Table 3.

Comparison of summarized data methods for uncorrelated IVs.

		Imprecise external weights (from 5000 individuals)					Precise external weights (from 50000 individuals)					Oracle weights
	α	Median	SD	SE	Coverage	Power	Median	SD	SE	Coverage	Power	Median	SD	SE	Coverage	Power
		Allele score method using summarized data
β U=+1	0.05	0.200	0.239	0.219	97.3	14.8	0.198	0.192	0.201	97.0	17.6	0.198	0.186	0.197	97.1	18.0
	0.10	0.200	0.093	0.102	97.1	52.1	0.201	0.089	0.098	97.0	55.6	0.201	0.088	0.098	97.0	56.0
	0.20	0.199	0.044	0.049	97.2	97.8	0.199	0.043	0.049	97.2	98.0	0.199	0.043	0.049	97.1	98.1
β U=−1	0.05	0.201	0.232	0.186	93.1	20.6	0.201	0.196	0.165	92.5	24.9	0.201	0.191	0.161	92.5	25.7
	0.10	0.200	0.092	0.084	92.7	68.1	0.200	0.088	0.081	93.3	71.2	0.199	0.087	0.080	93.4	71.7
	0.20	0.200	0.044	0.041	93.0	99.7	0.200	0.043	0.040	92.8	99.8	0.200	0.043	0.040	92.8	99.8
		Summary statistic methoda
β U=+1	0.05	0.148	0.168	0.162	93.1	14.8	0.191	0.191	0.187	94.8	17.6	0.199	0.193	0.190	95.0	18.0
	0.10	0.183	0.094	0.091	94.2	52.1	0.199	0.096	0.095	94.8	55.6	0.201	0.096	0.095	95.0	56.0
	0.20	0.194	0.048	0.047	94.9	97.8	0.198	0.048	0.047	95.0	98.0	0.199	0.048	0.047	94.9	98.1
β U=−1	0.05	0.147	0.139	0.133	93.0	20.6	0.194	0.158	0.154	94.7	24.9	0.201	0.161	0.157	94.8	25.7
	0.10	0.181	0.077	0.075	94.0	68.1	0.198	0.078	0.078	95.3	71.2	0.200	0.079	0.078	95.3	71.7
	0.20	0.196	0.040	0.039	94.3	99.7	0.200	0.040	0.039	94.9	99.8	0.200	0.040	0.039	95.0	99.8
		Likelihood‐based methoda
β U=+1	0.05	0.203	0.242	0.195	94.0	18.3	0.194	0.182	0.191	97.3	17.0
	0.10	0.201	0.104	0.097	94.4	54.3	0.200	0.096	0.095	95.0	55.6
	0.20	0.197	0.050	0.048	94.5	97.7	0.198	0.052	0.048	93.4	97.8
β U=−1	0.05	0.204	0.201	0.161	94.3	25.4	0.198	0.153	0.157	97.1	24.4
	0.10	0.198	0.086	0.080	94.5	70.1	0.199	0.079	0.079	95.2	71.3
	0.20	0.199	0.042	0.040	94.3	99.7	0.200	0.046	0.039	92.0	99.7

Median estimates over 10000 simulations of β _X=0.2, standard deviation (SD) of estimates, median standard error (SE) of estimates, coverage (%) of nominal 95% confidence interval for the causal parameter and empirical power (%) based on nominal 95% confidence interval to detect a causal effect from simulation study with 15 uncorrelated instrumental variables (IVs) varying the direction of confounding (β _U) and average strength of IV (α) using three summarized data methods: allele score, summary statistic and likelihood‐based methods, with weights taken from an external source corresponding to an independent sample of size 5000, 50000 and using the true (oracle) weights.

^aThe ‘weights’ for the summary statistic and likelihood‐based methods are used as the ${\widehat{\beta }}_{\mathrm{Xk}}$ association estimates in equations (4) and (6). In the allele score method, the weights (w _j) and the association estimates ( ${\widehat{\beta }}_{\mathrm{Xk}}$) in the data under analysis are both used to provide the causal estimate.