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Joint betweensample normalization and differential expression detection through ℓ_{0}regularized regression
BMC Bioinformatics volume 20, Article number: 593 (2019)
Abstract
Background
A fundamental problem in RNAseq data analysis is to identify genes or exons that are differentially expressed with varying experimental conditions based on the read counts. The relativeness of RNAseq measurements makes the betweensample normalization of read counts an essential step in differential expression (DE) analysis. In most existing methods, the normalization step is performed prior to the DE analysis. Recently, Jiang and Zhan proposed a statistical method which introduces samplespecific normalization parameters into a joint model, which allows for simultaneous normalization and differential expression analysis from logtransformed RNAseq data. Furthermore, an ℓ_{0} penalty is used to yield a sparse solution which selects a subset of DE genes. The experimental conditions are restricted to be categorical in their work.
Results
In this paper, we generalize Jiang and Zhan’s method to handle experimental conditions that are measured in continuous variables. As a result, genes with expression levels associated with a single or multiple covariates can be detected. As the problem being highdimensional, nondifferentiable and nonconvex, we develop an efficient algorithm for model fitting.
Conclusions
Experiments on synthetic data demonstrate that the proposed method outperforms existing methods in terms of detection accuracy when a large fraction of genes are differentially expressed in an asymmetric manner, and the performance gain becomes more substantial for larger sample sizes. We also apply our method to a real prostate cancer RNAseq dataset to identify genes associated with preoperative prostatespecific antigen (PSA) levels in patients.
Introduction
A fundamental problem in RNAseq data analysis is to identify genes or exons that are differentially expressed with varying experimental conditions based on the read counts. Some widely used methods for differential expression analysis in RNAseq data are edgeR [1, 2], DESeq2 [3] and limmavoom [4, 5]. In edgeR and DESeq2, the read counts are assumed to follow negative binomial (NB) distributions; while in limmavoom, the logarithmic transformation is taken on the data which compresses the dynamic range of the read counts so that the outliers become more “normal”. Consequently, existing statistical methods that are designed for analyzing normally distributed data can be employed to analyze RNAseq data.
Due to the relative nature of RNAseq measurements for transcript abundances as well as differences in library sizes and sequencing depths across samples [6], betweensample normalization of read counts is essential in differential expression (DE) analysis with RNAseq data. A widely used approach for data normalization in RNAseq is to employ a samplespecific scaling factor, e.g., CPM/RPM (counts/reads per million) [7], upperquartile normalization [8], trimmed mean of M values [7] and DESeq normalization [9]. A review of normalization methods in RNAdata data analysis is given in [6]. In most existing methods for DE analysis in RNAseq, the normalization step is performed prior to the DE detection step, which is suboptimal because ideally normalization should be based on nonDE genes for which the complete list is unknown until after the DE analysis.
In [10], a statistical method for robust DE analysis using logtransformed RNAseq data is proposed, where samplespecific normalization factors are introduced as unknown parameters. This allows for more accurate and reliable detection of DE genes by simultaneously performing betweensample normalization and DE detection. An ℓ_{0} penalty is introduced to enforce that a subset of genes are selected as being differentially expressed. The experimental conditions are restricted to be categorical (e.g., 0 and 1 for control and treatment groups, respectively), and a oneway analysis of variance (ANOVA) type model is employed to detect differentially expressed genes across two or more experimental conditions.
In [11], the model of [10] is generalized to continuous experimental conditions, and the sparsityinducing ℓ_{0} penalty is relaxed as the ℓ_{1} penalty. An alternating direction method of multipliers (ADMM) algorithm is developed to solve the resultant convex problem. Due to the relaxation of the ℓ_{0} regularization, the method in [11] may not be as robust against noise and efficient in inducing sparse solutions as that in [10]. In this paper, we again generalize the model in [10] from categorical to continuous experimental conditions. But different from [11], we retain the ℓ_{0} penalty in our model to efficiently induce sparsity. We formulated two hypothesis tests suited to different applications: the first hypothesis test is that considered in [10] and answers the question of whether the expression of a gene is significantly affected by any covariate; and in addition, a second hypothesis is formulated to test whether the expression of a gene is significantly affected by a particular covariate, when all other covariates in the regression model are adjusted for.
