## 1 Introduction

Multivariate meta-analysis is a fairly recent methodological development (e.g. van Houwelingen et al., 1993, 2002; Berkey et al., 1998), which is becoming more commonly applied in medical statistics (Jackson et al., 2011). Multivariate meta-analysis is used to synthesise multiple outcome effects from separate studies (e.g. overall and disease free survival), whilst allowing for their correlation. Two types of correlations may exist: *within-study correlations*, which indicate the association between outcome effect estimates in each study, and *between-study correlations*, which indicate how the true outcome effects are associated across studies. The within-study correlations arise when the same patients contribute data to both outcomes in a study. The between-study correlation arises when (unknown) factors causing between-study heterogeneity induce a correlation in the true outcome effects across studies; for example studies with a larger than average treatment effect on overall survival will typically have a larger than average treatment effect on disease free survival.

Multivariate meta-analysis possesses many advantages over its more established univariate counterpart, including the potential for inferences for different outcomes to ‘borrow strength’ (Riley et al., 2007) from each other. Jackson et al. 2011 discuss the advantages, and limitations, of multivariate compared to univariate meta-analysis. Software has been produced in Stata to fit the random effects meta-analysis model (White, 2009, and has recently been extended to multivariate meta-regression models (White, 2011, and the R package mvmeta (Gasparrini, 2011) is now available.

Here, we take the multivariate random effects model as the standard model. The fixed effect model assumes that common underlying effects apply to all studies. We find this generally implausible: it is a very strong assumption to assume that there is no between-study heterogeneity in any of the outcomes included in the analysis. When fitting the multivariate random effects meta-analysis model, however, we must estimate the between-study covariance matrix, which increases the computational demands. We assume that within-study covariance matrices are available for all studies but recognise that obtaining the within-study correlations is often a practical difficulty and that these values are important (Riley, 2009). See Jackson et al. 2011 for a variety of methods for handling unknown within-study correlations and Riley et al. 2008 for an alternative random effects model that does not require them.

Several fully parametric approaches to estimation have been developed. These include maximum likelihood, restricted maximum likelihood (REML; e.g. van Houwelingen et al., 2002; Jackson et al., 2011 and Bayesian estimation (Nam et al., 2003. Maximum likelihood methods are invariant to linear transformations but, especially in high dimensions, are much more computationally intensive.

Semi-parametric alternatives therefore have their advantages, such as the method based on *U* statistics (Ma and Mazumdar, 2011. The method proposed by DerSimonian and Laird 1986 has also been extended to the multivariate setting (Jackson et al., 2010; Chen et al., 2012). By estimating the between-study covariance matrix by matching moments a valid, but not optimal, analysis may be performed without requiring the assumption of between-study normality. The more general validity of the non-likelihood-based methods may be considered advantageous because we can only invoke the Central Limit Theorem to justify this assumption by the notion that the unobserved random effects are the sum of several different factors. Despite this lack of optimality, the simulation studies performed by Ma and Mazumdar 2011, Jackson et al. 2010 and Chen et al. (2012) suggest that the semi-parametric methods perform well compared with likelihood-based methods when making inferences about the treatment effect. However, the method proposed by Jackson et al. 2010 is not invariant to linear transformations and the procedure described by Chen et al. (2012) cannot handle covariates or missing outcome data. Since missing outcome data are a very common occurrence, it is vitally important that estimation procedures handle them in an appropriate way. The aim of this paper is to provide a new estimation method that overcomes the problems associated with the existing methodologies.

This paper presents a multivariate generalisation of DerSimonian and Laird's extremely popular univariate method. The new method can handle missing data and can adjust for covariates in a meta-regression, and reduces to the method of Chen et al. (2012) with complete data and no covariates. Like the method by Chen et al. (2012), the new method is based on matrix operations and is invariant to linear transformations. The rest of the paper is set out as follows. In Section 'A new method of moments for multivariate meta-analysis and meta-regression', we present our new method and derive its properties. In Section 'Simulation study', we present some results from a simulation study and in Section 'Example: Treatment for hypertension', we apply our methods to an example. We conclude with a discussion in Section 'Discussion'.