  Valid estimation of a causal effect using instrumental variables requires
that all of the instruments are independent of the outcome conditional on the
risk factor of interest and any confounders. In Mendelian randomization studies
with large numbers of genetic variants used as instruments, it is unlikely that
this condition will be met. Any given genetic variant could be associated with
a large number of traits, all of which represent potential pathways to the
outcome which bypass the risk factor of interest. Such pleiotropy can be
accounted for using standard multivariable Mendelian randomization with all
possible pleiotropic traits included as covariates. However, the estimator
obtained in this way will be inefficient if some of the covariates do not truly
sit on pleiotropic pathways to the outcome. We present a method which uses
regularization to identify which out of a set of potential covariates need to
be accounted for in a Mendelian randomization analysis in order to produce an
efficient and robust estimator of a causal effect. The method can be used in
the case where individual-level data are not available and the analysis must
rely on summary-level data only. It can also be used in the case where there
are more covariates under consideration than instruments, which is not possible
using standard multivariable Mendelian randomization. We show the results of
simulation studies which demonstrate the performance of the proposed
regularization method in realistic settings. We also illustrate the method in
an applied example which looks at the causal effect of urate plasma
concentration on coronary heart disease.
