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If you are interested in performing multivariate meta-analysis when the within-study covariances are unknown, you may find the following articles helpful:
Riley, R. D., J. R. Thompson, and K. R. Abrams. 2008. An alternative model for bivariate random-effects meta-analysis when the within-study correlations are unknown. Biostatistics 9: 172–186.
Wei, Y., and J. PT Higgins. 2013. Estimating within‐study covariances in multivariate meta‐analysis with multiple outcomes. Statistics in Medicine 32: 1191–1205.
Chen, Y., Y. Cai, C. Hong, and D. Jackson. 2016. Inference for correlated effect sizes using multiple univariate meta‐analyses. Statistics in medicine 35: 1405–1422.
Hong, C., R. D. Riley, and Y. Chen. 2018. An improved method for bivariate meta‐analysis when within‐study correlations are unknown. Research synthesis methods 9: 73–88.
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