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 1 Bias (SE) for exposures in Group 1 and \({R}^{2}\) with elastic net regression 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	β₁	β₂	β₃	R²	β₁	β₂	β₃	R²
Scenario 1
Complete case	-0.12 (0.26)	-0.13 (0.28)	0.02 (0.21)	0.74	-0.16 (0.65)	-0.12 (0.41)	-0.03 (0.42)	0.60
LOD/\(\sqrt{2}\)	0.03 (0.14)	-0.03 (0.14)	0.01 (0.11)	0.84	-0.03 (0.53)	-0.16 (0.23)	-0.04 (0.22)	0.79
MI	0.05 (0.14)	0.02 (0.20)	0.02 (0.20)	0.80	0.06 (0.52)	0.12 (0.50)	-0.03 (0.53)	0.82
Truncated MI	0.00 (0.14)	0.01 (0.15)	0.00 (0.12)	0.83	0.04 (0.52)	0.08 (0.46)	-0.01 (0.40)	0.85
F-AFT	0.02 (0.14)	0.01 (0.15)	0.00 (0.12)	0.84	0.02 (0.53)	-0.06 (0.41)	-0.01 (0.34)	0.84
Scenario 2A
Complete case	0.00 (0.26)		0.09 (0.22)	0.52	-0.17 (0.65)		0.00 (0.40)	0.56
LOD/\(\sqrt{2}\)	0.14 (0.14)		0.10 (0.13)	0.62	-0.01 (0.54)		-0.02 (0.21)	0.77
MI	0.15 (0.15)		0.12 (0.20)	0.61	0.09 (0.55)		0.05 (0.50)	0.77
Truncated MI	0.15 (0.15)		0.13 (0.20)	0.61	0.07 (0.54)		0.04 (0.48)	0.78
F-AFT	0.14 (0.14)		0.11 (0.14)	0.62	0.03 (0.54)		0.01 (0.32)	0.80
Scenario 2B
Complete case	-0.04 (0.24)	-0.26 (0.26)	0.06 (0.21)	0.64	-0.19 (0.63)	-0.27 (0.14)	-0.01 (0.39)	0.54
LOD/\(\sqrt{2}\)	0.09 (0.14)	-0.18 (0.17)	0.06 (0.12)	0.73	-0.05 (0.54)	-0.26 (0.12)	-0.02 (0.20)	0.76
MI	0.10 (0.14)	-0.17 (0.17)	0.06 (0.20)	0.73	0.05 (0.54)	-0.25 (0.14)	0.03 (0.48)	0.76
Truncated MI	0.09 (0.14)	-0.17 (0.17)	0.07 (0.19)	0.73	0.03 (0.54)	-0.26 (0.13)	0.02 (0.46)	0.78
F-AFT	0.09 (0.14)	-0.18 (0.17)	0.06 (0.13)	0.74	-0.01 (0.54)	-0.26 (0.13)	0.01 (0.31)	0.80
Scenario 3
Complete case	-0.10 (0.26)	-0.28 (0.30)	0.01 (0.21)	0.71	-0.17 (0.35)	-0.34 (0.35)	0.02 (0.26)	0.65
LOD/\(\sqrt{2}\)	0.00 (0.14)	0.13 (0.15)	-0.01 (0.11)	0.85	0.00 (0.14)	0.16 (0.15)	-0.01 (0.11)	0.86
MI	0.07 (0.15)	-0.03 (0.22)	0.03 (0.20)	0.78	0.07 (0.15)	0.00 (0.25)	0.04 (0.24)	0.77
Truncated MI	-0.03 (0.14)	0.18 (0.15)	-0.02 (0.12)	0.83	-0.04 (0.14)	0.22 (0.16)	-0.03 (0.12)	0.83
F-AFT	0.00 (0.14)	0.15 (0.15)	-0.02 (0.12)	0.84	0.00 (0.14)	0.19 (0.16)	-0.03 (0.12)	0.84
Scenario 4
Complete case	-0.12 (0.26)	-0.10 (0.30)	0.02 (0.22)	0.74	-0.19 (0.32)	-0.18 (0.35)	0.03 (0.28)	0.67
LOD/\(\sqrt{2}\)	0.00 (0.14)	0.10 (0.15)	0.00 (0.11)	0.86	0.00 (0.13)	0.14 (0.15)	0.00 (0.11)	0.86
MI	0.04 (0.14)	0.06 (0.22)	0.03 (0.18)	0.81	0.05 (0.14)	0.06 (0.23)	0.03 (0.22)	0.79
Truncated MI	-0.02 (0.14)	0.12 (0.15)	-0.01 (0.11)	0.84	-0.02 (0.13)	0.15 (0.15)	-0.02 (0.12)	0.84
F-AFT	0.00 (0.14)	0.12 (0.16)	-0.01 (0.12)	0.85	0.00 (0.13)	0.15 (0.16)	-0.02 (0.12)	0.85

Bias (SE) was reported for exposures in Group 1 (\({\beta }_{1},{\beta }_{2}\) and \({\beta }_{3}\)). All other results are provided in Table S1. 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, \({\beta }_{2}\) 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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