Skip to content  Chi-square FAIL when many cells have small
expected values  William Perkins, Mark Tygert, and Rachel Ward
write:  If a discrete probability distribution in a model being
tested for goodness-of-fit is not close to uniform, then forming
the Pearson χ2 statistic can involve division by nearly zero.
This often leads to serious trouble in practice — even in the
absence of round-off errors . . .  The problem is not merely
that the chi-squared statistic doesn’t have the advertised chi-
squared distribution—a reference distribution can always be
computed via simulation, either using the posterior predictive
distribution or by conditioning on a point estimate of the cell
expectations and then making a degrees-of-freedom sort of
adjustment.  Rather, the problem is that, when there are lots of
cells with near-zero expectation, the chi-squared test is mostly
noise.  And this is not merely a theoretical problem. It comes
up in real examples.  Here’s one, taken from the classic 1992
genetics paper of Guo an