Linear combination test for gene set analysis of a continuous phenotype
 Irina Dinu^{1}Email author,
 Xiaoming Wang^{1, 2},
 Linda E Kelemen^{3, 4},
 Shabnam Vatanpour^{1} and
 Saumyadipta Pyne^{5, 6}
DOI: 10.1186/1471210514212
© Dinu et al.; licensee BioMed Central Ltd. 2013
Received: 5 March 2013
Accepted: 13 June 2013
Published: 1 July 2013
Abstract
Background
Gene set analysis (GSA) methods test the association of sets of genes with a phenotype in gene expression microarray studies. Many GSA methods have been proposed, especially methods for use with a binary phenotype. Equally, if not more importantly however, is the ability to test the enrichment of a gene signature or pathway against the continuous phenotypes which are routinely and commonly observed in, for example, clinicopathological measurements. It is not always easy or meaningful to dichotomize continuous phenotypes into two classes, and attempting to do this may lead to the inaccurate classification of samples, which would affect the downstream enrichment analysis. In the present study, we have build on recent efforts to incorporate correlation structure within gene sets and pathways into the GSA test statistic. To address the issue of continuous phenotypes directly without the need for artificial discrete classification and thus increase the power of the test while ensuring computational efficiency and rigor, new GSA methods that can incorporate a covariance matrix estimator for a continuous phenotype may present an effective approach.
Results
We have designed a new method by extending the GSA approach called Linear Combination Test (LCT) from a binary to a continuous phenotype. Simulation studies and a real microarray dataset were used to compare the proposed LCT for a continuous phenotype, a modification of LCT (referred to as LCT_{2}), and two publicly available GSA methods for continuous phenotypes.
Conclusions
We found that the LCT methods performed better than the other two GSA methods; however, this finding should be understood in the context of our specific simulation studies and the real microarray dataset that were used to compare the methods. Free Rcodes to perform LCT for binary and continuous phenotypes are available at http://www.ualberta.ca/~yyasui/homepage.html. The Rcode to perform LCT for a continuous phenotype is available as Additional file 1.
Background
Gene set enrichment analysis (GSEA) has greatly advanced highthroughput gene expression studies and a number of methods have been proposed to perform this type of analysis (see reviews by Goeman and Buhlmann [1] and Nam and Kim [2]). An important methodological challenge of GSA is the need to deal with large gene sets and small sample sizes. While most GSA methods employ a permutationbased approach to evaluate the significance gene sets, Kim and Volsky [3] gave a parametric view of the test statistic by assuming that the averages of fold changes across the genesets are distributed approximately normally. However, the majority of work in this field has focused on testing the enrichment of gene sets against binary, and sometimes categorical, phenotypes. Equally, if not more importantly, is the ability of the method to test the enrichment of a gene signature or a molecular pathway against a continuous phenotype. Such continuous variables are measured routinely and many important clinicopathological observations such as tumor size or the measurement of marker proteins are continuous. It may not always be technically easy or meaningful to categorize continuous phenotypes into two or more discrete classes. Indeed such artificial categorization may lead to inaccurate classification of the samples, which will eventually affect the downstream enrichment analysis.
We observed an important methodological distinction between the competitive and selfcontained GSA approaches [1, 4]. Competitive methods use gene permutation to test whether or not the association of the phenotype with a gene set is similar to its association with the other gene sets (the “Q1 hypothesis”), while selfcontained methods employ sample permutation to test the equality of the two mean vectors of geneset expressions which correspond to the two phenotype groups (the “Q2 hypothesis”). Here, we focused on the selfcontained methods because, unlike the gene permutation approaches, sample permutation preserves correlations within gene sets; a property that we have used to design the proposed method for continuous phenotypes.
