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Table 4 Summary measures for a strong direct effect (β6 = 0.5). Confounding and strength of IV remained as in the baseline scenario with α1 = 0.5, α2 = 0.3, β5 = 1.0. The causal treatment effect was β1 = 3.0. Results are across 200 simulated data sets; values are: sample mean (Monte Carlo SD) unless otherwise stated

From: Assessing causal treatment effect estimation when using large observational datasets

  N = 2000 N = 20,000 N = 200,000
Adjusted Linear Model and Propensity score IPTW
 Effect Estimate 3.55 (0.05) 3.55 (0.02) 3.55 (0.01)
 Bias 0.55 (0.05) 0.55 (0.02) 0.55 (0.01)
 Mean Square Error 0.30 (0.05) 0.30 (0.02) 0.30 (0.01)
 Z Statistic 10.76 (0.94) 34.18 (1.04) 108.28 (1.07)
 Coverage: n (%) 0 (0.00) 0 (0.00) 0 (0.00)
 Power: n (%) 200 (100.00) 200 (100.00) 200 (100.00)
2SLS IV
 Effect Estimate 5.67 (2.52) 6.10 (0.72) 6.01 (0.25)
 Bias 2.67 (2.52) 3.10 (0.72) 3.01 (0.25)
 Mean Square Error 19.83 (17.89) 10.65 (4.41) 9.17 (1.49)
 Z Statistic 0.99 (0.89) 3.69 (0.84) 11.39 (0.90)
 Coverage: n (%) 177 (88.50) 8 (4.00) 0 (0.00)
 Power: n (%) 118 (59.00) 200 (100.00) 200 (100.00)
Adjusted 2SLS IV
 Effect Estimate 6.07 (0.44) 6.04 (0.16) 6.02 (0.05)
 Bias 3.07 (0.44) 3.04 (0.16) 3.02 (0.05)
 Mean Square Error 9.82 (2.81) 9.32 (0.97) 9.15 (0.30)
 Z Statistic 6.03 (0.65) 18.93 (0.73) 60.01 (0.66)
 Coverage: n (%) 0 (0.00) 0 (0.00) 0 (0.00)
 Power: n (%) 200 (100.00) 200 (100.00) 200 (100.00)