Meta-analysis of genetic studies using Mendelian randomization—a multivariate approach

In traditional epidemiological studies the association between phenotype (risk factor) and disease is often biased by confounding and reverse causation. As a person's genotype is assigned by a seemingly random process, genes are potentially useful instrumental variables for adjusting for such bias. This type of adjustment combines information on the genotype-disease association and the genotype-phenotype association to estimate the phenotype-disease association and has become known as Mendelian randomization. The information on genotype-disease and genotype-phenotype may well come from a meta-analysis. In such a synthesis, a multivariate approach needs to be used whenever some studies provide evidence on both the genotype-phenotype and genotype-disease associations. This paper presents two multivariate meta-analytical models, which differ in their treatment of the heterogeneities (between-study variances). Heterogeneities on the genotype-phenotype and genotype-disease associations may be highly correlated, but a multivariate model that parameterizes the heterogeneity directly is difficult to fit because that correlation is poorly estimated. We advocate an alternative model that treats the heterogeneities on genotype-phenotype and phenotype-disease as being independent. This model fits readily and implicitly defines the correlation between the heterogeneities on genotype-phenotype and genotype-disease. We show how either maximum likelihood or a Bayesian approach with vague prior distributions can be used to fit the alternative model.