Boulesteix, AnneLaure; Strobl, Carolin
(2006):
Maximally selected chisquare statistics and umbrella orderings.
Collaborative Research Center 386, Discussion Paper 476

Preview 

272kB 
Abstract
Binary outcomes that depend on an ordinal predictor in a nonmonotonic 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 chisquare 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 chisquare statistic, referring the obtained chisquare statistic to the standard chisquare distribution is an inappropriate approach. It is then necessary to correct the pvalue for multiple comparisons by considering the distribution of the maximally selected chisquare statistic instead of the nominal chisquare distribution. Here, we derive the exact distribution of the chisquare 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.