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Table 1 Results of the simulation study of different methods to control for confounding

From: Adjustment for unmeasured confounding through informative priors for the confounder-outcome relation

Scenario Parameter settings Frequentist model Bayesian model
   Unadjusted Adjusted for Z Adjusted for Z, τ = 1000 Adjusted for Z, τ = 10
β zu β xz β xu β yu Bias SD MSE Bias SD MSE Bias SD MSE Bias SD MSE
1 0 1 0 0 0.50 0.027 0.25 0.00 0.034 0.0011 0.00 0.025 0.0006 0.00 0.033 0.0011
2 1 1 0 0 0.67 0.027 0.45 0.00 0.035 0.0012 0.00 0.022 0.0005 0.00 0.034 0.0012
3 0 1 1 0 0.33 0.024 0.11 0.00 0.024 0.0006 0.00 0.021 0.0004 0.00 0.024 0.0006
4 1 1 1 0 0.50 0.016 0.25 0.00 0.031 0.0009 0.00 0.017 0.0003 0.00 0.030 0.0009
5 0 1 2 0 0.17 0.017 0.029 0.00 0.014 0.0002 0.00 0.013 0.0002 0.00 0.014 0.0002
6 1 1 2 0 0.36 0.012 0.13 0.00 0.020 0.0004 0.00 0.011 0.0001 0.00 0.020 0.0004
7 0 1 0 1 0.50 0.034 0.25 0.00 0.043 0.0019 0.00 0.032 0.001 0.00 0.043 0.0018
8 1 1 0 1 1.00 0.036 1.00 0.00 0.034 0.0012 0.23 0.026 0.055 0.00 0.034 0.0012
9 0 1 1 1 0.67 0.023 0.45 0.50 0.029 0.25 0.38 0.024 0.15 0.50 0.029 0.25
10 1 1 1 1 0.83 0.018 0.69 0.33 0.028 0.11 0.33 0.017 0.11 0.33 0.028 0.11
11 0 1 2 1 0.50 0.016 0.25 0.40 0.016 0.16 0.36 0.015 0.13 0.40 0.015 0.16
12 1 1 2 1 0.64 0.011 0.41 0.33 0.018 0.11 0.29 0.010 0.085 0.33 0.017 0.11
13 0 1 0 2 0.50 0.049 0.25 −0.01 0.066 0.0044 0.00 0.047 0.0022 −0.01 0.063 0.004
14 1 1 0 2 1.34 0.045 1.79 0.01 0.057 0.0033 0.57 0.036 0.32 0.034 0.055 0.0042
15 0 1 1 2 1. 01 0.032 1.00 1.00 0.037 1.00 0.72 0.035 0.52 0.99 0.037 0.97
16 1 1 1 2 1.17 0.024 1.36 0.67 0.038 0.45 0.67 0.021 0.45 0.67 0.037 0.45
17 0 1 2 2 0.83 0.019 0.70 0.80 0.020 0.64 0.71 0.019 0.50 0.80 0.020 0.64
18 1 1 2 2 0.91 0.013 0.83 0.67 0.023 0.45 0.58 0.012 0.33 0.66 0.022 0.44
  1. Bias refers to the bias in the estimator of the relation between X and Y, compared to the true X-Y relation (βyx = 0). τ indicates the precision of the prior distribution of the Z-Y relation in the Bayesian model and is proportional to the sample size of each generated data set (n = 1000). Abbreviations: SD – standard deviation of the empirical distributions of the parameter estimates; MSE – mean squared error of the parameter estimates. See text for details on simulation study