Table 1.

Summary of the 15 methods we compare based on simulation studies, including Ideal, Unadjusted, and 13 variants of PCA, SVA, PEER, and HCP (Additional file 1: Section S4). Out of the 15 methods, we select a few representative methods (Section 5.2) for detailed comparison in Simulation Design 2, the abbreviations of which are shown in (D). Y denotes the gene expression matrix, $Y_{\text{resid}}$ denotes the residual matrix outputted by PEER, $X_1$ denotes the known covariate matrix, and $X_2$ denotes the hidden covariate matrix. In Line 3, PCA is run on Y directly; in Line 4, PCA is run after the effects of $X_1$ are regressed out from Y (Additional file 1: Section S4). The addition signs in (C) denote column concatenation. “filtered” means that we filter out the known covariates that are captured well by the inferred covariates (unadjusted $R^2\ge 0.9$); this filtering is only needed when the hidden variable inference method in (A) does not explicitly take the known covariates into account

	Inference method	Method	Response, covariates	Method abbr. (if selected)
	(A)	(B)	(C)	(D)
1		Ideal	Y, $X_1$ + $X_2$	Ideal
2		Unadjusted	Y, $X_1$	Unadjusted
3	PCA_direct	PCA_direct_screeK	Y, $X_1$ (filtered) + top PCs	PCA
4	PCA_resid	PCA_resid_screeK	Y, $X_1$ + top PCs
5	SVA_trueK	SVA_trueK	Y, $X_1$ + SVs
6	SVA_BE	SVA_BE	Y, $X_1$ + SVs	SVA
7	PEER_noCov_trueK	PEER_noCov_trueK_factors	Y, $X_1$ (filtered) + PEER factors
8	PEER_noCov_trueK	PEER_noCov_trueK_residuals	$Y_{\text{resid}}$, NULL
9	PEER_noCov_largeK	PEER_noCov_largeK_factors	Y, $X_1$ (filtered) + PEER factors
10	PEER_noCov_largeK	PEER_noCov_largeK_residuals	$Y_{\text{resid}}$, NULL
11	PEER_withCov_trueK	PEER_withCov_trueK_factors	Y, $X_1$ + PEER factors	PEER, true K, factors
12	PEER_withCov_trueK	PEER_withCov_trueK_residuals	$Y_{\text{resid}}$, NULL
13	PEER_withCov_largeK	PEER_withCov_largeK_factors	Y, $X_1$ + PEER factors
14	PEER_withCov_largeK	PEER_withCov_largeK_residuals	$Y_{\text{resid}}$, NULL	PEER, large K, residuals
15	HCP_trueK	HCP_trueK	Y, $X_1$ + HCPs	HCP