Due to the use of the ℓ_{0} penalty, the resulting problem is highdimensional, nondifferentiable and nonconvex. To fit the proposed model, we study the optimality conditions of the problem and develop an efficient algorithm for its solution. We also propose a simple rule for the selection of tuning parameters. Experiments on synthetic data demonstrate that the proposed method outperforms existing ones in terms of detection accuracy when a large fraction of genes are differentially expressed in an asymmetric manner, and the performance gain becomes more substantial for larger sample sizes. We also apply our method to a real prostate cancer RNAseq dataset to identify genes associated with preoperative prostatespecific antigen (PSA) levels in patients.
Methods
Given m genes and n samples, let y_{ij},i=1,…,m,j=1,…,n, be the logtransformed gene expression values of the ith gene in the jth sample. A small positive constant can be added prior to taking the logarithm to avoid taking logarithm of zeros. We formulate the following model:
where α_{i} is the intercept,
is the regression coefficient vector of the linear model for gene i, and
is a vector of p predictor variables for sample j representing its experimental conditions (drug dosage, blood pressure, age, BMI, etc.), d_{j} represents the normalization factor for sample j, and \(\varepsilon _{ij} \sim \mathcal {N}\left (0,\sigma _{i}^{2}\right)\) is i.i.d. Gaussian noise. Our goal is for each gene to determine whether its expression level is significantly associated with the experimental conditions or not.
Remark 1
The α_{i} and d_{j} in (1) model genespecific factors (e.g., gene length) and samplespecific factors (i.e., sequencing depth), respectively. Thus, model (1) can accommodate any gene expression levels summarized in the form of c_{ij}/(l_{i}·q_{j}), where c_{ij} is the read count, l_{i} is the genespecific scaling factor (e.g., gene length) associated with gene i and q_{j} is the samplespecific scaling factor (e.g., sequencing depth) associated with sample j. Special cases are read count (i.e., l_{i}=q_{j}=1), CPM/RPM (i.e., l_{i}=1) [7], RPKM/FPKM [12,13] and TPM [14].
Since the random noise in gene expression measurements are independent across genes and samples, the likelihood is given by
The negative loglikelihood is
where C depends on \(\left \{\sigma _{i}^{2}\right \}\) but not on {α_{i}},{β_{i}} and {d_{j}}. In “Maximum likelihood estimation of noise variance” section, we will describe how to estimate \(\sigma _{i}^{2}, i=1,\dots,m\). Hereafter, we assume that \(\sigma _{i}^{2}\)’s are known and simply denote the negative loglikelihood as \(l\left (\boldsymbol {\alpha }, \{\boldsymbol {\beta }_{i}\}_{i=1}^{m}, \boldsymbol {d}; \boldsymbol {Y} \right)\).
In practice, typically only a subset of genes are differentially expressed. We introduce a sparse penalty to the negative log likelihood function:
where λ_{i}’s are tuning parameters that control the sparsity level of the solution, and p(β_{i}) is a penalty function
In this paper, we use the following two types of penalty functions.
 i)
Type I penalty:
$$ p\left(\boldsymbol{\beta}_{i}\right) = 1_{\boldsymbol{\beta}_{i} \neq \boldsymbol{0}}. $$(7)This penalty function applies to applications where all covariates are of interest and we want to identify genes for which at least one covariate is associated with its expression.
 ii)
Type II penalty:
$$ p\left(\boldsymbol{\beta}_{i}\right) = 1_{\beta_{ip} \neq 0}. $$(8)This penalty applies to applications where only one (the pth) covariate is of main interest (e.g., treatment) while we want to adjust for all other covariates (e.g., age, sex, etc).
Algorithm development
Note that without d_{j}, model (1) would be decoupled as m independent linear regression models, one for each gene. The first step of our algorithm is to solve for d_{j} and express it as a function of β_{i}’s.
Note that the optimization problem (6) is convex in (α,d). Therefore, the minimizer of (α,d) is one of its stationary points.
