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Table 2 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

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
  1. Mean, standard deviation (sd) and coverage 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