  In early detection of disease, a single biomarker often has inadequate
classification performance, making it important to identify new biomarkers to
combine with the existing marker for improved performance. A biologically
natural method to combine biomarkers is to use logic rules, e.g. the OR/AND
rules. In our motivating example of early detection of pancreatic cancer, the
established biomarker CA19-9 is only present in a subclass of cancer; it is of
interest to identify new biomarkers present in the other subclasses and declare
disease when either marker is positive. While there has been research on
developing biomarker combinations using the OR/AND rules, the inference
regarding the incremental value of the new marker within this framework is
lacking and challenging due to a statistical non-regularity. In this paper, we
aim to answer the inferential question of whether combining the new biomarker
achieves better classification performance than using the existing biomarker
alone, based on a nonparametrically estimated OR rule that maximizes the
weighted average of sensitivity and specificity. We propose and compare various
procedures for testing the incremental value of the new biomarker and
constructing its confidence interval, using bootstrap, cross-validation, and a
novel fuzzy p-value-based technique. We compare the performance of different
methods via extensive simulation studies and apply them to the pancreatic
cancer example.
