Meta-analyses that simultaneously compare multiple treatments (usually referred to as network meta-analyses or mixed treatment comparisons) are becoming increasingly common. An important component of a network meta-analysis is an assessment of the extent to which different sources of evidence are compatible, both substantively and statistically. A simple indirect comparison may be confounded if the studies involving one of the treatments of interest are fundamentally different from the studies involving the other treatment of interest. Here, we discuss methods for addressing inconsistency of evidence from comparative studies of different treatments. We define and review basic concepts of heterogeneity and inconsistency, and attempt to introduce a distinction between ‘loop inconsistency’ and ‘design inconsistency’. We then propose that the notion of design-by-treatment interaction provides a useful general framework for investigating inconsistency. In particular, using design-by-treatment interactions successfully addresses complications that arise from the presence of multi-arm trials in an evidence network. We show how the inconsistency model proposed by Lu and Ades is a restricted version of our full design-by-treatment interaction model and that there may be several distinct Lu–Ades models for any particular data set. We introduce novel graphical methods for depicting networks of evidence, clearly depicting multi-arm trials and illustrating where there is potential for inconsistency to arise. We apply various inconsistency models to data from trials of different comparisons among four smoking cessation interventions and show that models seeking to address loop inconsistency alone can run into problems. Copyright © 2012 John Wiley & Sons, Ltd.

Background In this study, we aimed to evaluate the effects of tocilizumab in adult patients admitted to hospital with COVID-19 with both hypoxia and systemic inflammation. Methods This randomised, controlled, open-label, platform trial (Randomised Evaluation of COVID-19 Therapy [RECOVERY]), is assessing several possible treatments in patients hospitalised with COVID-19 in the UK. Those trial participants with hypoxia (oxygen saturation <92% on air or requiring oxygen therapy) and evidence of systemic inflammation (C-reactive protein ≥75 mg/L) were eligible for random assignment in a 1:1 ratio to usual standard of care alone versus usual standard of care plus tocilizumab at a dose of 400 mg–800 mg (depending on weight) given intravenously. A second dose could be given 12–24 h later if the patient's condition had not improved. The primary outcome was 28-day mortality, assessed in the intention-to-treat population. The trial is registered with ISRCTN (50189673) and ClinicalTrials.gov ( NCT04381936 ). Findings Between April 23, 2020, and Jan 24, 2021, 4116 adults of 21 550 patients enrolled into the RECOVERY trial were included in the assessment of tocilizumab, including 3385 (82%) patients receiving systemic corticosteroids. Overall, 621 (31%) of the 2022 patients allocated tocilizumab and 729 (35%) of the 2094 patients allocated to usual care died within 28 days (rate ratio 0·85; 95% CI 0·76–0·94; p=0·0028). Consistent results were seen in all prespecified subgroups of patients, including those receiving systemic corticosteroids. Patients allocated to tocilizumab were more likely to be discharged from hospital within 28 days (57% vs 50%; rate ratio 1·22; 1·12–1·33; p<0·0001). Among those not receiving invasive mechanical ventilation at baseline, patients allocated tocilizumab were less likely to reach the composite endpoint of invasive mechanical ventilation or death (35% vs 42%; risk ratio 0·84; 95% CI 0·77–0·92; p<0·0001). Interpretation In hospitalised COVID-19 patients with hypoxia and systemic inflammation, tocilizumab improved survival and other clinical outcomes. These benefits were seen regardless of the amount of respiratory support and were additional to the benefits of systemic corticosteroids. Funding UK Research and Innovation (Medical Research Council) and National Institute of Health Research.

Network meta-analysis (multiple treatments meta-analysis, mixed treatment comparisons) attempts to make the best use of a set of studies comparing more than two treatments. However, it is important to assess whether a body of evidence is consistent or inconsistent. Previous work on models for network meta-analysis that allow for heterogeneity between studies has either been restricted to two-arm trials or followed a Bayesian framework. We propose two new frequentist ways to estimate consistency and inconsistency models by expressing them as multivariate random-effects meta-regressions, which can be implemented in some standard software packages. We illustrate the approach using the mvmeta package in Stata. Copyright © 2012 John Wiley & Sons, Ltd.

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Network meta-analysis is becoming more popular as a way to analyse multiple treatments simultaneously and, in the right circumstances, rank treatments. A difficulty in practice is the possibility of ‘inconsistency’ or ‘incoherence’, where direct evidence and indirect evidence are not in agreement. Here, we develop a random-effects implementation of the recently proposed design-by-treatment interaction model, using these random effects to model inconsistency and estimate the parameters of primary interest. Our proposal is a generalisation of the model proposed by Lumley and allows trials with three or more arms to be included in the analysis. Our methods also facilitate the ranking of treatments under inconsistency. We derive R and I2 statistics to quantify the impact of the between-study heterogeneity and the inconsistency. We apply our model to two examples. © 2014 The Authors. Statistics in Medicine published by John Wiley & Sons, Ltd.