  Mendelian randomization is the use of genetic variants as instrumental
variables to assess whether a risk factor is a cause of a disease outcome.
Increasingly, Mendelian randomization investigations are conducted on the basis
of summarized data, rather than individual-level data. These summarized data
comprise the coefficients and standard errors from univariate regression models
of the risk factor on each genetic variant, and of the outcome on each genetic
variant. A causal estimate can be derived from these associations for each
individual genetic variant, and a combined estimate can be obtained by
inverse-variance weighted meta-analysis of these causal estimates. Various
proposals have been made for how to calculate this inverse-variance weighted
estimate. In this paper, we show that the inverse-variance weighted method as
originally proposed (equivalent to a two-stage least squares or allele score
analysis using individual-level data) can lead to over-rejection of the null,
particularly when there is heterogeneity between the causal estimates from
different genetic variants. Random-effects models should be routinely employed
to allow for this possible heterogeneity. Additionally, over-rejection of the
null is observed when associations with the risk factor and the outcome are
obtained in overlapping participants. The use of weights including second-order
terms from the delta method is recommended in this case.
