Both new estimation methods give results similar to those obtained using MCMC. The second uses “ear discharge" as an outcome, and exhibits substantial between-study heterogeneity and inconsistency. The first uses all-cause mortality as an outcome, and shows little evidence of between-study heterogeneity or inconsistency. We illustrate the application of the methods using two contrasting examples. The second new estimation method we describe uses a likelihood-based approach, implemented in the metafor package, which can be used to obtain (restricted) maximum-likelihood estimates of the model parameters and profile likelihood confidence intervals of the variance components. However, we confirm the accuracy of our importance sampling method by comparing the results to those obtained using MCMC as the gold standard. We fit the model using importance sampling and thereby avoid some of the difficulties that might be associated with using Markov Chain Monte Carlo (MCMC). Our first new estimation method uses a Bayesian framework with empirically-based prior distributions for both the heterogeneity and the inconsistency variances. The model we consider is an extension of the conventional random-effects model for meta-analysis to the network meta-analysis setting and allows for potential inconsistency using random inconsistency effects. We propose two new estimation methods for network meta-analysis models with random inconsistency effects. However, a network meta-analysis may exhibit inconsistency, whereby the treatment effect estimates do not agree across all trial designs, even after taking between-study heterogeneity into account. Network meta-analysis is becoming more widely used as a means to compare multiple treatments in the same analysis. Meta-analysis is a valuable tool for combining evidence from multiple studies.
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