Table 3.
Simulation 3 with binary outcome to validate bias and type 1 error rate formulae
| Mean observational | Mean IV | Relative | Empirical | Expected | ||||
|---|---|---|---|---|---|---|---|---|
| α | Mean F | Mean R 2 | estimate | estimate | bias | (Mean F)−1 | Type 1 | Type 1 |
| Risk factor measurements taken in controls only | ||||||||
| 0.01 | 1.1 | 0.4% | 0.481 | 0.001 | 0.002 | ‐ | 4.9% | ‐ |
| 0.02 | 1.4 | 0.6% | 0.481 | 0.000 | 0.000 | ‐ | 5.2% | ‐ |
| 0.03 | 2.0 | 0.8% | 0.479 | −0.003 | 0.007 | ‐ | 4.8% | ‐ |
| 0.04 | 2.7 | 1.1% | 0.478 | −0.001 | −0.001 | ‐ | 5.0% | ‐ |
| 0.05 | 3.7 | 1.5% | 0.476 | 0.000 | 0.000 | ‐ | 5.2% | ‐ |
| 0.08 | 7.9 | 3.1% | 0.469 | 0.000 | 0.001 | ‐ | 4.7% | ‐ |
| Risk factor measurements taken in all participants | ||||||||
| 0.01 | 1.2 | 0.3% | 0.481 | 0.360 | 0.748 | 0.837 | 24.4% | 29.1% |
| 0.02 | 1.8 | 0.4% | 0.481 | 0.237 | 0.493 | 0.561 | 17.4% | 21.1% |
| 0.03 | 2.8 | 0.5% | 0.479 | 0.149 | 0.311 | 0.363 | 12.8% | 15.2% |
| 0.04 | 4.1 | 0.8% | 0.478 | 0.099 | 0.207 | 0.242 | 10.0% | 11.7% |
| 0.05 | 5.9 | 1.2% | 0.476 | 0.068 | 0.142 | 0.170 | 8.4% | 9.6% |
| 0.08 | 13.6 | 2.6% | 0.469 | 0.030 | 0.064 | 0.074 | 6.3% | 6.9% |
Notes: Mean instrumental variable (IV) estimates and empirical Type 1 error rate (5% nominal significance level) from inverse‐variance weighted method with binary outcome for null causal effect () and six values of genetic associations with the risk factor (α) in a case‐control setting, with the risk factor measurements taken in control participants only and with the risk factor measurements taken in all participants. Observational estimates are log odds ratios from logistic regression of the outcome on the risk factor, and IV estimates are log odds ratios calculated using logistic regression for the IV–outcome association and linear regression for the IV–risk factor association.