Assessing evidence inconsistency in mixed treatment comparisons

Randomized comparisons among several treatments give rise to an incomplete-blocks structure known as mixed treatment comparisons (MTCs). To analyze such data structures, it is crucial to assess whether the disparate evidence sources provide consistent information about the treatment contrasts. In this article we propose a general method for assessing evidence inconsistency in the framework of Bayesian hierarchical models. We begin with the distinction between basic parameters, which have prior distributions, and functional parameters, which are defined in terms of basic parameters. Based on a graphical analysis of MTC structures, evidence inconsistency is defined as a relation between a functional parameter and at least two basic parameters, supported by at least three evidence sources. The inconsistency degrees of freedom (ICDF) is the number of such inconsistencies. We represent evidence consistency as a set of linear relations between effect parameters on the log odds ratio scale, then relax these relations to allow for inconsistency by adding to the model random inconsistency factors (ICFs). The number of ICFs is determined by the ICDF. The overall consistency between evidence sources can be assessed by comparing models with and without ICFs, whereas their posterior distribution reflects the extent of inconsistency in particular evidence cycles. The methods are elucidated using two published datasets, implemented with standard Markov chain Monte Carlo software.