Maximally selected chi-square statistics and umbrella orderings

Binary outcomes that depend on an ordinal predictor in a
non-monotonic way are common in medical data analysis. Such
patterns can be addressed in terms of cutpoints: for example, one
looks for two cutpoints that define an interval in the range of the
ordinal predictor for which the probability of a positive outcome
is particularly high (or low). A chi-square test may then be
performed to compare the proportions of positive outcomes in and
outside this interval. However, if the two cutpoints are chosen to
maximize the chi-square statistic, referring the obtained
chi-square statistic to the standard chi-square distribution is an
inappropriate approach. It is then necessary to correct the p-value
for multiple comparisons by considering the distribution of the
maximally selected chi-square statistic instead of the nominal
chi-square distribution. Here, we derive the exact distribution of
the chi-square statistic obtained by the optimal two cutpoints. We
suggest a combinatorial computation method and illustrate our
approach by a simulation study and an application to varicella
data.