## 1 Introduction

Systematic reviews that compare multiple treatments are clinically more useful than those making pairwise comparisons alone because, in the right circumstances, they allow competing interventions to be ranked. Network meta-analyses involve the simultaneous analysis of both direct and indirect comparisons among multiple treatments across multiple studies, usually randomized trials (Gleser and Olkin, 1994; Higgins and Whitehead, 1996; Lumley, 2002; Lu and Ades, 2004; Caldwell *et al*., 2005; Lu and Ades, 2006; Salanti *et al*., 2008; Ioannidis, 2009). An important component of a network meta-analysis is an assessment of the extent to which different sources of evidence are comparable, 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. Statistical conflicts of this nature have been termed incoherence (Lumley, 2002; Ioannidis, 2009) or inconsistency (Lu and Ades, 2006; Salanti *et al*., 2008) in the literature and are the principal topic of this paper.

This is the first of a pair of papers addressing inconsistency in network meta-analysis, with particular reference to studies with more than two treatments. In this paper, we address some fundamental conceptual issues and statistical models that allow for inconsistency. In the second paper, we address parameter estimation and provide a detailed application (White *et al*., 2012). In the present paper, we first review the concept of inconsistency, discussing its relationship with the familiar notion of heterogeneity in meta-analysis. We attempt to distinguish between different types of inconsistency, which we call ‘loop inconsistency’ and ‘design inconsistency’. The distinction between these, however, is not clearly defined when there are multi-arm trials. In particular, loop inconsistency cannot be defined unambiguously when such studies are included in the analysis. Therefore, we propose a general model to allow for inconsistency that encompasses both of these types of inconsistency but does not distinguish between them. The principal message of the paper is that this general model provides an approach to identifying inconsistencies that is free of arbitrary assumptions about the nature of any evidence inconsistency. We illustrate the models using an example of a network involving smoking cessation treatments. Within the paper, 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.