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Scientific RepoRts | 7: 13587 | DOI:10.1038/s41598-017-13665-w
www.nature.com/scientificreports
Controlling for Confounding Eects
in Single Cell RNA Sequencing
Studies Using both Control and
Target Genes
Mengjie Chen
1,2
& Xiang Zhou
3,4
Single cell RNA sequencing (scRNAseq) technique is becoming increasingly popular for unbiased
and high-resolutional transcriptome analysis of heterogeneous cell populations. Despite its many
advantages, scRNAseq, like any other genomic sequencing technique, is susceptible to the inuence
of confounding eects. Controlling for confounding eects in scRNAseq data is a crucial step for
accurate downstream analysis. Here, we present a novel statistical method, which we refer to as scPLS
(single cell partial least squares), for robust and accurate inference of confounding eects. scPLS takes
advantage of the fact that genes in a scRNAseq study often can be naturally classied into two sets: a
control set of genes that are free of eects of the predictor variables and a target set of genes that are
of primary interest. By modeling the two sets of genes jointly using the partial least squares regression,
scPLS is capable of making full use of the data to improve the inference of confounding eects. With
extensive simulations and comparisons with other methods, we demonstrate the eectiveness of
scPLS. Finally, we apply scPLS to analyze two scRNAseq data sets to illustrate its benets in removing
technical confounding eects as well as for removing cell cycle eects.
Single-cell RNA sequencing (scRNAseq) has emerged as a powerful tool in genomics. While the traditional RNA
sequencing, known as the bulk RNAseq, measures gene expression levels averaged across many dierent cells in
a sample of potentially heterogeneous cell population, scRNAseq can measure gene expression levels directly at
the single cell resolution. As a result, scRNAseq is less inuenced by the variation of cell type and cell composition
across dierent samples–a major confounding in the analyses of bulk RNAseq studies. Because of this benet and
its high resolution, scRNAseq provides unprecedented insights into many basic biological questions that are pre-
viously dicult to address. For example, scRNAseq has been applied to classify novel cell subtypes
1,2
and cellular
states
3,4
, reconstruct cell lineage and quantify progressive gene expression during development
5–8
, perform spatial
mapping and re-localization
9,10
, identify dierentially expressed genes and gene expression modulars
11–13
, and
investigate the genetic basis of gene expression variation by detecting heterogenic allelic specic expressions
14,15
.
Like any other genomic sequencing experiment, scRNAseq studies are inuenced by many factors that can
introduce unwanted variation in the sequencing data and confound the down-stream analysis
16
. However, such
unwanted variation are oen exacerbated in scRNAseq experiments due to a range of scRNAseq specic con-
ditions that include amplication bias, low amount of input material and low transcript capture eciency
17
;
dropout events that are driven by both biological and technical factors
18,19
; global changes in expression due to
transcriptional bursts
20
; as well as changes in cell cycle and cell size
21
. Indeed, adjusting for confounding fac-
tors in scRNAseq data has been shown to be crucial for accurate estimation of gene expression levels and suc-
cessful down-stream analysis
16–18,22,23
. However, depending on the source, adjusting for confounding factors in
scRNAseq can be non-trivial. Some confounding eects, such as read sampling noise and drop-out events, are
direct consequences of low sequencing-depth, which are random in nature and can be readily addressed by prob-
abilistic modeling using existing statistical methods
18,22–25
. Other confounding eects are inherent to a particular
1
Department of Medicine, University of Chicago, Chicago, IL 60637, USA.
2
Department of Human Genetics,
University of Chicago, Chicago, IL 60637, USA.
3
Department of Biostatistics, University of Michigan, Ann Arbor, MI
48109, USA.
4
Center for Statistical Genetics, University of Michigan, Ann Arbor, MI 48109, USA. Correspondence and
requests for materials should be addressed to M.C. (email: mengjiechen@uchicago.edu) or X.Z. (email: xzhousph@
umich.edu)
Received: 18 May 2017
Accepted: 29 September 2017
Published: xx xx xxxx
OPEN
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Scientific RepoRts | 7: 13587 | DOI:10.1038/s41598-017-13665-w
experimental protocol and can cause amplication bias, but can be easily mitigated by using new protocols
26
. Yet
other confounding eects are due to observable batches and can be adjusted for by including batch labels and
technician ids as covariates or dealt with other statistical methods
27,28
. However, many confounding factors are
hidden and are dicult or even impossible to measure. Common hidden confounding factors include various
technical artifacts during library preparation and sequencing, and unwanted biological confounders such as cell
cycle status. ese hidden confounding factors can cause systematic bias, are notoriously dicult to control for,
and are the focus of the present study.