Correlations among gene expression measurements have long been observed, especially among measurements for functionally related gene set. Yet in the past, only the multivariate analysis of variance test for gene set analysis (MANOVAGSA) for categorical phenotypes [5] and the Linear Combination Test (LCT) for binary phenotypes [6] have used a covariance matrix estimator of gene expressions to compute the enrichment test statistic. The main challenges in using these methods are the relatively small sample sizes and large gene sets; a situation which is not uncommon in GSA, especially in small microarray studies. To overcome these difficulties, shrinkage methods [7] have been used to estimate the gene expressions covariance matrix. However, GSA has rarely been used for continuous phenotypes, and currently no methods that incorporate a covariance matrix estimator are available. Previously, when we compared the performances of various selfcontained GSA methods for binary phenotypes, we found that LCT was more computationally efficient than MANOVAGSA and approximated its superior power very well. Here, we propose both an extension of LCT to continuous phenotype (hereafter referred to as LCT) and a modified version of LCT (hereafter referred to as LCT_{2}). We compared the proposed methods with two existing GSA methods for continuous phenotype; namely, Significance Analysis of Microarrays for Gene Sets (SAMGS) [8] and Global Test [9]. We used simulations to compare the performances of the GSA methods with small sample sizes and large gene sets.
In addition, we analyzed the performances of the GSA methods using real microarray gene expression data from prostate tumor samples of AfricanAmerican prostate cancer patients [10]. Increased plasma or serum leptin levels have previously been found to be associated with development of prostate cancer [1113]. We, therefore, used the C2 catalog, an extensive collection of metabolic and signaling pathways and gene sets, from the Molecular Signatures Database (MSigDB) of Gene Set Enrichment Analysis application of Broad Institute of MIT and Harvard. The catalog was screened for associations with human leptin gene (LEP) expression, a wellstudied marker of adiposity, and various metabolic and inflammatory conditions, and we identified important molecular pathways that were associated with high expression of this marker in a prostate cancer cohort. In our comparative study, we focused on testing both the power and computational efficiency of the four GSA methods.
Methods
Linear combination test for a continuous phenotype
In this section we derive the LCT for a continuous phenotype. Our derivations follow the binary phenotype framework in that the correlation structure is accommodated in a similar way to the binary phenotype, and the shrinkage covariance matrix estimation is implemented to take care of the small sample size and large gene set problems.
Where A=Cov_{ Y,X }Cov_{ Y,X }^{ T } and $\mathit{{\rm B}}\mathit{=}\widehat{\mathbf{\Omega}}$. Thus, the solution to this optimization problem is the maximal eigenvector of AB^{1} and ${\rho}_{Y,Z}^{2}{}_{\left(\mathit{\beta}\mathit{*}\right)}$ is the corresponding eigenvalue [14].
When the size of the gene set is larger than the sample size (i.e., p > n) the matrix B is singular. Similar to the adjustment implemented in MANOVAGSA [5], a possible remedy for the singularity is to employ a shrinkage covariance matrix as proposed previously by Schafer and Strimmer [7]. Thus, the singular covariance matrix $\widehat{\mathbf{\Omega}}\phantom{\rule{0.25em}{0ex}}$ can be replaced with a shrinkage covariance matrix $\widehat{\mathbf{\Omega}}*$ given by ${\omega}_{\mathit{hh}\text{'}}^{*}={\rho}_{\mathit{hh}\text{'}}^{*}\sqrt{{\omega}_{\mathit{hh}}{\omega}_{\mathit{hh}\text{'}}}$ with shrinkage coefficients ${\rho}_{\mathit{hh}\text{'}}^{*}=1$ if h=h'and ${\rho}_{\mathit{hh}\text{'}}^{*}={\rho}_{\mathit{hh}\text{'}}min\left\{1\mathrm{max}\left(0,1\widehat{\lambda}\text{'}\right)\right\}$, if h≠h' where ρ_{hh '} is the sample correlation between the h th and h' th genes, and the optimal shrinkage intensity ${\widehat{\lambda}}^{*}$ can be estimated by ${\widehat{\lambda}}^{*}={\displaystyle \sum _{h\ne h\text{'}}\mathit{Var}(}{\rho}_{\mathit{hh}\text{'}})/{\displaystyle \sum _{h\ne h\text{'}}{\rho}_{\mathit{hh}\text{'}}{}^{2}}$.
where c is a constant. The statistical significance can be evaluated against the null hypothesis with a permutation test (permuting phenotype labels) using this test statistic. The constant c can be ignored in the permutation test. This approach is advantageous computationally because ${\widehat{\mathbf{\Omega}}}^{\mathbf{*}}=\mathit{UD}{\mathit{U}}^{T}$ is evaluated only once for the original data, and then there is no need to evaluate it for each permuted version of the data.
