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 odds ratios are similar to estimates of the likelihood of participation in UK Biobank for individuals with any educational or vocational qualifications and for non–smokers, respectively (see Box 2), and indicate a difference in mean phenotype/outcome of 0.2 SD, 0.4 SD and 0.6 SD between those participating and those not participating. We varied the correlation between the allele score and the phenotype (between r = 0.05 and r = 0.30) to simulate genetic instruments explaining between 0.25% and 9% of the variance in phenotypes. These values are in the typical range for the association between common genetic variants, or polygenic risk scores comprising multiple common variants, and complex phenotypes. For example, the rs16969968 variant accounts for approximately 1% of the phenotypic variance in cigarette consumption,17 whereas the polygenic risk score for height captures approximately 9% of phenotypic variance.18 We controlled the baseline risk of selection into the sample, resulting in a selected sample of approximately 500 000 people. The analysis