To eectively infer and control for hidden confounding factors in scRNAseq studies, we develop a novel sta-
tistical method, which we refer to as scPLS (single cell partial least squares). scPLS takes advantage of the fact
that genes in a scRNAseq study can oen be naturally classied into two sets: a control set of genes that are free of
eects of the predictor variables and a target set of genes that are of primary interest. By modeling the two sets of
genes jointly using the partial least squares regression, scPLS is capable of making full use of the data to improve
the inference of confounding factors. scPLS is closely related to and bridges between two existing subcategories
of methods for transcriptome analysis: a subcategory of methods that treat control and target genes in the same
fashion (e.g. PCA
29–32
and LMM
33–35
), and another subcategory of methods that use control genes alone for infer-
ring confounding factors (e.g. RUV
29,36
and scLVM
37
). By bridging between the two subcategories of methods,
scPLS enjoys robust performance across a range of application scenarios. scPLS is also computationally ecient:
with a new block-wise expectation maximization (EM) algorithm, it is scalable to thousands of cells and tens of
thousands of genes. Using simulations and two real data applications, we show how scPLS can be used to remove
confounding eects and enable accurate down-stream analysis in scRNAseq studies. Our method is implemented
as a part of the Citrus project and is freely available at: http://chenmengjie.github.io/Citrus/.
e paper is organized as follows. In the Review of Previous Methods section, we provide a brief review of
existing statistical methods for removing confounding eects in transcriptome analysis and describe how scPLS
is related to and motivated from these methods. In the Method Overview Section, we provide a methodologi-
cal description of the scPLS model, with inference details provided in the Methods Section. In the Simulations
section we present comparisons between scPLS and several existing methods using simulations. In Real Data
Applications section, we apply scPLS to two real scRNAseq data sets to remove technical confounding eects or
cell cycle eects. Finally, we conclude the paper with a summary and discussion.
Review of Previous Methods
Many statistical methods have been developed in sequencing- and array-based genomic studies to infer hidden
confounding factors and control for hidden confounding eects. Based on their targeted application, these statis-
tical methods can be generally classied into two categories.
e rst category of methods are supervised and application-specic: these methods are designed to infer
the confounding factors in the presence of a known predictor variable, and to correct for the confounding eects
without removing the eects of the predictor variable. For example, scientists are oen interested in identifying
genes that are dierentially expressed between two pre-determined treatment conditions or that are associated
with a measured predictor variable of interest. To remove the confounding eects in these applications, meth-
ods, include SVA
30
, sparse regression models
38,39
, and, more recently, RUV
40,41
, are developed. Although these
application-specic methods are widely applied in many genomics studies, their usage is naturally restricted to
cases where the primary variable of interest is known. e application-specic methods become inconvenient in
cases where there are multiple variables of interest (e.g. in eQTL mapping problems). ey also become inappli-
cable when the primary variable of interest is not observed (e.g. in clustering problems).
e second category of methods are unsupervised, and are designed to infer the confounding factors without
knowing or using the predictor variable of interest. Our scPLS belongs to this category. Notable applications of
unsupervised methods in scRNAseq studies include cell type clustering and classication
1–8
. Existing unsuper-
vised statistical methods can be further classied into two subcategories. e rst subcategory of methods treat
all genes in the same fashion and use all of them to infer the confounding factors. For example, the principal
component analysis (PCA) or the factor model extracts the principal components or factors from all genes (or all
highly variable genes) as surrogates for the confounding factors
29–32
. e inferred factors are treated as covariates
whose eects are further removed from gene expression levels before downstream analyses. Similarly, the linear
mixed models (LMMs) construct a sample relatedness matrix based on all genes to capture the inuence of the
confounding factors
33–35
. e relatedness matrix are then included in the downstream analyses to control for the
confounding eects. In contrast, the second subcategory of unsupervised methods are recently developed to take
advantage of a set of control genes for inferring the confounding factors
29,37
. ese methods divide genes into
two sets: a control set of genes that are known to be free of eects of interest a priori and a target set of genes that
are of primary interest. Unlike the rst subcategory, the second subcategory of methods treat the two gene sets
dierently in inferring the confounding factors: the confounding factors are only inferred from the control set,
and are then used to remove the confounding eects in the target genes for subsequent downstream analysis. For
example, scRNAseq studies oen add ERCC spike-in controls prior to the PCR amplication and sequencing
steps. e spike-in controls can be used to capture the hidden confounding technical factors associated with the
experimental procedures, which are further used to remove technical confounding eects (e.g. reverse transcrip-
tion or PCR amplication confounding eects) from the target genes
33
. Similarly, most scRNAseq studies include
a set of control genes that are known to have varying expression levels across cell cycles. ese cell cycle genes
can be used to capture the unmeasured cell cycle status of each cell, which are further used to remove cell cycle
eects in the target genes
37
. Prominent methods in the second subcategory include the unsupervised version of
RUV
29,36
and scLVM
37
.
