Table 1.

Simulation results for genetic effect estimation of our method and the gold standard.

MAF	Individual level data frequency (gold standard)		Proposed method full dataset		Proposed method subset dataset (20%)		Proposed method within ± 20% of true value
	$\widehat{\beta }$	SE($\widehat{\beta }$)	$\widehat{\beta }$	SE($\widehat{\beta }$)	$\widehat{\beta }$	SE($\widehat{\beta }$)	$\widehat{\beta }$	SE($\widehat{\beta }$)
Continuous ${\mathbf{Y}}_1$ and continuous ${\mathbf{Y}}_2$
2%	0.710	0.146	0.710	0.146	0.709	0.145	0.709	0.145
5%	0.460	0.092	0.460	0.092	0.461	0.092	0.460	0.092
10%	0.332	0.065	0.332	0.065	0.331	0.065	0.332	0.065
25%	0.231	0.046	0.231	0.046	0.23	0.046	0.231	0.046
Continuous ${\mathbf{Y}}_1$ and binary ${\mathbf{Y}}_2$
2%	0.709	0.143	0.707	0.143	0.708	0.143	0.707	0.143
5%	0.460	0.091	0.455	0.091	0.457	0.09	0.455	0.090
10%	0.333	0.064	0.329	0.064	0.329	0.064	0.328	0.064
25%	0.228	0.046	0.222	0.046	0.223	0.046	0.221	0.046
Continuous ${\mathbf{Y}}_2$ and binary ${\mathbf{Y}}_1$
2%	0.597	1.072	0.561	1.101	0.571	1.144	0.541	1.057
5%	0.752	0.288	0.722	0.284	0.727	0.309	0.695	0.277
10%	0.809	0.203	0.781	0.200	0.782	0.215	0.750	0.192
25%	0.862	0.150	0.843	0.149	0.843	0.155	0.811	0.143
Binary ${\mathbf{Y}}_1$ and binary ${\mathbf{Y}}_2$
2%	0.867	0.364	0.786	0.340	0.784	0.342	0.784	0.340
5%	0.886	0.234	0.808	0.219	0.806	0.222	0.806	0.220
10%	0.842	0.172	0.782	0.162	0.784	0.165	0.780	0.162
25%	0.764	0.133	0.744	0.130	0.748	0.132	0.742	0.130

Number in the table represent averages over all simulation replicates.

MAF: minor allele frequency. Individual level data analysis is the gold standard for estimation. “Full dataset” means the relationship between the outcome and the covariate is estimated in the full sample of individuals, but the effect is estimated using our approximate approach. “Subset dataset” means the relationship between the traits is estimated by randomly selecting 200 individuals, or 20% of the total sample size. “Proposed method within ± 20% of true value” means the relationship between the outcome, and the covariate is a random estimate falling with 20% the true covariance between the traits. The latter scenario reflects what might happen when using estimates from published reports.