  The $X^2$ and $G^2$ tests are the most frequently applied tests for testing
the independence of two categorical variables. However, no one, to the best of
our knowledge has compared them, extensively, and ultimately answer the
question of which to use and when. Further, their applicability in cases with
zero frequencies has been debated and (non parametric) permutation tests are
suggested. In this work we perform extensive Monte Carlo simulation studies
attempting to answer both aforementioned points. As expected, in large sample
sized cases ($>1,000$) the $X^2$ and $G^2$ are indistinguishable. In the small
sample sized cases ($\leq 1,000$) though, we provide strong evidence supporting
the use of the $X^2$ test regardless of zero frequencies for the case of
unconditional independence. Also, we suggest the use of the permutation based
$G^2$ test for testing conditional independence, at the cost of being
computationally more expensive. The $G^2$ test exhibited inferior performance
and its use should be limited.