A modification of the linear combination test for a continuous phenotype
We also considered an alternative form of LCT (LCT_{2}) which we derived in the linear regression context. A least squares estimate of the regression function is given by $\widehat{f}=X{\left({X}^{T}X\right)}^{1}{X}^{T}Y$ where X represents the gene expression matrix and Y represents the vector of phenotype values. In case of singularities, a shrinkage version of the regression function estimate analogous to LCT can be obtained. An alternative version of LCT, can be derived as the square of the L_{ 2 } norm of the shrinkage regression function $\mathrm{LC}{\mathrm{T}}_{2}=\left\right\widehat{f}{}_{2}^{2}$.
Simulation study design
Type I errors for four GSA methods: LCT, LCT _{ 2 } , SAMGS and GlobalTest
Type I  0.005  0.01  0.05  

n  10  10  10  
p  20  20  20  
p1 = p2  5  5  5  
Method  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9 
LCT  .005  .003  .003  .002  .012  .012  .008  .009  .045  .049  .044  .052 
LCT_{2}  .006  .007  .004  .003  .013  .013  .009  .009  .048  .057  .048  .046 
SAMGS  .004  .005  .006  .004  .014  .012  .007  .008  .044  .051  .048  .050 
Global  .007  .008  .011  .011  .013  .019  .0117  .015  .053  .052  .054  .053 
n  20  20  20  
p  100  100  100  
p1 = p2  20  20  20  
Method  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9 
LCT  .010  .009  .010  .010  .012  .014  .014  .016  .056  .053  .051  .058 
LCT_{2}  .011  .011  .009  .007  .014  .017  .017  .014  .056  .055  .055  .054 
SAMGS  .010  .011  .010  .007  .012  .017  .018  .014  .056  .053  .052  .051 
Global  .004  .004  .005  .006  .010  .005  .006  .012  .033  .041  .043  .052 
n  50  50  50  
p  200  200  200  
p1 = p2  40  40  40  
Method  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9 
LCT  .010  .010  .010  .011  .016  .016  .015  .015  .057  .055  .051  .053 
LCT_{2}  .008  .010  .011  .008  .016  .016  .017  .013  .053  .058  .049  .059 
SAMGS  .010  .008  .011  .009  .015  .015  .015  .019  .056  .057  .051  .050 
Global  .004  .004  .004  .001  .009  .012  .012  .010  .040  .059  .052  .051 
n  100  100  100  
p  400  400  400  
p1 = p2  60  60  60  
Method  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9  ρ = 0.0  ρ = 0.3  ρ = 0.6  ρ = 0.9 
LCT  .008  .006  .003  .005  .012  .010  .010  .007  .049  .046  .043  .034 
LCT_{2}  .009  .006  .004  .004  .012  .011  .011  .010  .055  .054  .042  .034 
SAMGS  .006  .007  .005  .008  .012  .009  .010  .011  .047  .050  .047  .040 
Global  .002  .003  .003  .002  .009  .005  .007  .006  .056  .036  .034  .034 
The Rpackage to implement Global Test is available for download from http://www.bioconductor.org. The LCT tests and SAMGS for continuous phenotypes were implemented by us using the R statistical software [15].
