We simulated data on an allele score, a phenotype and an outcome, where both the phenotype and the outcome influence selection into the study, but there was no association between the allele score and the outcome in the intended study population (see Figure 2). The simulation scenario was based loosely on the UK Biobank, and we simulated selection into the study, so all the data on non-selected individuals are missing and therefore imputation is not a potential solution (see below), because this requires some data on which to base the imputation.16 All variables were Normally distributed, with a standard deviation of 1, and the sample size of the intended study population was 9 000 000. We assumed that phenotype and outcome had independent effects (i.e. no interaction on the additive scale) on the odds of selection into the sample, and for convenience we set these effects to be equal, and examined a weak association [odds ratio (OR) of 1.2 for missingness for a one standard deviation (SD) increase in phenotype/outcome] and two stronger associations (ORs of 1.5 and 1.8). These