Taking partial derivatives of \(f\left (\boldsymbol {\alpha }, \{\boldsymbol {\beta }_{i}\}_{i=1}^{m}, \boldsymbol {d}\right)\) with respect to d_{j},j=1,…,n, and setting them to zeros, we have
The solution to model (1) is not unique because an arbitrary constant can be added to d_{j}’s and subtracted from α_{i}’s, while having the same model fit. To address this issue, we fix d_{1}=0. Therefore
where
Here the superscript ^{(w)} denotes “weighted mean".
Calculating the partial derivatives of \(f\left (\boldsymbol {\alpha }, \{\boldsymbol {\beta }_{i}\}_{i=1}^{m}, \boldsymbol {d}\right)\) with respect to α_{i},i=1,…,m, and setting them to zeros, we have
where
From (10) it follows
where
Substituting (16) into (13) yields
The sum of (10) and (18) yields
Substituting (19) into (6), the problem becomes an ℓ_{0}regularized linear regression problem with \(\{\boldsymbol {\beta }_{i}\}_{i=1}^{m}\) being the only variables to be optimized:
where
It is easy to see that
In the next two sections, we will describe algorithms to solve Problem (20) with type I and type II penalties, respectively.
Fitting the model with type I penalty
Denote \(\boldsymbol {\delta }=\boldsymbol {\bar {\beta }}^{(w)}\), and let
where β_{i}’s are considered as functions of δ. The objective in Problem (20) can be written as \(f\left (\boldsymbol {\beta }\right) = {\sum _{i=1}^{m}} g_{i}\left (\boldsymbol {\beta }_{i}\right)\). Assume that δ is fixed, f can be minimized by minimizing each g_{i}(β_{i}) separately.
Next we express the minimizing solution of g_{i}(β_{i}) as a function of δ.
When β_{i}=0,
When β_{i}≠0,
Taking partial derivatives of (26) with respect to β_{i},i=1,…,m, and setting them to zeros yields
where the superscript ^{(ols)} indicates an ordinary least squares estimate for the model,
and \(\boldsymbol {\tilde {y}}_{i}\) is a column vector containing the centered expression of gene i in all samples, i.e., the ith row of \(\boldsymbol {\tilde {Y}}\):
The objective function value at \(\boldsymbol {\beta }_{i}=\boldsymbol {\beta }_{i}^{\text {(ols)}}\) is
The change in the objective value g_{i}(β_{i}) from \(\boldsymbol {\beta }_{i}=\boldsymbol {\beta }_{i}^{\text {(ols)}} \neq 0\) in Eq. (30) to β_{i}=0 in Eq. (25) is
Therefore, the solution is
Now we only need to solve for δ. We have
where the second equality is due to the fact that \(g_{i}\left (\boldsymbol {\beta }_{i}^{\text {(ols)}}\right)\) is a constant independent of δ, and the third equality follows from (31). Problem (33) can be solved exactly using an exhausted grid search for p=1 or 2, and approximately using a general global optimization algorithm (e.g., the optim function in R) for larger p. A more efficient algorithm proposed in [15] can also be used.
After we obtain the estimate of δ, we substitute it into (32) to get the estimate of β_{i}. Algorithm 1 describes the complete model fitting procedure.
Fitting the model with type II penalty
Denote \(\boldsymbol {\delta }=\boldsymbol {\bar {\beta }}^{(w)}\), and let
where β_{i}’s are considered as functions of δ. The objective function in Eq. (20) is \(f\left (\boldsymbol {\beta }\right) \,=\, {\sum \limits _{i=1}^{m}} h_{i}\left (\boldsymbol {\beta }_{i}\right)\). Assume that δ is fixed, f can be optimized by minimizing each h_{i}(β_{i}) separately.
Next we find the solution for β_{i}’s as a function of δ by minimizing h_{i}(β_{i}).