e two subcategories of unsupervised methods use dierent strategies to infer the confounding factors.
erefore, these two sets of methods are expected to perform well in dierent settings. Specically, the rst
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Scientific RepoRts | 7: 13587 | DOI:10.1038/s41598-017-13665-w
subcategory of methods have the advantage of using information contained in all genes to accurately infer the
confounding eects. However, when the predictor variable of interest inuences a large number of genes, then
this subcategory of methods may incorrectly remove the primary eects of interest. On the other hand, the sec-
ond subcategory of methods infer confounding factors only from the control genes and are thus not prone to
mistakenly removing the primary eects of interest. However, these methods overlook one important fact–that
the hidden confounding factors not only inuence the control genes but also the target genes, i.e. the exact reason
that we need to remove such confounding eects in the rst place. Because the confounding factors inuence
both control and target genes, using control genes alone to infer the confounding factors can be suboptimal as it
fails to use the information from target genes.
To more eectively infer and control for hidden confounding factors in scRNAseq studies, we develop a novel
statistical method, which we refer to as scPLS (single cell partial least squares). scPLS bridges between the two
subcategories of unsupervised methods and eectively includes each as a special case. Like the rst subcategory
of methods, scPLS models both control and target genes jointly to infer the confounding factors. Like the second
subcategory of methods, scPLS is capable of taking advantage of a control set to guild the inference of confound-
ing factors. scPLS builds upon the partial least squares regression model and relies on a key modeling assumption
that only target genes contain the primary eects of interest or other systematic biological variations. By incor-
porating such systematic variations in the target genes only, we can jointly model both control and target genes
to infer the confounding eects while avoiding mis-removing the primary eects of interest. erefore, scPLS
has the potential to make full use of the data to improve the inference of confounding factors and the removal of
confounding eects.
Results
scPLS Method Overview. We provide modeling details for scPLS here. While the formulation of scPLS is
general, we focus on its application in scRNAseq. e scRNAseq data resembles that of the bulk RNAseq data and
consists of a gene expression matrix on n cells and
p q+
genes. We consider dividing the genes into two sets: a con-
trol set that contains q control genes and a target set that contains p genes of primary interest. e control genes are
selected based on the purpose of the analysis. For example, the control set would contain ERCC spike-ins if we want
to remove technical confounding factors, and would contain cell cycle genes if we want to remove cell cycle eects.
We use the following partial least squares regression to jointly model both control and target genes:
ΛεεΨ=+
~
xz,MVN(0,)
(1)
xxixixiii
~yz u ,MVN(0,)
(2)
yuyi yi yi
i
ii
ΛΛ εε Ψ=+ +
where for
i
'th individual cell,
x
i
is a q-vector of expression levels for q control genes;
y
i
is a p-vector of expression
levels for p target genes;
z
i
is
k
z
-vector of unknown confounding factors that aect both control and target genes;
the coecients of the confounding factors are represented by the
q
by
k
z
loading matrix
Λ
x
for the control genes
and the
p
by
k
z
loading matrix
y
Λ
for the target genes;
u
i
is a
k
u
-vector of unknown factors in the target genes and
potentially represents the predictors of interest or other structured variations (see below);
Λ
u
is a
p
by
k
u
loading
matrix;
ε
xi
is a
q
-vector of idiosyncratic error with covariance
diag(, ,)
xi xxq1
22
σσΨ =
;
yi
ε
is a
p
-vector of idiosyn-
cratic error with covariance
diag(, ,)
yi yyp1
22
σσΨ =
; MVN denotes the multivariate normal distribution. We
assume that
ε
xi
,
ε
yi
,
z
i
, and
u
i
are all independent from each other. Following standard latent factor models, we
further assume that
z IMVN(0, )
i
~
and
~u IMVN(0, )
i
. We model transformed data instead of the raw read
counts. We also assume that the expression levels of each gene have been centered to have mean zero, which
allows us to ignore the intercept.
