This paper was published as Statistics in Medicine, 2005, 24 (14), pp. 2241-2254. It is available from http://www3.interscience.wiley.com/journal/110491700/abstract. Doi: 10.1002/sim.2100Metadata only itemIn 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
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