Denote
When β_{ip}=0,
Taking derivatives of (35) with respect to \(\boldsymbol {\beta }_{i}^{}, i=1,\ldots,m,\) and setting them to zeros yields
where
Denote \(\boldsymbol {\beta }_{i}^{\text {(r)}} = \left [\begin {array}{c} \boldsymbol {\beta }_{i}^{} \\ 0 \\ \end {array}\right ]\), where the superscript ^{(r)} denotes the reduced model. Substituting \(\boldsymbol {\beta }_{i} = \boldsymbol {\beta }_{i}^{\text {(r)}}\) into (35) and after some matrix algebraic manipulation, we have
When β_{ip}≠0,
The minimizing solution of h_{i}(β_{i}) is \(\boldsymbol {\beta }_{i}^{\text {(ols)}}\) shown in (27), and its pth coordinate is
where the second equality follows from \(\boldsymbol {\tilde {X}}= \begin {bmatrix} \boldsymbol {\tilde {X}}^{} & \boldsymbol {x}^{p} \\ \end {bmatrix}\) and the inverse formula for the partitioned matrix of \(\boldsymbol {\tilde {X}}^{\mathrm {T}} \boldsymbol {\tilde {X}}\). The value of h_{i}(β_{i}) at \(\boldsymbol {\beta }_{i}=\boldsymbol {\beta }_{i}^{\text {(ols)}}\) is
The decrease in the objective value from β_{ip}=0 in Eq. (38) to β_{ip}≠0 in Eq. (41) is
where the second equality employs the following equality:
which is obtained by partitioning \(\boldsymbol {\tilde {X}}^{\mathrm {T}} \boldsymbol {\tilde {X}}\) into a 2×2 block matrix and then substituting the formula for its inverse, and \(\beta _{ip}^{\text {(ols)}}\) is defined in Eq. (40).
Therefore, the solution is
Now we only need to solve for δ_{p}. We have
where the second equality is due to the fact that \(h_{i}\left (\boldsymbol {\beta }_{i}^{\text {(ols)}}\right)\) is a constant independent of δ.
Substituting (42) into (44) yields
where the \(\beta _{ip}^{\text {(ols)}} \left (\delta _{p}\right)\) as a function of δ_{p} is defined in Eq. (40).
After \(\hat {\delta }_{p}\) is estimated, the estimate of β_{ip} is obtained by substituting \(\delta _{p}=\hat {\delta }_{p}\) into (43). Algorithm 2 describes the complete model fitting procedure.
Next, we introduce a simple method for the selection of the tuning parameters in our model, which is based on the property of the solution (32) or (43).
Tuning parameter selection: regression with type I penalty
Substituting (19) into (1) and assuming that \(\boldsymbol {\delta }=\boldsymbol {\bar {\beta }}^{(w)}\) is fixed, we have
where \(\tilde {y}_{ij} + \boldsymbol {\delta }^{\mathrm {T}} \boldsymbol {\tilde {x}}_{j} \) are the normalized data, which we use here as the response variables, and \(\varepsilon _{ij} \sim \mathcal {N}\left (0,\sigma _{i}^{2}\right)\).
The condition for β_{i}=0 in (32) can be rewritten as
Under the null hypothesis, β_{i}=0; the lefthand side of (47) follows a chisquared distribution with p degrees of freedom, i.e., \(\chi _{p}^{2}\). This suggests us choose λ_{i}=1/2·F^{−1}(1−q; p)=1/2·{x:F(x; p)=1−q}, where F(x; p) is the cumulative distribution function of \(\chi _{p}^{2}\), and q is a prespecified significance level.
Tuning parameter selection: regression with type II penalty
Let \(\tilde {y}_{ij} + \boldsymbol {\delta }^{\mathrm {T}} \boldsymbol {\tilde {x}}_{j} \) denote the normalized data:
where \(\varepsilon _{ij} \sim \mathcal {N}\left (0,\sigma _{i}^{2}\right)\).
The condition for β_{ip}=0 in (43) can be rewritten as
where \(\beta _{ip}^{\text {(ols)}}\) is defined in (40) and
is the standard error of the estimate \(\beta _{ip}^{\text {(ols)}}\).
Under the null hypothesis, β_{ip}=0; the lefthand side of (49) follows the standard Gaussian distribution. This suggests us choose λ_{i}=1/2·[Φ^{−1}(1−q/2)]^{2}, where Φ(·) is the cumulative distribution function of the standard Gaussian distribution, and q is a prespecified significance level.