scPLS includes two types of unknown latent factors. e rst set of factors,
z
i
, represents the unknown con-
founding factors that aect both control and target genes. e eects of
z
i
on the control and target genes are
captured in the loading matrices
Λ
x
and
y
Λ
, respectively. We call
z
i
the confounding factors throughout the text,
and we aim to remove the confounding eects
Λ z
y i
from the target genes. e second set of factors,
u
i
, aims to
capture a low dimensional structure of the expression level of
p
target genes. e factors
u
i
can represent the
unknown predictor variables of interest, specic experimental perturbations, cell subpopulations, gene signatures
or other intermediate factors that coordinately regulate a set of genes. erefore, the factors
u
i
can be interpreted
as cell subtypes, treatment status, transcription factors or regulators of biological pathways in dierent stud-
ies
42–46
. Although
u
i
could be of direct biological interest in many data sets, we do not explicitly examine the
inferred
u
i
here. Rather, we view modeling
u
i
in the target genes as a way to better capture the complex variance
structure there and to facilitate the precise estimation of confounding factors
z
i
. For simplicity, we call
u
i
the
biological factors throughout the text, though we note that
u
i
could well represent non-biological processes such
as treatment or environmental eects. us, the expression levels of the control genes can be described by a linear
combination of the confounding factors
z
i
and residual errors; the expression levels of the target genes can be
described by a linear combination of the confounding factors
z
i
, the biological factors
u
i
and residual errors. For
both types of confounding factors, we are interested in inferring the factor eects
z
y i
Λ
and
Λ u
u i
rather than the
individual factors
z
i
and
u
i
. erefore, unlike in standard factor models, we are not concerned with the identia-
bility of the factors. Figure1 shows an illustration of scPLS.
scPLS is closely related to the two subcategories of unsupervised methods described in the previous Section.
Specically, without the biological eects term
Λ u
u i
, scPLS eectively reduces to the rst subcategory of methods
that treat all genes in the same fashion for inferring the confounding factors. Without the Equation2 term, scPLS
eectively reduces to the second subcategory of methods that use only control genes for inference. (Note that,
aer inferring the confounding factors
z
i
from Equation1, the second subcategory of methods still use a reduced
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Scientific RepoRts | 7: 13587 | DOI:10.1038/s41598-017-13665-w
version of Equation2 without the biological eects term
Λ u
u i
to remove the confounding eects.) By including
both modeling terms, scPLS can robustly control for confounding eects across a range of scenarios. erefore,
scPLS provides a exible modeling framework that eectively includes the two subcategories of unsupervised
methods as special cases and has the potential to outperform these previous methods.
Simulations. We performed a simulation study to compare scPLS with other methods. Specically, we sim-
ulated gene expression levels for 50 control genes and 1,000 target genes for 200 cells. ese 200 cells come from
two equal-sized groups, representing two treatment conditions or two cell subpopulations. Among the 1,000
target genes, only 200 of them are dierentially expressed (DE) between the two groups and thus represent the
signature of the two groups. e eect sizes of the DE genes were simulated from a normal distribution and we
scaled the eects further so that the group label explains twenty percent of phenotypic variation (PVE) in expres-
sion levels in the DE genes. In addition to the group eects, we set
k k2, 5
zu
==
and simulated each element of
z
i
and
u
i
from a standard normal distribution. We simulated each element of
x
Λ
from
σ−.N(025, )
l
2
and each
element of
y
Λ
from
σ.N(0 25, )
l
2
. Note that
Λ
x
and
Λ
y
were simulated dierently to capture the fact that the eect
sizes of the confounding factors could be dierent for control and target genes. We simulated each element of
Λ
u
from
N(0,)
b
2
σ
. We simulated each element of
ε
xi
and
ε
yi
from a standard normal distribution. We set
σ =.04
l
2
and
σ =.06
b
2
to ensure that, in non-DE genes, the confounding factors
z
i
explain 20% PVE in either the control or the
target genes; the biological factors
u
i
explain 30% PVE of the target genes; and the residual errors to explain the
rest of PVE. To vary signal strength in the data, we also created a series of sub data sets by varying the number of
non DE genes in the data, so that the proportion of variance explained by DE genes in total equal to a xed per-
centage (PDE, in the range of 20–100%, with 10% increments). Aer we simulated gene expression levels, we
further converted these continuous values into count data by using a Poisson distribution: the nal observation
for ith cell and jth gene
c
ij
is from
µ +~c NwPoi( exp( ))
ij ij
, with
w
ij
being the continuous gene expression levels
simulated above and
N 500000, log(10/500000)µ==
, which ensures an average read count of 10. Note that,
because of the residual errors, the resulting count data are over-dispersed with respect to a Poisson distribution.
