CONTEXT: Many statistical studies aim to assess the causal effect of a phenotype or exposure, X, on an outcome Y. In many such studies an experimental design is unfeasible, and the only remaining option is to work on the basis of observational data. Unfortunately in this case, no matter how impeccable is the study design, how accurate are the observations and smart the inference algorithm, there is no guarantee that the result will not be biased due to unobserved confounding or reverse causation. A useful approach to this situation is Mendelian randomisation (MR) (Katan 1986, Davey 2003, Lawlor 2008). The bare bones of the idea are that, under certain assumptions, for a phenotype X to be a causal influence on an outcome Y, we expect a genetic variant Z that modulates X to likewise affect Y. Information about Z can then be used as an instrument to assess the causal effect of X on Y, despite confounding. In such a context, the variable Z will often be called an instrumental variable, or briefly, an instrument.
RELEVANCE: The potential impact of MR on science cannot be underestimated (Robinson 2017). In various occasions, an MR study based on observational data has predicted the outcome of a clinical trial, thereby supporting or casting doubt on the motivating causal hypothesis (see, for example, Voight 2012). Furthermore, MR can help biologists reconstruct a disease process from its molecular causes to its phenotypic manifestations, and unravel causal relationships of pharmacological relevance through the analysis of biobank data.
STATE OF THE ART: Recent developments, mostly in the frequentist realm, have focused on methods for multi-instrument MR (MIMR), notably Egger regression (ER) and the weighted median estimator (WME) of Bowden (2016). These methods use multiple instruments, while allowing an unspecified subset of them to affect the outcome directly. In brief, these methods allow for "cryptic" pleiotropy.
MOTIVATION: The existing frequentist approaches to MR do not coherently account for important sources of uncertainty, such as the uncertainty arising from the estimation of the instrument-exposure (i-e) associations. Hence our concern that these methods may yield over-optimistic results. We attempt to remedy this by proposing a Bayesian approach to MR (Burgess 2010; Didelez 2007; Jones 2012) which deals with cryptic pleiotropy, and can in principle acknowledge uncertainty at all levels of the model.
PROPOSED BAYESIAN FRAMEWORK FOR MR: Our method allows the user to shape the prior of the entertained MR model on the basis of external information, for stronger and more accurate inferences, although it will work with vague prior specifications, too. The method combines the power of Bayesian analysis with that of Markov chain Monte Carlo (MCMC) inference, for an exceptional freedom in elaborating the basic model. At the current point in time, this repository contains methods that do not deal with non-linear dependencies and model uncertainty, and assume that the exposure variable X and and the noutcome variable Y are continuous.
The current version of the proposed method relaxes the exclusion-restriction assumption, thereby allowing an instrument to affect the outcome both indirectly (via the exposure) and through a direct (pleiotropic) effect, provided no instrument is correlated with the exposure-outcome confounders. The method also allows the instruments to be correlated, for an increased power to detect the causal effect of interest. By having the pleiotropic effects represented in the model by unknown parameters, and by imposing a Bayesian shrinkage (Carvalho et al, 2010) prior distribution that assumes an unspecified subset of these parameters to be zero, we obtain a proper posterior distribution for the causal effect of interest. This posterior we sample via Markov chain Monte Carlo to obtain point and interval estimates. The model priors require a minimal input from the user.
An upcoming article on Biostatistics, by Carlo Berzuini, Hui Guo, Stephen Burgess and Luisa Bernardinelli, describes the method and reports about a simulation experiment that assesses its performance in terms of type-1 error and power in various situations, including the presence of correlated instruments and instrument weakness.
AIMS OF BMR PROJECT: The aim of the present project is to collaboorate in developing Mendelian randomisation methods for situations of practical or scientific interest, within a Bayesian modelling framework. Possible directions of development are incorporating multiple types of external information in the prior, handling multiple sources of uncertainty, and elaborating the basic model to tackle a new or extended class of problems, within the field of genomic epidemiology or other related areas.
R: We assume the availability of the free software environment for statistical computing and graphics called R. R compiles and runs on a wide variety of UNIX platforms, Windows and MacOS. To download R, please access https://www.r-project.org and choose your preferred CRAN mirror.
