Table 2.

Comparison of allele score methods for uncorrelated IVs.

	α	R 2	F	OLS	Crudely weighted	Equally weighted	Externally weighted
					Using individual‐level data
β U=+1	0.05	0.010	3.3	0.697	0.346 (0.136)	0.198 (0.178)	0.199 (0.205)
	0.10	0.030	10.2	0.687	0.246 (0.080)	0.199 (0.089)	0.198 (0.090)
	0.20	0.102	37.9	0.650	0.212 (0.042)	0.199 (0.044)	0.199 (0.043)
β U=−1	0.05	0.010	3.3	−0.297	0.052 (0.135)	0.201 (0.178)	0.200 (0.205)
	0.10	0.030	10.2	−0.287	0.151 (0.080)	0.198 (0.089)	0.198 (0.090)
	0.20	0.102	37.9	−0.250	0.186 (0.042)	0.199 (0.044)	0.199 (0.043)
					Using summarized data
β U=+1	0.05	0.010	3.3	0.697	0.346 (0.171)	0.198 (0.204)	0.199 (0.235)
	0.10	0.030	10.2	0.687	0.246 (0.093)	0.199 (0.101)	0.198 (0.102)
	0.20	0.102	37.9	0.650	0.212 (0.048)	0.199 (0.050)	0.199 (0.049)
β U=−1	0.05	0.010	3.3	−0.297	0.052 (0.133)	0.201 (0.168)	0.200 (0.194)
	0.10	0.030	10.2	−0.287	0.151 (0.076)	0.198 (0.083)	0.198 (0.084)
	0.20	0.102	37.9	−0.250	0.186 (0.039)	0.199 (0.042)	0.199 (0.041)

Median estimates over 10 000 simulations of β _X=0.2 (median standard errors) from simulation study with 15 uncorrelated instrumental variables (IVs) varying direction of confounding (β _U) as shown by median observational estimate (OLS) and average strength of IV (α; strength is also expressed by the mean values of the R ² and F statistics), using allele score methods with crude weights (derived from the data under analysis), equal weights (unweighted analysis) and external weights (equivalent to estimates derived from an independent sample of equal size to the data under analysis), calculated from individual‐level and summarized data.