We considered three dierent simulation scenarios. In scenario I, the confounding factors
z
i
are independent
of group labels. In scenario II, the confounding factors are correlated with group labels. To simulate correlated
data, we simulated each element of
z
i
from
N(0,1)
if the corresponding sample belongs to the rst group, but
from
−.N(025,1)
if the corresponding sample belongs to the second group. Finally, we also considered a scenario
III where there is no biological factor (i.e. data were simulated eectively under the PCA modeling assumption
and all genes could be used to infer the confounding factors). We performed 10 simulation replicates for each
scenario. For scenario I and II, we further introduced dropout events that are commonly observed in scRNAseq
data. is was done by going through one gene at a time and setting the expression level for
j
th gene
c
ij
to zero
with probability
ij
π
that depends on the expression level through
l
og c
ij
1
ij
ij
=
π
π−
.
We compared scPLS to four dierent methods: (1) PCA and (2) LMM (implemented in GEMMA
47,48
) use all
genes to infer the confounding eects; while (3) RUVseq (version 1.2.0); which we simply refer to as RUV in the
following text) and (4) scLVM (version 0.99.1) use only control genes to infer the confounding eects. We note
that while some of these methods are developed not specically for scRNAseq, these methods represent a range
of strategies to deal with confounding factors. We used default settings in all the above methods. We used the
Figure 1. Illustration of scPLS. We model the expression level of genes in the control set
X()
and genes in the
target set
Y()
jointly. Both control and target genes are aected by the common confounding factors (Z) with
eects
Λ
x
and
y
Λ
in the two gene sets, respectively. e target genes are also inuenced by biological factors (U)
with eects
u
Λ
. e biological factors represent intermediate factors that coordinately regulate a set of genes,
and are introduced to better capture the complex variance structure in the target genes.
E
x
and
E
y
represent
residual errors. scPLS aims to remove the confounding eects
ΛZ
y
in the target genes.
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Scientific RepoRts | 7: 13587 | DOI:10.1038/s41598-017-13665-w
count data directly for RUV and used log transformed data (i.e.
+clog(1)
ij
) for all other methods. For PCA and
RUV, we set the number of latent factors to be the true number (i.e. 2). Such number is determined automatically
by the soware itself for scLVM, and is not needed for LMM. We compared dierent methods based on clustering
performance. In particular, for each of these methods, we obtained corrected data and applied k-means method
to cluster cells into two subpopulations. We the compared the clusters inferred from the corrected data with the
truth and used adjusted rand index (ARI) to measure clustering performance. ARI is computed across a range of
signal strength that is measured as PDE explained above. Intuitively, if a method performs well in removing con-
founding factors, then the corrected data from this method can be used to better infer the two cell subpopulations
and thus yields a higher ARI score.
Overall, scPLS performs the best in both scenarios I and II, with or without dropout events (Fig.2a). e
addition of dropout events in either of the two scenarios reduces the performance of all methods but does not
change their relative rank of performance. e superior performance of scPLS also suggests that properly using
both control and target genes can lead to eective removal of confounding eects. Among the rest of the methods,
PCA and LMM performs better than RUV and scLVM, suggesting that target genes contain a substantial amount
of information for removing confounding eects. Beside the comparison of clustering performance, for each gene
in turn, we also used dierent methods to estimate the proportion of gene expression variance contributed by
confounding factors. Consistent with the clustering performance comparison, we found that scPLS also yielded
more accurate proportion of variance estimates (Fig.2b).
