  Mendelian randomization uses genetic variants as instrumental variables to
make causal inferences about the effects of modifiable risk factors on diseases
from observational data. One of the major challenges in Mendelian randomization
is that many genetic variants are only modestly or even weakly associated with
the risk factor of interest, a setting known as many weak instruments. Many
existing methods, such as the popular inverse-variance weighted (IVW) method,
could be biased when the instrument strength is weak. To address this issue,
the debiased IVW (dIVW) estimator, which is shown to be robust to many weak
instruments, was recently proposed. However, this estimator still has
non-ignorable bias when the effective sample size is small. In this paper, we
propose a modified debiased IVW (mdIVW) estimator by multiplying a modification
factor to the original dIVW estimator. After this simple correction, we show
that the bias of the mdIVW estimator converges to zero at a faster rate than
that of the dIVW estimator under some regularity conditions. Moreover, the
mdIVW estimator has smaller variance than the dIVW estimator.We further extend
the proposed method to account for the presence of instrumental variable
selection and balanced horizontal pleiotropy. We demonstrate the improvement of
the mdIVW estimator over the dIVW estimator through extensive simulation
studies and real data analysis.
