  Inspired by the recently theoretical development relevant to the experimental
data of TlCuCl$_3$, particularly those associated with the universal scaling
between the N\'eel temperature $T_N$ and the staggered magnetization density
$M_s$, we carry a detailed investigation of 3-dimensional (3D) dimerized
quantum antiferromagnets using the first principles quantum Monte Carlo
calculations. The motivation behind our study is to better understand the
microscopic effects on these scaling relations of $T_N$ and $M_s$, hence to
shed some light on some of the observed inconsistency between the theoretical
and the experimental results. Remarkably, for the considered 3D dimerized
models, we find that the established universal scaling relations can indeed be
categorized by the amount of stronger antiferromagnetic couplings connected to
a lattice site. Convincing numerical evidence is provided to support this
conjecture. The relevance of the outcomes presented here to the experiments of
TlCuCl$_3$ is briefly discussed as well.