To examine the robustness of scPLS, we applied scPLS to the same data but with a reduced number of control
genes (Fig.3a). Because scPLS does not completely rely on the information contained in the control genes, it
achieves robust performance even if we only use a much smaller subset of control genes. We also examined the
performance of scPLS in Scenario III where there is no biological eects (Fig.3b) and found that scPLS performs
well there. As it is oen unknown whether a low-rank structural variation exists in a real data set, our simulation
suggests that we can always include the biological factors
u
i
in the model even in the absence of such factors. In
addition, scPLS is not sensitive with respect to the number of biological factors used in tting the model, and
achieves similar power for a range of reasonable
k
u
values (Fig.3c).
Real Data Applications. Next, we applied scPLS to two real data sets. e rst dataset is used to demonstrate
the eectiveness of scPLS in removing the technical confounding eects by using ERCC spike-ins. Removing
technical confounding eects is a common and important task in transcriptome analysis. e second dataset is
used to demonstrate the eectiveness of scPLS in removing cell cycle eects by using a known set of cell cycle
genes. Removing cell cycle eects can reveal gene expression heterogeneity that is otherwise obscured.
Removing Technical Confounding Factors. e rst dataset consists of 251 samples from
22
. Among
these, 119 are mouse embryonic stem cells (mESCs), including 74 mESCs cultured in a two-inhibitor (2i)
medium and 45 mESCs cultured in a serum medium. e remaining 132 cells are control “cells” that are obtained
by mixing single cells cultured in each condition (i.e. these control “cells” are similar to bulk seq data in terms of
consisting a mixture of cell types, but are prepared and sequenced using single cell protocol). e control cells
include 76 cells cultured in 2i and 56 cells cultured in serum. Because the control cells are homogeneous within
each culture condition, when we cluster these cell, we would expect the only true cluster detectable among these
cells is the culture condition. erefore, we decide to focus on these control cells to compare the performance of
dierent methods for removing technical eects.
We obtained the raw UMI counts data directly from the authors. e data contains measurements for 92
ERCC spike-ins and 23,459 genes. Due to the low coverage of this dataset (median coverage equals one), we
ltered out lowly expressed genes and selected only genes that have at least ve counts and spike-ins that have at
least one count in more than a third of the cells. is ltering step resulted in a total of 32 ERCC spike-ins that
were used as the controls and 2,795 genes that were used as the targets.
As in the simulations, we log transformed the count data and centered the transformed values for scPLS, PCA,
LMM and scLVM. We used the count data for RUV. In this data, scPLS infers
k 1
z
=
confounding factors and
k 1
u
=
biological factors. In the target genes, the confounding factors and structured biological factors explain a
median of 18% and 30% of gene expression variance, respectively. e PVE by the confounding and biological
factors can be as high as 73.7% and 77.9%, respectively, in the target genes.
We applied scPLS and the other four methods to remove confounding eects in the data. Since control cells
are homogeneous within each culture condition, we reasoned that if the method is eective in removing con-
founding eects, then the corrected data from the corresponding method could be used to better reveal two
clusters that correspond to the two known culture conditions. For the clustering analysis, we applied the four
dierent clustering approaches on the uncorrected or corrected data from dierent methods. e four clustering
approaches include: (1) kmeans, where we applied the k-means method directly on the uncorrected or corrected
data; (2) PCA, where we extracted the top ve PCs from either the uncorrected or corrected data and then applied
the k-means method using the top PCs; (3) tSNE, where we used tSNE to either the uncorrected or corrected data
and then applied the k-means method on the extracted tSNE factors; (4) SC3, where we used a recently devel-
oped state-of-the-art single cell clustering method single cell census clustering (SC3)
49
. For all these clustering
approaches, we set the number of clusters to two and measured clustering performance by the adjusted Rand
Index (ARI). e results are shown in Table1 and are overall consistent with the simulations. Specically, scPLS
outperforms the other methods in three out of the four clustering approaches. scPLS performs slightly worse
than RUV when tSNE was used to cluster data–but tSNE works extremely poorly in this data presumably because
tSNE’s non-linearity assumption does not t the data well.