Table 1.

The results of performing a simple bidirectional Mendelian randomization analysis for various degrees of pleiotropy.

δ	${\stackrel{\wedge }{\beta }}_{X\to Y}^{\mathit{IV}}$	${\stackrel{\wedge }{\sigma }}_{X\to Y}^{\mathit{IV}}$	${\stackrel{\wedge }{\beta }}_{Y\to X}^{\mathit{IV}}$	${\stackrel{\wedge }{\sigma }}_{Y\to X}^{\mathit{IV}}$	$\stackrel{\wedge }{p}({\mathcal{M}}_{X\to Y}|\mathbf{D})$
−0.5	0.517	0.0346	−0.8715	0.0415	0.167
−0.4	0.615	0.0348	−0.5853	0.0352	0.211
−0.3	0.712	0.0350	−0.3746	0.0305	0.516
−0.2	0.810	0.0353	−0.2130	0.0270	0.632
−0.1	0.908	0.0356	−0.0851	0.0242	0.894
0.0	1.006	0.0359	0.0187	0.0219	0.907
0.1	1.105	0.0362	0.1045	0.0200	0.774
0.2	1.204	0.0366	0.1767	0.0184	0.864
0.3	1.303	0.0370	0.2383	0.0170	0.803
0.4	1.402	0.0374	0.2914	0.0159	0.750
0.5	1.501	0.0378	0.3377	0.0149	0.774

Note: In the first step, we used G_X as an instrument for X to estimate the causal effect ${\beta }_{X\to Y}$ (the correct direction). We then used G_Y as an instrument for Y to estimate the causal effect ${\beta }_{Y\to X}$ (the wrong direction). We compared these estimates against our posterior probability estimate of ${\mathcal{M}}_{X\to Y}$ given the data, in which analysis we used both instruments concomitantly.