  We compare predictions of a number of empirical models and numerical
simulations of galaxy formation to the conditional stellar mass functions
(CSMF)of galaxies in groups of different masses obtained recently by Lan et al.
to test how well different models accommodate the data. The observational data
clearly prefer a model in which star formation in low-mass halos changes
behavior at a characteristic redshift $z_c\sim 2$. There is also tentative
evidence that this characteristic redshift depends on environment, becoming
$z_c\sim 4$ in regions that eventually evolve into rich clusters of galaxies.
The constrained model is used to understand how galaxies form and evolve in
dark matter halos, and to make predictions for other statistical properties of
the galaxy population, such as the stellar mass functions of galaxies at high
$z$, the star formation and stellar mass assembly histories in dark matter
halos. A comparison of our model predictions with those of other empirical
models shows that different models can make vastly different predictions, even
though all of them are tuned to match the observed stellar mass functions of
galaxies.
