Estimation of total mediation effect for high-dimensional omics mediators

Table 1 Bias and standard deviation under high-dimensional settings (Simulation setting I): bias in the first row, and standard deviation in the second row for each scenario

	\(R^2_{Mediated}\)	SOS	ab	ab (Lasso)	prop	prop (Lasso)	ratio	ratio (Lasso)
H1	0.0006	0.0013	\(-0.0084\)	\(-0.0324\)	0.0001	0.0069	\(-0.0107\)	\(-0.0231\)
(\(\hat{{\mathbf {M}}} = \mathbf{M }\))	(0.0181)	(0.0370)	(0.2846)	(0.2744)	(0.0833)	(0.0795)	(0.1161)	(0.1117)
H2	0.0146	0.0299	0.1602	\(-0.0359\)	\(-0.0493\)	0.0058	0.0075	\(-0.0212\)
(\(\hat{{\mathbf {M}}} = [\mathbf{M }, \mathbf{M }^{{(2)}}]\))	(0.0184)	(0.0375)	(0.6463)	(0.2604)	(0.1960)	(0.0777)	(0.2886)	(0.1165)
H3	0.0006	0.0053	0.0923	0.0547	\(-0.0552\)	\(-0.0520\)	\(-0.0013\)	\(-0.0315\)
(\(\hat{{\mathbf {M}}} = [\mathbf{M }, \mathbf{M }^{{(1)}}]\))	(0.0071)	(0.0653)	(0.7443)	(0.7547)	(0.2520)	(0.2495)	(0.2983)	(0.3392)
H4	0.0047	0.0095	0.1421	\(-0.0347\)	\(-0.0447\)	0.0025	0.0498	\(-0.0196\)
(\(\hat{{\mathbf {M}}} = [\mathbf{M }, noise]\))	(0.0198)	(0.0403)	(0.2613)	(0.2519)	(0.0785)	(0.0689)	(0.0982)	(0.1055)
H5	\(-0.0000\)	\(-0.0000\)	\(-0.0867\)	\(-0.3449\)	\(-0.0173\)	\(-0.0293\)	\(-0.0532\)	\(-0.2327\)
(\(\hat{{\mathbf {M}}} = \mathbf{M }\))	(0.0095)	(0.0109)	(0.0956)	(0.1618)	(0.0158)	(0.0184)	(0.0482)	(0.0295)

ab: product measure; prop: proportion measure. (Lasso) indicates that the estimation is based on the Lasso regression; otherwise, it is estimated by a mixed-effect model. The true values are presented in Additional file 1: Table S2. The set of variables included in the model is denoted as \(\hat{{\mathbf {M}}}\). The set of true mediators is denoted as \(\mathbf{M} \), the set of variables associated with exposure but not with outcome is denoted as \(\mathbf {M}^{\mathbf {(1)}}\), and the set of variables associated with outcome but not the exposure is denoted as \(\mathbf {M^{(2)}}\). Variables in \(\mathbf {M}^{\mathbf {(1)}}\) and \(\mathbf {M^{(2)}}\) are non-mediators falsely included in the putative mediator set \(\hat{{\mathbf {M}}}\)