When is a metric not a metric? Remarks on direct curve comparison in bioequivalence studies

The majority of measures proposed to date for direct curve comparison in bioequivalence studies were investigated. These measures have often been called metrics, but in most cases this was incorrect in the mathematical sense. It was demonstrated, with a set of counter-examples, that the axioms of a metric are fulfilled only for the integral p-metric and some of its transforms. The Rescigno index and two other measures devised by Polli and McLean are the semi-metrics, lacking the triangle inequality, while others also lack symmetry. The use of the p-metric is therefore recommended, and statistical analysis is suggested as a point at which the scaling of differences might be carried out.