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Table 1 Causal effects estimated from simulated data using our Bayesian method and classic methods (2SLS when sample overlap = 100% and IVW otherwise) when β=(β1,β2,β3)=0.3

From: Overlapping-sample Mendelian randomisation with multiple exposures: a Bayesian approach

    \(\hat {\beta }_{1}\) \(\hat {\beta }_{2}\) \(\hat {\beta }_{3}\)
overlap α δ Bayesian Classic Bayesian Classic Bayesian Classic
    mean sd coverage power mean sd coverage power mean sd coverage power mean sd coverage power mean sd coverage power mean sd coverage power
100% 0.5 1 0.300 0.001 0.96 1 0.301 0.020 0.96 1 0.300 0.002 0.97 1 0.298 0.025 0.99 1 0.300 0.001 0.99 1 0.301 0.019 0.99 1
   0.5 0.301 0.006 0.96 1 0.300 0.017 0.98 1 0.301 0.004 0.99 1 0.302 0.024 0.97 1 0.300 0.002 0.99 1 0.299 0.018 0.97 1
   0.1 0.301 0.006 0.96 1 0.300 0.015 0.95 1 0.301 0.009 0.98 1 0.300 0.022 0.97 1 0.300 0.006 0.95 1 0.300 0.016 0.96 1
  0.1 1 0.302 0.006 0.97 1 0.286 0.100 0.98 0.72 0.303 0.010 0.97 1 0.309 0.141 0.98 0.47 0.300 0.006 0.98 1 0.292 0.106 0.97 0.76
   0.5 0.306 0.009 0.95 1 0.302 0.087 0.96 0.92 0.309 0.015 0.98 1 0.305 0.130 0.96 0.65 0.301 0.009 0.99 1 0.294 0.092 0.98 0.86
   0.1 0.319 0.029 0.95 1 0.301 0.071 0.97 0.97 0.324 0.050 0.94 1 0.304 0.104 0.97 0.79 0.300 0.029 0.98 1 0.296 0.079 0.95 0.96
80% 0.5 1 0.304 0.020 0.93 1 0.297 0.108 0.95 0.78 0.304 0.022 0.95 1 0.311 0.117 0.99 0.66 0.300 0.013 0.97 1 0.294 0.100 0.96 0.83
   0.5 0.302 0.010 0.96 1 0.302 0.099 0.97 0.87 0.302 0.015 0.96 1 0.300 0.101 1 0.71 0.299 0.011 0.96 1 0.306 0.098 0.97 0.89
   0.1 0.302 0.011 0.95 1 0.307 0.098 0.92 0.91 0.300 0.016 0.97 1 0.311 0.092 0.97 0.85 0.301 0.011 0.96 1 0.298 0.093 0.95 0.90
  0.1 1 0.306 0.048 0.98 1 0.319 0.393 0.96 0.11 0.307 0.072 0.98 0.98 0.317 0.577 0.96 0.11 0.302 0.060 0.97 1 0.284 0.406 0.94 0.12
   0.5 0.320 0.041 0.98 1 0.276 0.335 0.95 0.17 0.320 0.060 0.98 1 0.301 0.461 0.97 0.12 0.303 0.047 1 1 0.320 0.355 0.96 0.15
   0.1 0.320 0.050 0.97 1 0.296 0.296 0.94 0.22 0.320 0.075 0.97 1 0.310 0.437 0.93 0.14 0.301 0.052 0.98 1 0.295 0.301 0.94 0.18
60% 0.5 1 0.302 0.020 0.95 1 0.295 0.078 0.94 0.97 0.299 0.027 0.95 1 0.310 0.081 0.99 0.93 0.302 0.018 0.94 1 0.295 0.073 0.94 0.99
   0.5 0.299 0.014 0.97 1 0.305 0.068 0.95 1 0.301 0.018 0.96 1 0.303 0.070 1 0.99 0.301 0.013 0.96 1 0.298 0.069 0.96 1
   0.1 0.300 0.012 0.98 1 0.304 0.060 0.96 1 0.303 0.018 0.96 1 0.301 0.070 0.98 0.98 0.298 0.012 0.95 1 0.298 0.065 0.96 1
  0.1 1 0.283 0.066 0.96 0.98 0.304 0.283 0.94 0.22 0.285 0.106 0.95 0.77 0.336 0.376 0.96 0.14 0.307 0.086 0.95 0.97 0.299 0.298 0.95 0.20
   0.5 0.300 0.057 0.97 1 0.304 0.236 0.92 0.27 0.307 0.085 0.97 0.93 0.324 0.319 0.95 0.17 0.304 0.066 0.98 1 0.287 0.229 0.96 0.26
   0.1 0.301 0.061 0.97 1 0.306 0.203 0.94 0.41 0.303 0.085 0.97 0.95 0.318 0.246 0.97 0.20 0.300 0.059 0.96 1 0.280 0.183 0.94 0.32
