From: Overlapping-sample Mendelian randomisation with multiple exposures: a Bayesian approach
 |  |  | \(\hat {\beta }_{1}\) | \(\hat {\beta }_{2}\) | \(\hat {\beta }_{3}\) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
overlap | α | δ | Bayeisan | Classic | Bayeisan | Classic | Bayesian | Classic | ||||||||||||
 |  |  | mean | sd | coverage | mean | sd | coverage | mean | sd | coverage | mean | sd | coverage | mean | sd | coverage | mean | sd | coverage |
100% | 0.5 | 1 | 0.000 | 0.001 | 0.98 | 0.000 | 0.015 | 0.94 | 0.000 | 0.002 | 0.99 | -0.002 | 0.020 | 0.95 | 0.000 | 0.001 | 0.99 | 0.001 | 0.014 | 0.97 |
 |  | 0.5 | 0.000 | 0.003 | 0.98 | 0.000 | 0.014 | 0.97 | 0.001 | 0.005 | 0.96 | 0.002 | 0.020 | 0.95 | 0.000 | 0.003 | 0.99 | -0.001 | 0.014 | 0.95 |
 |  | 0.1 | 0.001 | 0.006 | 0.97 | 0.000 | 0.014 | 0.95 | 0.001 | 0.010 | 0.94 | 0.000 | 0.021 | 0.96 | 0.001 | 0.006 | 0.99 | 0.000 | 0.015 | 0.96 |
 | 0.1 | 1 | 0.002 | 0.006 | 0.98 | -0.009 | 0.072 | 0.95 | 0.003 | 0.010 | 0.98 | 0.005 | 0.105 | 0.95 | 0.000 | 0.006 | 0.98 | -0.005 | 0.074 | 0.94 |
 |  | 0.5 | 0.007 | 0.009 | 0.94 | 0.004 | 0.073 | 0.95 | 0.011 | 0.015 | 0.93 | 0.000 | 0.114 | 0.91 | -0.001 | 0.009 | 0.99 | -0.002 | 0.076 | 0.95 |
 |  | 0.1 | 0.019 | 0.029 | 0.95 | -0.001 | 0.069 | 0.97 | 0.024 | 0.050 | 0.95 | 0.004 | 0.101 | 0.94 | 0.000 | 0.029 | 0.98 | -0.003 | 0.075 | 0.94 |
80% | 0.5 | 1 | 0.005 | 0.012 | 0.97 | -0.001 | 0.047 | 0.97 | 0.005 | 0.011 | 0.98 | 0.000 | 0.063 | 0.95 | 0.001 | 0.006 | 0.99 | -0.002 | 0.044 | 0.95 |
 |  | 0.5 | 0.003 | 0.010 | 0.98 | 0.003 | 0.048 | 0.94 | 0.004 | 0.011 | 0.98 | -0.002 | 0.067 | 0.95 | 0.000 | 0.007 | 0.97 | 0.002 | 0.050 | 0.95 |
 |  | 0.1 | 0.001 | 0.009 | 0.98 | 0.004 | 0.047 | 0.96 | 0.002 | 0.016 | 0.97 | 0.003 | 0.060 | 0.97 | 0.000 | 0.010 | 0.98 | 0.000 | 0.043 | 0.97 |
 | 0.1 | 1 | 0.028 | 0.026 | 0.95 | -0.008 | 0.248 | 0.96 | 0.027 | 0.042 | 0.96 | 0.002 | 0.372 | 0.96 | 0.002 | 0.033 | 0.99 | -0.018 | 0.263 | 0.94 |
 |  | 0.5 | 0.029 | 0.030 | 0.94 | -0.035 | 0.261 | 0.94 | 0.032 | 0.049 | 0.95 | 0.000 | 0.336 | 0.96 | 0.002 | 0.036 | 0.98 | 0.011 | 0.246 | 0.96 |
 |  | 0.1 | 0.024 | 0.049 | 0.97 | -0.002 | 0.254 | 0.94 | 0.025 | 0.077 | 0.96 | 0.003 | 0.376 | 0.92 | 0.001 | 0.049 | 1 | -0.020 | 0.249 | 0.96 |
60% | 0.5 | 1 | 0.004 | 0.011 | 0.97 | 0.001 | 0.034 | 0.95 | 0.004 | 0.015 | 0.98 | 0.005 | 0.045 | 0.96 | 0.000 | 0.009 | 0.98 | -0.003 | 0.035 | 0.94 |
 |  | 0.5 | 0.002 | 0.010 | 0.96 | 0.003 | 0.034 | 0.96 | 0.002 | 0.014 | 0.98 | 0.000 | 0.043 | 0.96 | 0.000 | 0.010 | 0.99 | 0.001 | 0.033 | 0.95 |
 |  | 0.1 | 0.000 | 0.010 | 0.99 | 0.001 | 0.036 | 0.91 | -0.001 | 0.017 | 0.97 | -0.002 | 0.048 | 0.94 | 0.001 | 0.012 | 0.97 | 0.000 | 0.030 | 0.97 |
 | 0.1 | 1 | 0.034 | 0.037 | 0.92 | -0.008 | 0.160 | 0.96 | 0.032 | 0.057 | 0.97 | 0.019 | 0.258 | 0.93 | -0.001 | 0.045 | 0.99 | -0.015 | 0.185 | 0.97 |
