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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

From: PCA outperforms popular hidden variable inference methods for molecular QTL mapping

 

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