Table 2.

Bias (SE) for exposures in Group 1 and $R^2$ with WQS and each LOD accommodation approach compared to using full dataset

LOD accommodation	Moderate correlation ($\mathit{\sigma }\mathbf{=}\mathbf{1}\mathbf{/}\mathbf{2}$)					High correlation ($\mathit{\sigma }\mathbf{=}\mathbf{1}\mathbf{/}\mathbf{8}$)
	$\mathit{\psi }$	${\mathit{w}}_{\mathbf{1}}$	${\mathit{w}}_{\mathbf{2}}$	${\mathit{w}}_{\mathbf{3}}$	${\mathit{R}}^{\mathbf{2}}$	$\mathit{\psi }$	${\mathit{w}}_{\mathbf{1}}$	${\mathit{w}}_{\mathbf{2}}$	${\mathit{w}}_{\mathbf{3}}$	${\mathit{R}}^{\mathbf{2}}$
Scenario 1
Complete case	-0.34 (0.33)	-0.03 (0.10)	-0.04 (0.10)	0.02 (0.06)	0.73	-0.03 (0.33)	-0.02 (0.13)	0.01 (0.13)	0.01 (0.10)	0.87
LOD/$\sqrt{2}$	0.15 (0.22)	-0.02 (0.06)	0.00 (0.08)	0.01 (0.05)	0.86	0.01 (0.26)	0.00 (0.13)	-0.02 (0.11)	0.00 (0.10)	0.92
MI	-0.06 (0.18)	0.03 (0.06)	-0.05 (0.06)	0.00 (0.04)	0.79	0.01 (0.24)	0.00 (0.13)	0.00 (0.11)	0.00 (0.09)	0.93
Truncated MI	0.00 (0.18)	0.00 (0.06)	0.00 (0.07)	0.00 (0.04)	0.84	0.01 (0.24)	0.00 (0.13)	0.00 (0.11)	0.00 (0.10)	0.93
F-AFT	-0.01 (0.18)	0.00 (0.06)	0.00 (0.07)	0.00 (0.04)	0.85	-0.01 (0.24)	0.00 (0.13)	-0.01 (0.11)	0.00 (0.10)	0.93
Scenario 2A
Complete case	-0.49 (0.29)	0.08 (0.12)		0.05 (0.08)	0.54	-0.05 (0.30)	0.02 (0.14)		0.03 (0.11)	0.83
LOD/$\sqrt{2}$	-0.17 (0.19)	0.08 (0.07)		0.05 (0.07)	0.65	-0.01 (0.25)	0.03 (0.14)		0.02 (0.11)	0.88
MI	-0.28 (0.17)	0.11 (0.07)		0.02 (0.05)	0.63	-0.01 (0.23)	0.03 (0.14)		0.02 (0.10)	0.90
Truncated MI	-0.28 (0.17)	0.12 (0.07)		0.03 (0.05)	0.63	-0.01 (0.23)	0.04 (0.14)		0.02 (0.11)	0.90
F-AFT	-0.28 (0.17)	0.11 (0.07)		0.04 (0.06)	0.65	-0.02 (0.23)	0.04 (0.14)		0.02 (0.12)	0.90
Scenario 2B
Complete case	-0.33 (0.34)	-0.14 (0.04)	-0.20 (0.05)	0.04 (0.07)	0.45	-0.01 (0.34)	-0.06 (0.04)	-0.06 (0.08)	0.01 (0.10)	0.82
LOD/$\sqrt{2}$	0.09 (0.22)	-0.11 (0.04)	-0.22 (0.03)	0.04 (0.06)	0.57	0.04 (0.29)	-0.06 (0.05)	-0.07 (0.07)	0.01 (0.11)	0.88
MI	-0.10 (0.19)	-0.15 (0.02)	-0.21 (0.03)	0.01 (0.04)	0.50	0.03 (0.25)	-0.08 (0.01)	-0.07 (0.07)	0.01 (0.09)	0.89
Truncated MI	-0.12 (0.19)	-0.15 (0.02)	-0.21 (0.03)	0.02 (0.05)	0.50	0.03 (0.26)	-0.08 (0.01)	-0.07 (0.07)	0.01 (0.10)	0.89
F-AFT	-0.12 (0.19)	-0.16 (0.02)	-0.21 (0.03)	0.03 (0.05)	0.52	0.01 (0.26)	-0.08 (0.01)	-0.07 (0.08)	0.01 (0.10)	0.89
Scenario 3
Complete case	-0.46 (0.35)	0.00 (0.10)	-0.11 (0.10)	0.03 (0.06)	0.69	-0.54 (0.44)	-0.03 (0.11)	-0.12 (0.11)	0.04 (0.07)	0.66
LOD/$\sqrt{2}$	0.16 (0.23)	-0.01 (0.06)	-0.01 (0.08)	0.01 (0.04)	0.85	0.20 (0.24)	-0.02 (0.06)	0.01 (0.07)	0.01 (0.04)	0.86
MI	-0.14 (0.20)	0.05 (0.07)	-0.10 (0.07)	0.01 (0.04)	0.75	-0.14 (0.21)	0.05 (0.07)	-0.10 (0.07)	0.01 (0.04)	0.74
Truncated MI	0.03 (0.19)	-0.01 (0.06)	0.01 (0.07)	0.00 (0.04)	0.83	0.05 (0.19)	-0.01 (0.06)	0.02 (0.07)	0.00 (0.04)	0.83
F-AFT	0.01 (0.20)	0.00 (0.06)	0.00 (0.07)	0.00 (0.04)	0.84	0.01 (0.20)	0.00 (0.06)	0.00 (0.07)	0.00 (0.04)	0.84
Scenario 4
Complete case	-0.35 (0.34)	-0.02 (0.11)	-0.06 (0.10)	0.03 (0.06)	0.72	-0.45 (0.42)	-0.05 (0.11)	-0.08 (0.10)	0.04 (0.07)	0.69
LOD/$\sqrt{2}$	0.18 (0.22)	-0.02 (0.06)	0.01 (0.08)	0.01 (0.04)	0.86	0.20 (0.22)	-0.03 (0.06)	0.02 (0.08)	0.01 (0.04)	0.87
MI	-0.08 (0.18)	0.03 (0.07)	-0.07 (0.07)	0.00 (0.04)	0.78	-0.08 (0.18)	0.04 (0.07)	-0.07 (0.06)	0.00 (0.04)	0.76
Truncated MI	0.02 (0.17)	0.00 (0.07)	0.01 (0.07)	0.00 (0.04)	0.84	0.03 (0.18)	-0.01 (0.06)	0.02 (0.07)	0.00 (0.04)	0.84
F-AFT	0.00 (0.18)	0.00 (0.07)	0.00 (0.07)	0.00 (0.04)	0.85	0.00 (0.18)	0.00 (0.07)	0.00 (0.07)	0.00 (0.04)	0.85

Bias (SE) was reported for the total effect ($\psi$) and exposures in group 1 ($w_1,w_2$ and $w_3$). All other results are provided in Table S2. All comparisons were made to the parameters with full datasets without LOD. $R^2$ was calculated by regression $\widehat{h}$ from each LOD accommodation on $\widehat{h}$ with the full dataset. In Scenario 2A, $w_2$ was not estimated because $Z_2$ was not included in the analysis

Abbreviations: Imputation by LOD/$\sqrt{2}$ (LOD/$\sqrt{2}$), MI Conventional multiple imputation, Truncated MI Truncated multiple imputation, F-AFT Imputation by estimates using the AFT model