RSTUDIO: We assume the availability of the RStudio software, that allows the user to run R in a more user-friendly environment. Rstudio is open-source (which means free), and available at http://www.rstudio.com/
SWEAVE: R includes a powerful and flexible system called Sweave for creating dynamic reports and reproducible research using LaTeX. Sweave enables the embedding of R code within a LaTeX document to generate a PDF file that includes narrative and analysis, graphics, code, and the results of computations. Find detailed information about the use of Sweave within RStudio on https://support.rstudio.com/hc/en-us/articles/200552056-Using-Sweave-and-knitr In order to run Sweave, you will need a LaTeX installation.
STAN: We finally assume the availability of the probabilistic modelling language STAN. For links to up-to-date code, examples, manuals, bug reports, feature requests, and everything else related to STAN, see the STAN home page: http://mc-stan.org/. Our current implementation uses the R interface to STAN called Rstan. The Rstan home page is http://mc-stan.org/rstan.html.
THREE FILES ARE CURRENTLY INCLUDED IN THIS REPOSITORY:
FILE 1: The file SweaveDocument contains a basic Sweave template. The document contains both LaTeX text sections and chunks of R code, interweaved with one another. Running this from within Rstudio will generate a PDF file that includes narrative and analysis, graphics, code, and the results of computations. You can use the document as a basis for developing more elaborate analyses, by altering both the LaTeX narrative and the R code. Currently, the document contains two functions R that perform an essential role in the implementation of a Mendelian Randomisation analysis, and an illustrative implementation of an analysis where a dataset is simulated from a basic Mendelian Randomisation model, in accord with modifiable parameter values, and then analysed by the method discussed in the above mentioned paper. When using Rstudio, you can insert an R chunk via the Chunks menu at the top right of the Rstudio source editor. Once the Sweave document is ready, you can run it from within Rstudio to generate the output PDF file.
FILE 2: The file LaTeXDocument contains the LaTeX output from running SweaveDocument.
FILE 3: The file PDFDocument has been generated by compiling LaTeXDocument.
No external data are provided.
REFERENCES
Berzuini, Dawid and Bernardinelli (eds.) (2012). Causality: Statistical Perspectives and Applications. J. Wiley and Sons.
Bowden, J., Davey~Smith, G., Haycock, P.C. and Burgess, S. (2016). Consistent Estimation in Mendelian Randomization with Some Invalid Instruments Using a Weighted Median Estimator. Genetic Epidemiology. 40, 304-314.
Burgess, S. and Thompson, S.G.} (2010). Bayesian methods for meta-analysis of causal relationships estimated using genetic instrumental variables. Statistics in Medicine, 29, 1298-1311.
Carvalho, C.M., Polson, N.G. and Scott, J.G. (2010). The horseshoe estimator for sparse signals. Biometrika, asq017.
Davey Smith, G. and Ebrahim, S. (2003). Mendelian Randomization: Can Genetic Epidemiology Contribute to Understanding Environmental Determinants of Disease? International Journal of Epidemiology. 32}, 1-22.
Didelez, V. and Sheehan, N.A. (2007). Mendelian Randomisation as an Instrumental Variable Approach to Causal Inference. Statistical Methods in Medical Research. 16, 309-330.
Egger, M., Smith, G.D., Schneider, M. and Minder, C. (1997). Bias in meta-analysis detected by a simple, graphical test. British Medical Journal. 315}(7109), 629-634.
Jones, E.M., Thompson, J.R., Didelez, V. and Sheehan, N.A. (2012). On the choice of parameterisation and priors for the Bayesian analyses of Mendelian randomisation studies. Statistics in Medicine, 31}(14), 1483-1501.
Katan, M.B. (1986). Apoupoprotein E-isoforms, serum cholesterol, and cancer. The Lancet. 327(8479), 507-508.
Lawlor, D.A, Harbord, R.M., Sterne, J.A., Timpson, N. and Davey~Smith, G. (2008). Mendelian randomization: using genes as instruments for making causal inferences in epidemiology. Statistics in Medicine, 27}(8), 1133-1163.
Robinson, P.C., Choi, H.K., Do, R. and Merriman, T.R. (2017). Insight into rheumatological cause and effect through the use of Mendelian randomization. Nature Reviews Rheumatology, 13(193), 486-496.
STAN Development Team (2014). STAN: A C++ library for probability and sampling, version 2.2. http://mc-stan.org/.
Voight, B.F., Ardissino, D. and colleagues. (2012). Plasma HDL cholesterol and risk of myocardial infarction: a Mendelian Randomisation study. The Lancet, 380(9841), 572-580.