Bayesian methods for meta‐analysis of causal relationships estimated using genetic instrumental variables

Stephen Burgess, Simon G Thompson, S Burgess, S G Thompson, G Andrews, N J Samani, A Hall, P Whincup, R Morris, D A Lawlor, G Davey Smith, N Timpson, S Ebrahim, Y Ben-Shlomo, G Davey Smith, N Timpson, M Brown, S Ricketts, M Sandhu, A Reiner, B Psaty, L Lange, M Cushman, J Hung, P Thompson, J Beilby, N Warrington, L J Palmer, B G Nordestgaard, A Tybjaerg-Hansen, J Zacho, C Wu, G Lowe, I Tzoulaki, M Kumari, M Sandhu, J F Yamamoto, B Chiodini, M Franzosi, G J Hankey, K Jamrozik, L Palmer, E Rimm, J Pai, B Psaty, S Heckbert, J Bis, S Anand, J Engert, R Collins, R Clarke, O Melander, G Berglund, P Ladenvall, L Johansson, J-H Jansson, G Hallmans, A Hingorani, S Humphries, E Rimm, J Manson, J Pai, H Watkins, R Clarke, J Hopewell, D Saleheen, R Frossard, J Danesh, N Sattar, M Robertson, J Shepherd, E Schaefer, A Hofman, J C M Witteman, I Kardys, Y Ben-Shlomo, G Davey Smith, N Timpson, U de Faire, A Bennet, N Sattar, I Ford, C Packard, M Kumari, J Manson, Debbie A Lawlor, George Davey Smith, S Anand, R Collins, J P Casas, J Danesh, G Davey Smith, M Franzosi, A Hingorani, D A Lawlor, J Manson, B G Nordestgaard, N J Samani, M Sandhu, L Smeeth, F Wensley, S Anand, J Bowden, S Burgess, J P Casas, E Di Angelantonio, J Engert, P Gao, T Shah, L Smeeth, S G Thompson, C Verzilli, M Walker, J Whittaker, A Hingorani, J Danesh

Genetic markers can be used as instrumental variables, in an analogous way to randomization in a clinical trial, to estimate the causal relationship between a phenotype and an outcome variable. Our purpose is to extend the existing methods for such Mendelian randomization studies to the context of multiple genetic markers measured in multiple studies, based on the analysis of individual participant data. First, for a single genetic marker in one study, we show that the usual ratio of coefficients approach can be reformulated as a regression with heterogeneous error in the explanatory variable. This can be implemented using a Bayesian approach, which is next extended to include multiple genetic markers. We then propose a hierarchical model for undertaking a meta-analysis of multiple studies, in which it is not necessary that the same genetic markers are measured in each study. This provides an overall estimate of the causal relationship between the phenotype and the outcome, and an assessment of its heterogeneity across studies. As an example, we estimate the causal relationship of blood concentrations of C-reactive protein on fibrinogen levels using data from 11 studies. These methods provide a flexible framework for efficient estimation of causal relationships derived from multiple studies. Issues discussed include weak instrument bias, analysis of binary outcome data such as disease risk, missing genetic data, and the use of haplotypes.