  Multivariable Mendelian randomization estimates the causal effect of multiple
exposures on an outcome, typically using summary statistics of genetic variant
associations. However, exposures of interest in Mendelian randomization
applications will often be measured with error. The summary statistics will
therefore not be of the genetic associations with the exposure, but with the
exposure measured with error. Classical measurement error will not bias genetic
association estimates but will increase their standard errors. With a single
exposure, this will result in bias toward the null in a two sample framework.
However, this will not necessarily be the case with multiple correlated
exposures. In this paper, we examine how the direction and size of bias, as
well as coverage, power and type I error rates in multivariable Mendelian
randomization studies are affected by measurement error on exposures. We show
how measurement error can be accounted for in a maximum likelihood framework.
We consider two applied examples. In the first, we show that measurement error
leads to the effect of body mass index on coronary heart disease risk to be
overestimated, and that of waist-to-hip ratio to be underestimated. In the
second, we show that the proportion of the effect of education on coronary
heart disease risk which is mediated by body mass index, smoking and blood
pressure may be underestimated if measurement error is not taken into account.
