Kullback–Leibler divergence for evaluating bioequivalence

Abstract

In this paper we propose a methodology for evaluating the bioequivalence of two formulations of a drug that encompasses not only average bioequivalence (ABE), but also the more recently introduced measures of population bioequivalence (PBE) and individual bioequivalence (IBE). The latter two measures are concerned with prescribability (PBE) and switchability (IBE). The main idea is to use the Kullback-Leibler divergence (KLD) as a measure of discrepancy between the distributions of the two formulations. Two formulations are declared bioequivalent if the upper bound of a level-alpha confidence interval for the KLD is less than a given goalpost to be set by a regulator. This new methodology overcomes many of the disadvantages of the corresponding measures recommended by the FDA. In particular the KLD: (i) possesses the natural hierarchical property that IBE => PBE => ABE; (ii) satisfies the properties of a true distance metric; (iii) is invariant to monotonic transformations of the data; (iv) generalizes easily to the multivariate case where equivalence on more than one parameter (for example, AUC, C(max) and T(max)) is required; and (v) is applicable over a wide range of distributions of the response variable (for example, those in the exponential family). The performance of the KLD relative to the metric proposed in guidance by the FDA for the evaluation of individual bioequivalence is evaluated using a simulation study. Previously published retrospective analyses using the FDA-proposed metric are contrasted with those based on the KLD. It is concluded that the KLD is a viable alternative to the FDA-proposed metric and that its mathematical and statistical properties make it a readily interpretable measure of the differences between formulations.