Maximum likelihood estimation of noise variance
To estimate \(\sigma _{i}^{2}, i=1,\dots,m\), consider the negative loglikelihood function with \(\sigma _{i}^{2}\)’s being unknown as well:
Setting partial derivatives of l(·) with respect to α_{i},β_{i},i=1,…,m, and d_{j},j=1,…,n to zeros, and after some mathematical manipulation, we obtain
where \(\boldsymbol {\tilde {X}}, \boldsymbol {\tilde {x}}_{j}\) and \(\bar {y}_{\cdot j}^{(w)}\) are defined in (28), (22) and (11), respectively.
Taking partial derivatives of l(·) with respect to \(\sigma _{i}^{2}, i=1,\ldots,m,\) and setting them to zeros gives
Substituting (19) into (52), we have
where \(\bar {y}_{i\cdot }\) and \(\bar {y}^{(w)}\) are as defined in (14) and (17), respectively.
Given initial estimates of \(\sigma _{i}^{2}\)’s and \(\boldsymbol {\bar {\beta }}^{(w)}, \{\boldsymbol {\beta }_{i}\}, \{\sigma _{i}^{2}\}\) and \(\boldsymbol {\bar {\beta }}^{(w)}\) can be updated alternately using Eqs. (51), (53), and (12) until convergence.
After \(\{\sigma _{i}^{2}\}_{i=1}^{m}\) are estimated, they can be robustified using a “shrinkage toward the mean” scheme [16]:
where
The noise variance estimates \(\hat {\hat {\sigma }}_{i}^{2}, i=1,\dots,m\), can then be used in Algorithm 1 or 2 to solve for \(\{\boldsymbol {\beta }_{i}\}_{i=1}^{m}\).
Remark 2
Note that when \( \sigma _{1}^{2} = \sigma _{2}^{2} = \dotsb = \sigma _{m}^{2} = \sigma ^{2} \), it is no longer needed to estimate σ^{2} since σ^{2} in (6) can be incorporated into the tuning parameters λ_{i}’s.
Results and discussion
We demonstrate the performance of our proposed method (named rSeqRobust) by comparing it with other existing methods for DE gene detection from RNAseq data: edgeRrobust [1,2], DESeq2 [3], limmavoom [4,5], and ELMSeq (which fits a similar model but with ℓ_{1} rather than ℓ_{0} penalty) [11]. We consider the simple regression model (p=1), in which case Algorithms 1 and 2 coincide. For ELMSeq, the tuning parameter is set as the 5th percentile of m values: \(\left \frac {1}{\hat {\sigma }_{1}^{2}} \boldsymbol {\tilde {x}}^{\mathrm {T}} \boldsymbol {\tilde {y}}_{1}\right , \left \frac {1}{\hat {\sigma }_{2}^{2}} \boldsymbol {\tilde {x}}^{\mathrm {T}} \boldsymbol {\tilde {y}}_{2}\right , \dots, \left \frac {1}{\hat {\sigma }_{m}^{2}} \boldsymbol {\tilde {x}}^{\mathrm {T}} \boldsymbol {\tilde {y}}_{m}\right \) [11]. The tuning parameters λ is set based on the significant level of q=0.01.
Simulations on synthetic data
We simulate both lognormally distributed and negativebinomially distributed read counts, with m=20,000 genes and n=7, 20 or 200 samples. The RNAseq read counts are generated as \(c_{ij} \sim \lceil \mathrm {e}^{\mathcal {N}(\log \mu _{ij},\sigma _{i}^{2})}\rceil \) under the lognormal (LN) distribution assumption, and as \(c_{ij} \sim \mathcal {NB}(\mu _{ij},\phi _{i})\) [2] under the NB distribution assumption, where μ_{ij} is the mean read counts of gene i in sample j. The generation of μ_{ij} is described in Table 1. The variance of the normal distribution is set as \(\sigma _{i}^{2}=0.01\), and the dispersion parameter of the NB distribution is set as ϕ_{i}=0.25.
After estimating the samplespecific normalization factors d_{j}’s using Algorithm 1, we substitute \(\hat {d}_{j}\)’s into model (1) to obtain m decoupled genewise linear regression models. For each gene i, we test the null hypothesis that β_{i}=0, and calculate the pvalue. We decide there is a significant linear association between the experimental variable x_{j} and the gene expression y_{ij} if the pvalue is less than a predefined threshold (e.g., 0.05). With the pvalue for each gene, we rank the genes and vary the pvalue threshold from 0 to 1 to determine significant DE genes and calculate the area under the ROC curve (AUC).
