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. 2013 Jul 17;178(7):1177–1184. doi: 10.1093/aje/kwt084

Table 1.

A Comparison of Different Methods of Estimating 95% Confidence Intervals for Selected Simulated Data Setsa

Strong IV (R2 = 0.025)
(Theoretical F = 50)b
 Moderate IV (R2 = 0.005)
(Theoretical F = 10)
 Weak IV (R2 = 0.002)
(Theoretical F = 5)
β SE CI β SE CI β SE CI
Subsample IV approach  
 Delta method 0.148 0.057 0.037, 0.259 0.152 0.132 −0.108, 0.411 0.081 0.161 −0.234, 0.397
 Sequential regressionc 0.055 0.039, 0.256 0.128 −0.099, 0.403 0.159 −0.231, 0.394
 Fieller's theorem N/A 0.040, 0.272 N/A −0.108, 0.562 N/A −0.291, 0.602
 Bootstrapd 0.068 0.014, 0.280 0.137 −0.117, 0.421 0.551 −0.999, 1.162
 Bayesian 0.143 0.056 0.040, 0.258 0.174 0.289 −0.161, 0.778 0.089 0.443 −0.563, 0.975
2-sample IV approach  
 Delta method 0.117 0.068 −0.015, 0.250 0.051 0.119 −0.182, 0.284 −0.086 0.163 −0.405, 0.232
 Sequential regressionc 0.065 −0.011, 0.245 0.118 −0.181, 0.282 0.160 −0.440, 0.227
 Fieller's theorem N/A −0.012, 0.267 N/A −0.201, 0.336 N/A −0.610, 0.280
 Bootstrapd 0.071 −0.023, 0.257 0.138 −0.221, 0.322 0.997 −2.041, 1.868
 Bayesian 0.119 0.072 −0.013, 0.273 0.055 0.169 −0.232, 0.390 −0.100 0.456 −1.012, 0.554

Abbreviations: CI, confidence interval; IV, instrumental variable; N/A, not applicable; SE, standard error.

a The simulated data sets consisted of 10,000 persons with data on G and Y and 2,000 persons with data on G and X. The true effect of X on Y was set to 0.1, and a confounding variable U had the effect of 0.2 on both X and Y.

b Theoretical F values were obtained using the following equation: F = R2 (nX − 1)/(1 − R2).

c For the second-stage regression (of sequential regression), robust SEs are reported.

d Bootstrapping was conducted using 1,000 replications, with samples of size nX and nY randomly selected (with replacement) from the original samples of size nX and nY.