Local inconsistency detection using the Kullback-Leibler divergence measure

The standard approach to local inconsistency assessment typically relies on testing the conflict between the direct and indirect evidence in selected treatment comparisons. However, statistical tests for inconsistency have low power and are subject to misinterpreting a p-value above the significance threshold as evidence of consistency.

We propose a simple framework to interpret local inconsistency based on the average Kullback-Leibler divergence (KLD) from approximating the direct with the corresponding indirect estimate and vice versa. Our framework uses directly the mean and standard error (or posterior mean and standard deviation) of the direct and indirect estimates obtained from a local inconsistency method to calculate the average KLD measure for selected comparisons. The average KLD values are compared with a semi-objective threshold to judge the inconsistency as acceptably low or material. We exemplify our novel interpretation approach using three networks with multiple treatments and multi-arm studies.

Almost all selected comparisons in the networks were not associated with statistically significant inconsistency at a significance level of 5%. The proposed interpretation framework indicated 14%, 66%, and 75% of the selected comparisons with an acceptably low inconsistency in the corresponding networks. Overall, information loss was more notable when approximating the posterior density of the indirect estimates with that of the direct estimates, attributed to indirect estimates being more imprecise.

Using the concept of information loss between two distributions alongside a semi-objectively defined threshold helped distinguish target comparisons with acceptably low inconsistency from those with material inconsistency when statistical tests for inconsistency were inconclusive.