Table 2 shows the AUCs of the five methods on lognormally distributed data with sample size n=20. We observe the followings: i) In relatively easy scenarios where a small percent of genes are differentially expressed (i.e., DE% =1% or 10%) or the up and downregulated genes are equal in portions (i.e., Up% =50%), all five methods perform equally well (within one standard error of AUC difference); ii) In challenging scenarios where a large percent of genes are differentially expressed (i.e., DE% ≥30%) and the up and downregulated genes are different in portions (i.e., Up% =75% or 100%), rSeqRobust outperforms ELMSeq, which in turn outperforms the rest; iii) In the most challenging scenarios with 70% or 90% DE genes that are all overexpressed (i.e., Up% =100%), only rSeqRobust achieves good results (AUC=0.9638 or 0.8276).
Table 3 shows the AUCs of different methods for negativebinomially distributed data. The same observations are obtained as in Table 2: In relatively easy settings with small percent of DE genes or symmetric over and underexpression pattern, rSeqRobust performs as well as other methods; In challenging settings with large percent of DE genes (DE% ≥10%) and asymmetric over and underexpression pattern (Up% =75% or 100%), rSeqRobust consistently performs the best, and ELMSeq ranks second in all except extreme cases: (Up%, DE%) =(70%, 100%), (90%, 75%) or (90%, 100%) where most methods suffer from severe performance degradation or complete failure.
In Tables 4 and 5, the sample size is reduced to n=7. Again, we observe similar patterns: when a small subset of genes are differentially expressed (i.e., DE% =1% or 10%), or the up and downregulated DE genes are imbalanced in numbers, rSeqRobust and other methods perform equally well; when most genes are differentially expressed (i.e., DE% = 50% or 70%) in an asymmetric manner (i.e., Up% =75% or 100%), rSeqRobust outperforms all other methods. Note that in the presence of 70% DE genes that are all upregulated, only rSeqRobust achieves good results (AUC =0.9059 for LN data and AUC =0.8623 for NB data).
In Tables 6 and 7, the sample size is increased to n=200. As the sample size increases from n=20 to n=200, the AUCs of edgeRrobust, DESeq2 and limmavoom increase for easy cases (small percent of DE genes or symmetric over and underexpression patterns). However, for challenging cases (i.e., DE% =50%, 70% or 90%, Up% =75% or 100%), the AUCs decrease. On the contrary, the AUC of rSeqRobust increases consistently in all cases. The performance gain of rSeqRobust over other methods is more significant for more challenging cases. Note that rSeqRobust performs nearly as well in the most challenging cases (Up%, DE%) =(50%, 100%), (70%, 100%), (90%, 75%) or (90%, 100%) as in easy cases. In contrast, ELMSeq only works for (Up%, DE%) =(50%, 100%) and edgeRrobust, DESeq2, limmavoom completely fail in all these cases. This indicates that rSeqRobust is more robust than ELMSeq, which in turn is more robust than edgeRrobust, DESeq2 and limmavoom.
Table 8 shows the average running times (in seconds) of the five methods on an Intel Core i3 processor with 8GB of memory and a clock frequency of 3.9GHz. We can see that rSeqRobust is slower than limmavoom; however, it scales well for large sample sizes and is much faster than ELMSeq.
Application to a real RNAseq dataset
We further assess the proposed method on a real RNAseq dataset from The Cancer Genome Atlas (TCGA) project [17], which contains 20,531 genes from 187 prostate adenocarcinoma patient samples. The dataset was downloaded from the TCGA data portal (https://portal.gdc.cancer.gov). In this experiment, we aim at identifying genes associated with preoperative prostatespecific antigen (PSA), which is an important biomarker for prostate cancer. The data are preprocessed using the procedures described in [11]. We use the Bonferroni correction and determined DE genes using a pvalue threshold of 0.05/m. Figure 1 shows the Venn diagram based on the sets of differentially expressed genes discovered by five methods.
