## 1 Introduction

When the synthesis of data from studies reporting multiple outcomes is required, multivariate random effects meta-analysis (MRMA) can be used instead of performing meta-analyses on each outcome separately, which has an advantage of taking into account the correlations between the outcomes. Methods for MRMA have been developed for a number of purposes, for example, to estimate multiple outcomes from clinical trials [1], to model relationships between surrogate endpoints [2], or to evaluate diagnostic tests [3]. Attention has mainly focussed on frequentist approaches, but methods, which use a Bayesian framework, have also been developed [2, 4]. In a recent review, Jackson *et al.* [5] describe advances in the development of the methodology of multivariate meta-analysis and discuss the advantages and disadvantages of the use of these methods. One of the advantages of integrating data on multiple outcomes through MRMA is that of ‘borrowing of strength’ across studies as well as across outcomes, which can potentially lead to reduced uncertainty around the resulting effectiveness estimates. In a Bayesian framework, a wider range of sources of evidence can be integrated. For example, additional data (external data from observational studies, clinical trials, or systematic reviews) or experts’ opinions can be incorporated in the form of prior distributions, which can further inform a multivariate meta-analysis model. The use of external data can potentially lead to a further reduction of uncertainty around the estimates. The main advantage of Bayesian analysis, however, is that it enables an analyst to combine evidence from multiple sources, which is an important factor in evidence-based medicine [6-8].

The MRMA models typically have a hierarchical structure in which the correlated multiple outcomes estimate underlying true effects (within-study model) and the multiple true effects are correlated and follow the same common distribution (between-study model). The elements of the within-study covariance matrices are assumed to be known; however, in practice, often only the variances are available. The correlations between the estimates (due to sampling variability), however, are rarely reported. The within-study correlations could be obtained if individual patient data (IPD) for all of the studies in MRMA were available, but this is hardly ever the case, and therefore, estimation of the within-study correlation is one of the major challenges in multivariate meta-analysis. Previous studies, which addressed this issue, included, for example, those suggesting the use of the average correlation obtained from the subset of those studies, which listed IPD [2], implementation of an alternative formulation of bivariate meta-analysis for studies with unknown within-study correlation, which combined covariances from both the within-study and the between-study model in a single term [9], or recently, a method of approximating the within-study covariances, based on the bivariate delta method, was developed [10]. In contrast to the within-study covariance, the between-study covariance matrix is estimated from the model using study level data [11]. However, in a Bayesian framework, the between-study covariance matrix or its elements require prior distributions to be placed on them (as do all the parameters: the between-study variances and correlations as well as the pooled effects).

Some of the previously developed Bayesian MRMA models were restricted to the bivariate case [2, 12, 13]. Nam *et al.* [4] developed more general MRMA models; however, the examples used were also restricted to two outcomes. Arends *et al.* [14] proposed extension to trivariate meta-analysis, but the focus was mostly on a frequentist approach, and the use of Bayesian methods was only briefly discussed. In this article, we introduce a novel approach to MRMA in a Bayesian framework, which can be implemented for an extended number of outcomes and is presented here for a trivariate random-effects meta-analysis (TRMA) of mixed (continuous and binary) outcomes. This model allows us to incorporate data from a variety of sources, by not only including in the TRMA studies that are a mixture of those reporting all or only some of the outcomes but also by incorporating external data in the form of prior distributions. The use of informative prior distributions also extends previously developed methods, which used noninformative prior distributions, such as uniform distributions placed on correlations and inverse gamma distributions placed on variances, or Wishart prior distributions for precision matrices in models by Nam *et al.* [4]. By contrast, we use external sources of data to construct informative prior distributions for both the within-study and between-study correlations. In our model, we assume that none of the studies in TRMA report the within-study correlations. We employ external IPD to construct prior distributions for the within-study correlations. To obtain the distributions of the correlations between mixed outcomes from the IPD, we used a double bootstrap method [15]. We also describe methods of parameterization of the between-study model, which with an appropriate choice of prior distributions, allow the explicit incorporation of external information into the model through the Bayesian framework. Independent prior distributions are typically placed on the parameters of the multivariate model [2], but perhaps more appealingly, they can be placed on the correlations and other parameters in such a way as to account for their interdependence.

We apply this meta-analytical framework to an example from rheumatoid arthritis (RA). We carried out a meta-analysis of tumor necrosis factor-alpha inhibitors used sequentially in patients treated with RA. In studies assessing long-term effectiveness and/or cost-effectiveness of treatments in RA, the Health Assessment Questionnaire (HAQ) score, a self-reported measure of physical function, is of main interest as it is commonly used to estimate quality of life of patients following the treatment [16]. In a systematic review, Lloyd *et al*. [17] found that very few studies reported the HAQ score, but there were other instruments used to assess patients’ responses to treatment. In the multivariate evidence synthesis models, we take into account data from studies reporting the HAQ score, as well as studies reporting other measures of response to treatment with the aim of integrating all available evidence and potentially decreasing uncertainty around the HAQ. We have devised a Bayesian framework in which external IPD is used to construct prior distributions for the within-study correlations between the HAQ and the alternative outcomes whereas external summary data (ESD) from a systematic review allows us to derive prior distributions for the between-study correlations.

The outline of this article is as follows. In Section 2, we introduce the motivating example and describe the data and the logic behind the meta-analysis model. Section 3 contains the details of trivariate meta-analysis model, including methods of constructing the prior distributions for the correlations. In Section 4, we briefly describe the implementation of these models in winbugs [18]. We include the results of applying the model to the RA example in Section 5, which is followed by the discussion in Section 6. We describe further technical details, including the description of the multivariate model for any number of outcomes, in the Appendices.