Maximally selected chi-square statistics for at least ordinal scaled variables

The association between a binary variable Y and a variable X with
an at least ordinal measurement scale might be examined by
selecting a cutpoint in the range of X and then performing an
association test for the obtained 2x2 contingency table using the
chi-square statistic. The distribution of the maximally selected
chi-square statistic (i.e. the maximal chi-square statistic over
all possible cutpoints) under the null-hypothesis of no association
between X and Y is different from the known chi-square
distribution. In the last decades, this topic has been extensively
studied for continuous X variables, but not for non-continuous
variables with an at least ordinal measurement scale (which include
e.g. classical ordinal or discretized continuous variables). In
this paper, we suggest an exact method to determine the
distribution of maximally selected chi-square statistics in this
context. This novel approach can be seen as a method to measure the
association between a binary variable and variables with an at
least ordinal scale of different types (ordinal, discretized
continuous, etc). As an illustration, this method is applied to a
new data set describing pregnancy and birth for 811 babies.