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

Table 3 Summary measures of estimated \(h\left(Z\right)\)with BKMR and each LOD accommodation approach compared to using full dataset

LOD accommodation	Moderate correlation (\(\boldsymbol\sigma\boldsymbol=\mathbf1\boldsymbol/\mathbf2\))				High correlation (\(\boldsymbol\sigma\boldsymbol=\mathbf1\boldsymbol/\mathbf8\))
LOD accommodation	Intercept	Slope	\(\boldsymbol R^{\mathbf2}\)	SE	Intercept	Slope	\(\boldsymbol R^{\mathbf2}\)	SE
Scenario 1
Complete case	0.07	0.91	0.83	0.42	0.58	0.68	0.56	0.21
LOD/\(\sqrt{2}\)	0.11	0.94	0.96	0.28	0.44	0.77	0.75	0.16
MI	0.23	0.88	0.87	0.30	-0.23	1.14	0.81	0.18
Truncated MI	0.05	0.98	0.95	0.32	-0.21	1.13	0.85	0.18
F-AFT	0.08	0.97	0.96	0.28	0.12	0.95	0.82	0.16
Scenario 2A
Complete case	0.54	0.68	0.70	0.37	0.79	0.57	0.51	0.20
LOD/\(\sqrt{2}\)	0.47	0.72	0.82	0.25	0.50	0.74	0.73	0.16
MI	0.47	0.72	0.79	0.27	0.23	0.89	0.73	0.17
Truncated MI	0.45	0.72	0.81	0.27	0.28	0.86	0.75	0.17
F-AFT	0.47	0.72	0.82	0.25	0.39	0.80	0.77	0.15
Scenario 2B
Complete case	0.31	0.79	0.78	0.40	0.80	0.55	0.48	0.21
LOD/\(\sqrt{2}\)	0.31	0.81	0.89	0.27	0.55	0.71	0.71	0.16
MI	0.31	0.82	0.87	0.29	0.23	0.88	0.72	0.18
Truncated MI	0.29	0.82	0.88	0.29	0.29	0.85	0.74	0.18
F-AFT	0.31	0.82	0.89	0.27	0.45	0.77	0.75	0.16
Scenario 3
Complete case	0.29	0.87	0.81	0.43	0.36	0.80	0.70	0.50
LOD/\(\sqrt{2}\)	0.05	0.98	0.95	0.31	0.07	0.98	0.95	0.30
MI	0.30	0.84	0.83	0.33	0.33	0.83	0.81	0.33
Truncated MI	0.00	1.03	0.92	0.36	0.01	1.04	0.91	0.36
F-AFT	0.04	1.01	0.95	0.31	0.06	1.01	0.95	0.31
Scenario 4
Complete case	0.18	0.89	0.82	0.43	0.24	0.82	0.72	0.50
LOD/\(\sqrt{2}\)	0.05	0.98	0.96	0.28	0.05	0.98	0.96	0.28
MI	0.21	0.89	0.88	0.30	0.24	0.88	0.86	0.30
Truncated MI	-0.01	1.02	0.95	0.32	-0.01	1.02	0.94	0.32
F-AFT	0.03	1.01	0.97	0.28	0.03	1.01	0.96	0.28

Summary measures were obtained by regressing the estimated\(\widehat{h}\)of each LOD-accommodation approach on\(\widehat{h}\)using full datasets, and reported average intercept, slope, and\({R}^{2}\)across simulation iterations. Zero intercept and 1 of slope indicate LOD accommodation approach does not influence BKMR results. “SE” denotes the posterior standard deviation of the\(\widehat{h}\)
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

ISSN: 1476-069X