Accommodating detection limits of multiple exposures in environmental mixture analyses: an overview of statistical approaches

Lee, Myeonggyun; Saha, Abhisek; Sundaram, Rajeshwari; Albert, Paul S.; Zhao, Shanshan

doi:10.1186/s12940-024-01088-w

Environmental Health

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

From: Accommodating detection limits of multiple exposures in environmental mixture analyses: an overview of statistical approaches

LOD accommodation	Moderate correlation (\(\boldsymbol\sigma\boldsymbol=\mathbf1\boldsymbol/\mathbf2\))					High correlation (\(\boldsymbol\sigma\boldsymbol=\mathbf1\boldsymbol/\mathbf8\))
LOD accommodation	\(\boldsymbol\psi\)	\({\boldsymbol w}_{\mathbf1}\)	\({\boldsymbol w}_{\mathbf2}\)	\({\boldsymbol w}_{\mathbf3}\)	\(\boldsymbol R^{\mathbf2}\)	\(\boldsymbol\psi\)	\({\boldsymbol w}_{\mathbf1}\)	\({\boldsymbol w}_{\mathbf2}\)	\({\boldsymbol w}_{\mathbf3}\)	\(\boldsymbol R^{\mathbf2}\)
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

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