Profile-likelihood Bayesian model averaging for two-sample summary data Mendelian randomization in the presence of horizontal pleiotropy

Two-sample summary data Mendelian randomisation (MR) is a popular method for assessing causality in epidemiology, by using genetic variants as instrumental variables. If genes exert pleiotropic effects on the outcome not through the exposure of interest, this can lead to heterogeneous and (potentially) biased estimates of causal effect. We investigate the use of Bayesian model averaging (BMA) to preferentially search the space of models with the highest posterior likelihood. We develop a bespoke Metropolis-Hasting algorithm to perform the search using the recently developed Robust Adjusted Profile Likelihood (MR-RAPS) of Zhao et al as the basis for defining a posterior distribution that efficiently accounts for pleiotropic and weak instrument bias. We demonstrate how our general modelling approach can be extended from a standard one-parameter causal model to a two-parameter model, to allow a large proportion of SNPs to violate the Instrument Strength Independent of Direct Effect (InSIDE) assumption. We use Monte Carlo simulations to illustrate our methods and compare it to several related approaches. We finish by applying our approach in practice to investigate the changes in causal effect of their resulting high risk metabolite on the development age-related macular degeneration.

LIKELIHOOD INFERENCE IN AN AUTOREGRESSION WITH FIXED EFFECTS

We calculate the bias of the profile score for the regression coefficients in a multistratum autoregressive model with stratum-specific intercepts. The bias is free of incidental parameters. Centering the profile score delivers an unbiased estimating equation and, upon integration, an adjusted profile likelihood. A variety of other approaches to constructing modified profile likelihoods are shown to yield equivalent results. However, the global maximizer of the adjusted likelihood lies at infinity for any sample size, and the adjusted profile score has multiple zeros. Consistent parameter estimates are obtained as local maximizers inside or on an ellipsoid centered at the maximum likelihood estimator.