874 is consistent with that of the Milky Way while NGC~4889 presents
evidence for a significantly bottom-heavy IMF. For this sample, the IMF appears
to vary between objects in a way that is not explained by a single
metallicity-dependent prescription.
