Abstract
Random-effects meta-analyses are very commonly used in medical statistics. Recent methodological developments include multivariate (multiple outcomes) and network (multiple treatments) meta-analysis. Here, we provide a new model and corresponding estimation procedure for multivariate network meta-analysis, so that multiple outcomes and treatments can be included in a single analysis. Our new multivariate model is a direct extension of a univariate model for network meta-analysis that has recently been proposed. We allow two types of unknown variance parameters in our model, which represent between-study heterogeneity and inconsistency. Inconsistency arises when different forms of direct and indirect evidence are not in agreement, even having taken between-study heterogeneity into account. However, the consistency assumption is often assumed in practice and so we also explain how to fit a reduced model which makes this assumption. Our estimation method extends several other commonly used methods for meta-analysis, including the method proposed by DerSimonian and Laird (). We investigate the use of our proposed methods in the context of both a simulation study and a real example.
Highlights
In order to evaluate the expectations required, we will need to be able to compute expressions of the form btr(A(M ⊗ Σ)B), where A and B are np × np matrices, M is an n × n matrix and Σ is a p × p matrix
We continue to use the notation Ai,j to denote the ith by jth block of A, where these blocks are p × p matrices
This is just the law of matrix multiplication applied to blocks
Summary
In order to evaluate the expectations required, we will need to be able to compute expressions of the form btr(A(M ⊗ Σ)B), where A and B are np × np matrices, M is an n × n matrix and Σ is a p × p matrix. We continue to use the notation Ai,j to denote the ith by jth block of A, where these blocks are p × p matrices. This is just the law of matrix multiplication applied to blocks. To obtain the block trace, we sum the matrices along the main diagonal. To obtain the block trace we take l = k to obtain the matrices along the main diagonal and sum over k so obtain btr(A(M ⊗ Σ)B) =.
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