 |  | 0.5 | 0.033 | 0.039 | 0.95 | -0.003 | 0.173 | 0.94 | 0.030 | 0.061 | 0.96 | 0.002 | 0.252 | 0.93 | -0.001 | 0.049 | 0.97 | -0.009 | 0.169 | 0.95 |
 |  | 0.1 | 0.010 | 0.061 | 0.95 | 0.002 | 0.179 | 0.94 | 0.009 | 0.089 | 0.94 | 0.021 | 0.223 | 0.96 | 0.002 | 0.059 | 0.95 | -0.022 | 0.162 | 0.95 |
40% | 0.5 | 1 | 0.001 | 0.014 | 0.96 | 0.000 | 0.026 | 0.95 | 0.003 | 0.017 | 0.97 | 0.001 | 0.036 | 0.97 | 0.000 | 0.011 | 0.98 | -0.001 | 0.025 | 0.96 |
 |  | 0.5 | 0.002 | 0.011 | 0.97 | -0.003 | 0.029 | 0.93 | 0.002 | 0.017 | 0.97 | 0.001 | 0.036 | 0.97 | 0.000 | 0.012 | 0.95 | 0.001 | 0.026 | 0.95 |
 |  | 0.1 | 0.001 | 0.013 | 0.99 | 0.000 | 0.027 | 0.95 | 0.000 | 0.019 | 0.95 | 0.006 | 0.036 | 0.97 | 0.000 | 0.012 | 0.96 | -0.004 | 0.028 | 0.94 |
 | 0.1 | 1 | 0.013 | 0.045 | 0.96 | -0.021 | 0.135 | 0.95 | 0.019 | 0.077 | 0.96 | -0.010 | 0.204 | 0.92 | -0.001 | 0.060 | 0.96 | -0.003 | 0.152 | 0.97 |
 |  | 0.5 | 0.013 | 0.051 | 0.98 | 0.002 | 0.131 | 0.96 | 0.013 | 0.078 | 0.97 | 0.009 | 0.215 | 0.94 | -0.001 | 0.059 | 0.96 | -0.016 | 0.150 | 0.95 |
 |  | 0.1 | 0.005 | 0.064 | 0.95 | 0.008 | 0.130 | 0.97 | -0.002 | 0.082 | 0.98 | -0.004 | 0.190 | 0.94 | 0.000 | 0.061 | 0.97 | 0.003 | 0.141 | 0.93 |
20% | 0.5 | 1 | 0.002 | 0.014 | 0.95 | 0.002 | 0.021 | 0.96 | 0.004 | 0.019 | 0.95 | 0.001 | 0.033 | 0.94 | -0.001 | 0.012 | 0.99 | 0.000 | 0.023 | 0.96 |
 |  | 0.5 | -0.001 | 0.015 | 0.92 | -0.002 | 0.023 | 0.95 | 0.001 | 0.020 | 0.97 | 0.002 | 0.032 | 0.98 | 0.001 | 0.015 | 0.95 | 0.000 | 0.023 | 0.96 |
 |  | 0.1 | 0.000 | 0.014 | 0.98 | 0.001 | 0.022 | 0.98 | 0.001 | 0.019 | 0.97 | -0.002 | 0.033 | 0.94 | -0.001 | 0.014 | 0.96 | 0.002 | 0.023 | 0.95 |
 | 0.1 | 1 | 0.012 | 0.057 | 0.96 | -0.018 | 0.106 | 0.97 | -0.002 | 0.083 | 0.96 | 0.000 | 0.158 | 0.97 | 0.002 | 0.064 | 0.96 | -0.001 | 0.116 | 0.97 |
 |  | 0.5 | 0.010 | 0.061 | 0.98 | -0.004 | 0.115 | 0.95 | 0.007 | 0.091 | 0.97 | -0.010 | 0.157 | 0.95 | 0.000 | 0.068 | 0.96 | 0.001 | 0.114 | 0.96 |
 |  | 0.1 | 0.013 | 0.069 | 0.95 | 0.007 | 0.111 | 0.97 | 0.005 | 0.092 | 0.98 | 0.003 | 0.160 | 0.97 | 0.007 | 0.066 | 0.98 | -0.006 | 0.117 | 0.94 |
0% | 0.5 | 1 | 0.003 | 0.014 | 0.96 | 0.001 | 0.020 | 0.97 | 0.001 | 0.020 | 0.95 | 0.000 | 0.028 | 0.95 | 0.000 | 0.014 | 0.96 | 0.002 | 0.021 | 0.94 |
 |  | 0.5 | 0.002 | 0.013 | 0.97 | -0.001 | 0.021 | 0.96 | 0.001 | 0.023 | 0.93 | -0.002 | 0.029 | 0.95 | 0.000 | 0.014 | 0.96 | 0.000 | 0.021 | 0.94 |
 |  | 0.1 | 0.001 | 0.015 | 0.96 | 0.001 | 0.022 | 0.91 | 0.000 | 0.022 | 0.94 | -0.002 | 0.029 | 0.95 | 0.001 | 0.014 | 0.95 | 0.001 | 0.022 | 0.93 |
 | 0.1 | 1 | 0.011 | 0.080 | 0.92 | -0.005 | 0.100 | 0.97 | -0.008 | 0.109 | 0.93 | 0.003 | 0.158 | 0.93 | 0.008 | 0.075 | 0.96 | 0.000 | 0.108 | 0.96 |
 |  | 0.5 | 0.009 | 0.073 | 0.95 | -0.003 | 0.096 | 0.96 | -0.005 | 0.102 | 0.94 | -0.003 | 0.169 | 0.92 | 0.005 | 0.071 | 0.98 | 0.006 | 0.121 | 0.91 |
 |  | 0.1 | 0.010 | 0.072 | 0.96 | -0.004 | 0.106 | 0.96 | -0.003 | 0.105 | 0.97 | -0.012 | 0.149 | 0.95 | 0.000 | 0.077 | 0.96 | 0.001 | 0.106 | 0.94 |