The impact of imprecisely measured covariates on estimating gene-environment interactions

Table 2 The effect of measurement error in an exposure σ _u ² on estimated exposure ( $\hat{β}$ ₂) and interaction between the exposure and a perfectly measured genotype ( $\hat{β}$ ₃).

Coefficient for true effect of interaction(β ₃)	Ratio for true effect of interaction(β ₃+β ₂)/β ₂	$\hat{β}$ ₂(sd of estimate)				$\hat{β}$ ₃(sd of estimate)
		σ _u ² = 1	σ _u ² = 2	σ _u ² = 4	σ _u ² = 9	σ _u ² = 1	σ _u ² = 2	σ _u ² = 4	σ _u ² = 9
0.0	1.0	0.50 (0.05)	0.33 (0.04)	0.20 (0.03)	0.10 (0.02)	0.00 (0.12)	0.00 (0.10)	0.00 (0.08)	0.00 (0.06)
0.5	1.5	0.50 (0.05)	0.33 (0.04)	0.20 (0.03)	0.10 (0.02)	0.25 (0.13)	0.17 (0.11)	0.10 (0.09)	0.05 (0.06)
1.0	2.0	0.50 (0.05)	0.33 (0.05)	0.20 (0.03)	0.10 (0.02)	0.50 (0.14)	0.33 (0.12)	0.20 (0.09)	0.10 (0.07)
2.0	3.0	0.50 (0.05)	0.33 (0.04)	0.20 (0.04)	0.10 (0.03)	1.00 (0.16)	0.67 (0.14)	0.40 (0.11)	0.20 (0.08)

True values of coefficients β ₁ = 1 (binary genotype), β ₂ = 1 (continuous exposure), β ₃ (interaction term) given in first column. No confounding present, β ₄ = 0. Simulations based on 10,000 simulations of 1000 observations. Monte Carlo error is 1% of the empirical standard deviation of the estimates for $\hat{β}$ ₂ or $\hat{β}$ ₃; approximately 0.0005 for $\hat{β}$ ₂ and 0.0015 for $\hat{β}$ ₃.

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