Maximally selected chi-square statistics and binary splits of nominal variables

We address the problem of maximally selected chi-square statistics
in the case of a binary Y variable and a nominal X variable with
several categories. The distribution of the maximally selected
chi-square statistic has already been derived when the best
cutpoint is chosen from a continuous or an ordinal X, but not when
the best split is chosen from a nominal X. In this paper, we derive
the exact distribution of the maximally selected chi-square
statistic in this case using a combinatorial approach. Applications
of the derived distribution to variable selection and hypothesis
testing are discussed based on simulations. As an illustration, our
method is applied to a pregnancy and birth data set.