Using Normalized Entropy to Measure Uncertainty of Rankings for Network Meta-analyses

Ranking of treatments offers a straightforward interpretation of results derived from network meta-analysis. However, some published network meta-analyses have overemphasized treatment ranking without paying attention to its uncertainty. According to a review of 91 network meta-analyses, 52 reported treatment ranking, but 43 of them did not report the uncertainty of ranking. Without reporting the uncertainty, small differences in the ranking of treatments may be overinterpreted. Rankograms, cumulative rankograms, the credible/confidence interval of mean rank, the surface under the cumulative ranking curve (SUCRA), and the interquartile range of median rank have been used to show the uncertainty of rankings. However, it is not always straightforward to compare the differences in the distribution of probabilities by inspecting rankograms or to compare the intervals or ranges of treatment ranks. We therefore proposed normalized entropy, which transforms the distribution of ranking probabilities into a single quantitative measure, to facilitate a refined interpretation of uncertainty of treatment ranking. We used 4 real examples to demonstrate the uncertainty of ranking quantified by ranking probabilities, 95% confidence interval of SUCRA, and normalized entropy. We showed that as normalized entropy ranges from 0 to 1 and is independent of the number of treatments, it can be used to compare the uncertainty of treatment ranking within a network meta-analysis (NMA) and between different NMAs. Normalized entropy is an alternative tool for measuring the uncertainty of treatment ranking by improving the translation of results from NMAs to clinical practice and avoiding naïve interpretation of treatment ranking. We therefore recommend normalized entropy to be included in the presentation and interpretation of results from NMAs.