There are twelve genes that are detected by rSeqRobust and ELMSeq, but not by edgeR, DESeq2 and limmavoom: EPHA5, RNF126P1, BCL11A, RIC3, AJAP1, CDH3, WIT1, PRSS16, CEACAM1, DCHS2, CRHR1 and SRD5A2. For the majority of these twelve genes, there are existing publications reporting their associations with prostate cancer. For instance, EPHA5 is known to be underexpressed in prostate cancer [18]. CEACAM1 is known to suppress prostate cancer cell proliferation when overexpressed [19]. Two of the twelve genes, CRHR1 and SRD5A2, are identified only by rSeqRobust, where SRD5A2 is associated with racial/ethnic disparity in prostate cancer risk [20].
There are twelve genes that are detected by all five methods: KANK4, RHOU, TPT1, SH2D3A, EEF1A1P9, ZCWPW1, ZNF454, RACGAP1, PTPLA, POC1A, AURKA and TIMM17A. Similarly, there are existing publications reporting their associations with prostate cancer. For instance, RHOU is associated with the invasion, proliferation and motility of prostate cancer cells [21].
Conclusion & discussion
In this paper, we present a unified statistical model for joint normalization and differential expression detection in RNAseq. Different from existing methods, we explicitly model samplespecific normalization factors as unknown parameters, so that they can be estimated simultaneously together with detection of differentially expressed genes. Using an ℓ_{0}regularized regression approach, our method is robust against large proportion of DE genes and asymmetric DE pattern, and is shown in empirical studies to be more accurate in detecting differential gene expression patterns.
This model generalizes [10] from categorical experimental conditions using an ANOVAtype model to continuous covariates using a regression model. In addition, two hypothesis tests are formulated: i) Is the expression level of a gene associated with any covariates of the regression model? This is the test considered in [10]; ii) Is the expression level of a gene associated with a specific covariate of our interest, when all other variables in the regression model are adjusted for? Although the model is highdimensional, nondifferentiable and nonconvex due to the ℓ_{0} penalty, we manage to develop an efficient algorithms to find their its solution by making use of the optimality conditions of the ℓ_{0}regularized regression. It can be shown that for categorical experimental data, the developed algorithm for the first hypothesis test for the slopes in a regression model with p binary covariates reduces to that in [10] for the (p+1)group model.
Availability of data and materials
The computer programs for the algorithms described in this paper are available at http://wwwpersonal.umich.edu/~jianghui/rseqrobust.
Abbreviations
 ANOVA:

Analysis of variance
 AUC:

Area under the ROC curve
 CPM/RPM:

Counts/reads per million
 DE:

Differential expression
 LN:

Lognormal
 NB:

Negative binomial
 PSA:

prostatespecific antigen
 RNAseq:

RNA sequencing
 ROC:

Receiver operating characteristic
 RPKM/FPKM:

Reads/fragments per kilobase of exon per million mapped reads
 TCGA:

The Cancer Genome Atlas
 TPM:

Transcripts per million
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This article has been published as part of BMC Bioinformatics Volume 20 Supplement 16, 2019: Selected articles from the IEEE BIBM International Conference on Bioinformatics & Biomedicine (BIBM) 2018: bioinformatics and systems biology. The full contents of the supplement are available online at https://bmcbioinformatics.biomedcentral.com/articles/supplements/volume20supplement16.
Funding
HJ’s research was supported in part by a startup grant from the University of Michigan and the National Cancer Institute grants P30CA046592 and P50CA186786. The publication costs are funded by the University of Michigan (HJ’s startup fund). The funding body(s) played no role in the design or conclusions of the study.
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HJ conceived the study and designed the experiments. KL developed the algorithms and performed the experiments. KL, HJ and LS wrote the paper. All authors read and approved this version of the manuscript.
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Correspondence to Hui Jiang.
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Liu, K., Shen, L. & Jiang, H. Joint betweensample normalization and differential expression detection through ℓ_{0}regularized regression. BMC Bioinformatics 20, 593 (2019) doi:10.1186/s1285901930704
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Keywords
 Differential expression
 Betweensample normalization
 ℓ _{0}regularized regression
 RNAseq