40% 0.5 1 0.300 0.025 0.91 1 0.299 0.060 0.96 1 0.298 0.033 0.91 1 0.304 0.059 1 1 0.301 0.021 0.93 1 0.298 0.055 0.96 1
   0.5 0.301 0.017 0.93 1 0.297 0.058 0.94 1 0.300 0.023 0.95 1 0.303 0.055 0.99 1 0.298 0.015 0.94 1 0.297 0.057 0.94 1
   0.1 0.300 0.013 0.97 1 0.300 0.049 0.97 1 0.300 0.019 0.95 1 0.308 0.055 0.99 1 0.300 0.014 0.95 1 0.296 0.058 0.93 1
  0.1 1 0.270 0.085 0.94 0.90 0.265 0.200 0.97 0.24 0.277 0.143 0.97 0.57 0.298 0.323 0.94 0.18 0.301 0.111 0.92 0.82 0.291 0.237 0.95 0.29
   0.5 0.292 0.073 0.95 0.99 0.299 0.174 0.96 0.38 0.296 0.112 0.94 0.74 0.321 0.276 0.95 0.27 0.298 0.085 0.94 0.95 0.273 0.192 0.95 0.37
   0.1 0.301 0.063 0.98 1 0.313 0.154 0.97 0.55 0.296 0.097 0.98 0.84 0.302 0.213 0.94 0.27 0.301 0.067 0.97 1 0.307 0.155 0.96 0.51
20% 0.5 1 0.306 0.028 0.86 1 0.304 0.045 0.99 1 0.297 0.040 0.86 1 0.301 0.058 1 1 0.301 0.025 0.91 1 0.301 0.051 0.95 1
   0.5 0.301 0.018 0.96 1 0.298 0.050 0.95 1 0.301 0.029 0.93 1 0.304 0.050 0.99 1 0.299 0.021 0.93 1 0.299 0.044 0.97 1
   0.1 0.302 0.015 0.96 1 0.304 0.041 0.96 1 0.300 0.021 0.94 1 0.298 0.047 0.99 1 0.299 0.016 0.92 1 0.304 0.047 0.95 1
  0.1 1 0.281 0.097 0.93 0.82 0.267 0.171 0.96 0.28 0.273 0.147 0.95 0.49 0.298 0.235 0.98 0.21 0.286 0.124 0.93 0.74 0.308 0.177 0.97 0.41
   0.5 0.283 0.092 0.94 0.86 0.294 0.153 0.95 0.47 0.291 0.127 0.97 0.63 0.276 0.201 0.97 0.24 0.292 0.097 0.92 0.89 0.306 0.151 0.95 0.49
   0.1 0.316 0.073 0.95 0.99 0.304 0.132 0.97 0.64 0.308 0.099 0.99 0.82 0.307 0.178 0.96 0.38 0.308 0.071 0.97 0.99 0.290 0.130 0.97 0.57
0% 0.5 1 0.303 0.028 0.88 1 0.299 0.049 0.95 1 0.295 0.037 0.91 1 0.296 0.055 0.97 1 0.302 0.028 0.87 1 0.303 0.050 0.93 1
   0.5 0.304 0.021 0.93 1 0.301 0.042 0.97 1 0.302 0.030 0.91 1 0.300 0.047 0.98 1 0.300 0.020 0.94 1 0.308 0.043 0.95 1
   0.1 0.302 0.015 0.97 1 0.301 0.039 0.97 1 0.301 0.025 0.92 1 0.299 0.041 1 1 0.300 0.015 0.96 1 0.308 0.041 0.95 1
  0.1 1 0.323 0.135 0.93 0.74 0.289 0.158 0.98 0.41 0.289 0.188 0.91 0.43 0.294 0.280 0.89 0.31 0.309 0.140 0.90 0.74 0.311 0.194 0.91 0.46
   0.5 0.319 0.110 0.90 0.91 0.290 0.135 0.96 0.58 0.303 0.153 0.91 0.62 0.293 0.229 0.89 0.35 0.308 0.110 0.90 0.85 0.309 0.162 0.91 0.62
   0.1 0.307 0.081 0.95 0.97 0.316 0.120 0.95 0.74 0.286 0.108 0.93 0.75 0.312 0.184 0.92 0.45 0.307 0.082 0.94 0.95 0.299 0.135 0.89 0.73
  1. Mean, standard deviation (sd), coverage and power are displayed for the estimated causal effects of the exposures X1,X2 and X3 on the outcome Y, denoted as \(\hat {\beta }_{1}, \hat {\beta }_{2}\) and \(\hat {\beta }_{3}\) respectively. There are 36 configurations containing six sample overlapping rates (100%, 80%, 60%, 40%, 20%, 0%), two levels of IV strengths (α=(α1,α2,α3)=0.5 and 0.1) and three levels of effects of the confounder U on the exposures (δ=(δ1,δ2,δ3)=1,0.5 and 0.1). The effect of U on Y (δ4) is set to 1