Open Access Research article

Bivariate random-effects meta-analysis and the estimation of between-study correlation

Richard D Riley1*, Keith R Abrams2, Alexander J Sutton2, Paul C Lambert2 and John R Thompson2

Author Affiliations

1 Centre for Medical Statistics and Health Evaluation, School of Health Sciences, University of Liverpool, Shelley's Cottage, Brownlow Street, Liverpool, L69 3GS, UK

2 Centre for Biostatistics and Genetic Epidemiology, Department of Health Sciences, University of Leicester, 2nd Floor, Adrian Building, University Road, Leicester, LE1 7RH, UK

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BMC Medical Research Methodology 2007, 7:3  doi:10.1186/1471-2288-7-3

Published: 12 January 2007

Abstract

Background

When multiple endpoints are of interest in evidence synthesis, a multivariate meta-analysis can jointly synthesise those endpoints and utilise their correlation. A multivariate random-effects meta-analysis must incorporate and estimate the between-study correlation (ρB).

Methods

In this paper we assess maximum likelihood estimation of a general normal model and a generalised model for bivariate random-effects meta-analysis (BRMA). We consider two applied examples, one involving a diagnostic marker and the other a surrogate outcome. These motivate a simulation study where estimation properties from BRMA are compared with those from two separate univariate random-effects meta-analyses (URMAs), the traditional approach.

Results

The normal BRMA model estimates ρB as -1 in both applied examples. Analytically we show this is due to the maximum likelihood estimator sensibly truncating the between-study covariance matrix on the boundary of its parameter space. Our simulations reveal this commonly occurs when the number of studies is small or the within-study variation is relatively large; it also causes upwardly biased between-study variance estimates, which are inflated to compensate for the restriction on <a onClick="popup('http://www.biomedcentral.com/1471-2288/7/3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2288/7/3/mathml/M1">View MathML</a>B. Importantly, this does not induce any systematic bias in the pooled estimates and produces conservative standard errors and mean-square errors. Furthermore, the normal BRMA is preferable to two normal URMAs; the mean-square error and standard error of pooled estimates is generally smaller in the BRMA, especially given data missing at random. For meta-analysis of proportions we then show that a generalised BRMA model is better still. This correctly uses a binomial rather than normal distribution, and produces better estimates than the normal BRMA and also two generalised URMAs; however the model may sometimes not converge due to difficulties estimating ρB.

Conclusion

A BRMA model offers numerous advantages over separate univariate synthesises; this paper highlights some of these benefits in both a normal and generalised modelling framework, and examines the estimation of between-study correlation to aid practitioners.