  Selection bias is a common concern in epidemiologic studies. In the
literature, selection bias is often viewed as a missing data problem. Popular
approaches to adjust for bias due to missing data, such as inverse probability
weighting, rely on the assumption that data are missing at random and can yield
biased results if this assumption is violated. In observational studies with
outcome data missing not at random, Heckman's sample selection model can be
used to adjust for bias due to missing data. In this paper, we review Heckman's
method and a similar approach proposed by Tchetgen Tchetgen and Wirth (2017).
We then discuss how to apply these methods to Mendelian randomization analyses
using individual-level data, with missing data for either the exposure or
outcome or both. We explore whether genetic variants associated with
participation can be used as instruments for selection. We then describe how to
obtain missingness-adjusted Wald ratio, two-stage least squares and inverse
variance weighted estimates. The two methods are evaluated and compared in
simulations, with results suggesting that they can both mitigate selection bias
but may yield parameter estimates with large standard errors in some settings.
In an illustrative real-data application, we investigate the effects of body
mass index on smoking using data from the Avon Longitudinal Study of Parents
and Children.
