## Introduction

It is common for randomized controlled trials (RCTs) to report more than one outcome. For purposes of designing a trial, it is generally felt that a single outcome should be prespecified to be the “primary” outcome. But, it is also recognized that, when pooling results from trials in a meta-analysis, there are several reasons why it may be appropriate to combine information on different outcomes. First, one might wish to “gather strength” by combining several similar outcomes, or to be able to combine results from trials that report different, but similar, outcomes [1,2]. Second, in a decision-making context, the different outcomes recorded may each have separate implications for estimating quality of life or economic consequences of each treatment.

A key requirement in the synthesis of multiple outcomes is that the correlation structures are appropriately represented [1–4]. In a meta-analysis, the correlations may occur at either or both of two levels. At the between-patient within-trial level, a patient's outcome on one measure may be positively or negatively correlated with their outcome on another. At the between-trial level, trials in which there is a larger treatment effect on one measure may tend to be the trials on which there is a larger treatment effect on another (a positive correlation), or possibly a smaller one (a negative correlation).

Competing risk outcomes represent a special type of multiple outcome structure in which there are several different failure time outcomes that are considered mutually exclusive. Once a patient has reached any one of these end points, they are considered to be out of the risk set. Censoring may also be occurring. When results from these trials are pooled in a meta-analysis, the competing risk structure should be taken into account so that the statistical dependencies between outcomes are correctly reflected in the analysis. These dependencies are essentially within-trial, negative correlations between outcomes, applying in each arm of each trial. They arise because the occurrence of outcome events is a stochastic process, and if more patients should by chance reach one outcome, then fewer must reach the others. The importance of these correlations in the context of meta-analysis of competing risk outcomes has been recognized by Trikalinos and Olkin [5], who suggest an approach based on normal approximation of the variances and covariances arising from multinomial data, and illustrate it with an application to a two-treatment meta-analysis with two competing outcomes.

In this article, we present an alternative approach based on hazards rather than probabilities, to more appropriately take account of time at risk. We use Bayesian Markov Chain Monte Carlo (MCMC) estimation [6], which we believe is more flexible in a situation with large numbers of treatments and outcomes. We begin by describing the data set, and we then explain the “proportional competing risks” assumption that underlies our approach. We next propose three alternative models: one fixed effects, and two random effects analyses. We also suggest some methods for model selection. In our discussion of the results, we compare our proposed method to previously described approaches, and consider some possible extensions.