Table 1.

Empirical type I error estimates under the null model of a quantitative primary phenotype

Scenario	MAFX	CIEE	BS	G‐EST	MR	RR	SEM
1	0.05	5.45%	5.40%	5.03%	5.35%	5.29%	5.04%
	0.1	5.26%	5.18%	5.05%	5.34%	5.23%	5.05%
	0.2	4.83%	4.88%	4.76%	4.94%	4.87%	5.34%
	0.4	5.16%	5.17%	5.12%	5.06%	4.80%	5.24%
2	0.05	5.17%	5.12%	4.77%	4.71%	4.67%	5.57%
	0.1	5.16%	5.13%	5.02%	5.12%	4.99%	4.75%
	0.2	5.01%	4.91%	4.87%	5.16%	4.90%	5.46%
	0.4	5.14%	5.14%	5.06%	4.91%	4.54%	4.86%
3	0.05	5.37%	5.27%	4.89%	4.89%	4.82%	5.18%
	0.1	5.17%	5.11%	4.99%	5.00%	4.87%	4.85%
	0.2	4.89%	4.90%	4.81%	5.25%	4.95%	5.19%
	0.4	5.06%	4.96%	4.97%	4.77%	4.38%	5.21%
4	0.05	5.44%	5.30%	4.98%	5.15%	5.10%	5.32%
	0.1	5.25%	5.21%	4.99%	5.27%	5.13%	4.87%
	0.2	4.81%	4.79%	4.73%	6.03%	5.68%	4.98%
	0.4	5.09%	5.14%	5.03%	5.90%	5.48%	5.43%
5	0.05	5.26%	5.11%	4.83%	4.94%	4.82%	5.42%
	0.1	5.08%	5.02%	4.91%	5.42%	5.23%	5.24%
	0.2	4.91%	4.93%	4.88%	6.01%	5.69%	5.42%
	0.4	5.12%	5.14%	5.07%	6.11%	5.75%	5.40%
6	0.05	5.14%	5.21%	4.57%	5.35%	5.29%	5.62%
	0.1	5.29%	5.27%	5.10%	5.13%	5.08%	5.83%
	0.2	5.03%	4.99%	4.83%	5.25%	5.01%	6.01%
	0.4	5.09%	5.04%	4.96%	4.94%	4.68%	6.33%
7	0.05	5.06%	4.97%	4.61%	36.14%	30.45%	21.33%
	0.1	5.05%	5.16%	4.94%	56.37%	45.31%	33.26%
	0.2	4.97%	4.93%	4.94%	73.78%	54.86%	45.47%
	0.4	5.18%	5.23%	5.17%	82.96%	59.54%	55.24%

Data were generated for $n=1,000$ individuals and $m=10,000$ replicates. CIEE is the proposed method using estimating equations; BS is CIEE using nonparametric bootstrap standard errors; G‐EST is the sequential G‐estimation approach (Vansteelandt et al., 2009); MR is multiple regression; RR is residual regression; and SEM is structural equation modeling.