Results
Simulation study
Identifying gene sets associated with human leptin gene expression measurements
Leptin is a wellknown marker protein for human adiposity and the circulating levels of leptin in the blood are directly proportional to the total amount of body fat. Leptin is also associated with various metabolic and inflammatory conditions. We applied all four GSA methods to analyze a real Affymetrix microarray dataset consisting of genomewide transcriptomic measurements of prostate tumor samples from AfricanAmerican prostate cancer patients [10] against the continuous phenotype of the human leptin gene (LEP) expression values. The publicly available data were downloaded from Gene Expression Omnibus [17] [GEO:GSE6956]. The data that we used in the present study are part of a larger microarray study into immunobiological differences in prostate cancer tumors between AfricanAmerican and EuropeanAmerican men. Because LEP expression levels may be different between the two groups, we used only the data from the AfricanAmerican group to test the LCT methods. For our analysis, we used the expression values of 13,233 genes measured in tumor samples from 33 patients. The tumor samples were resected adenocarcinomas from patients who had not received any therapy before prostatectomy and were obtained from the National Cancer Institute Cooperative Prostate Cancer Tissue Resource (CPCTR) and the Department of Pathology at the University of Maryland. According to Wallace et al. [10], the macro dissected CPCTR tumor specimens were reviewed by a CPCTRassociated pathologist who confirmed the presence of tumors in the specimens. The tissues were collected between 2002 and 2004 at four different sites. The median age of prostatectomy was 61 and the median prostatespecific antigen (PSA) at diagnosis was 6.1 ng/mL. Fiftyfive percent of the tumors were stage pT2, and 45% were stage pT3 or more. Detailed RNA extraction, labeling and hybridization protocols were as described previously [10]. The gene expression values were centered and scaled across the samples before the four GSA methods were applied. The need for such standardization was pointed out in an earlier comparative study of GSA methods for a binary phenotype [18].
For comprehensive analysis, we used the C2 catalog from MsigDB [19] consisting of 1,892 gene sets, including metabolic and signaling pathways from major pathway databases, gene signatures from biomedical literature including 340 PubMed articles, as well as other gene sets compiled from published mammalian microarray studies. Following Subramanian et. al. 2005 [19], we restricted the size of gene sets to between 15 and 500 which gave us 1,403 gene sets for use in our analysis. Each gene set was tested for its association with the LEP expression measurements. A limitation of our study is that the findings come from a relatively small observational study and therefore cannot be generalized to other populations.
In terms of computational efficiency, we noted that LCT incorporated the covariance matrix into the estimations for only a small cost (CPU time of 413 seconds) compared with the cost using SAMGS (CPU time of 397 seconds). In contrast, Global Test was very computationally attractive (CPU time of 92 seconds). The CPU times were recorded on our PC (Processor: x86 Family 6 Model 23 Stepping 10 Genuine Intel 3Ghz; 4GB RAM).
Percentages of gene sets with pvalues less than 0.005, 0.01, 0.05 and 0.10
Method  Pvalue  

≤.005  ≤.01  ≤.05  ≤.10  
LCT  3  4  27  63 
LCT_{2}  3  4  20  46 
SAMGS  1  2  14  27 
Global Test  0  2  15  34 
Gene sets and pathways associated with LEP gene expression measurements
Gene set  Size  LCT  LCT_{2}  SAMGS  Global 

NADLER_OBESITY_UP  46  0  0.004  0.108  0.098 
HSA04920_ADIPOCYTOKINE_SIGNALING_PATHWAY  68  0.003  0.003  0.042  0.032 
HSA04140_REGULATION_OF_AUTOPHAGY  26  0.004  0.007  0.003  0.002 
HIF1_TARGETS  32  0.006  0.025  0.027  0.03 
DORSEY_DOXYCYCLINE_UP  29  0.011  0.063  0.174  0.174 
SHIPP_DLBCL_CURED_UP  28  0.013  0.003  0.01  0.02 
JNK_UP  24  0.015  0.026  0.083  0.074 
PROSTAGLANDIN_SYNTHESIS_REGULATION  28  0.016  0.04  0.165  0.155 
CARDIACEGFPATHWAY  16  0.019  0.018  0.02  0.01 
CITED1_KO_HET_UP  23  0.022  0.023  0.036  0.031 
XU_CBP_DN  32  0.022  0.027  0.06  0.064 
CHREBPPATHWAY  16  0.027  0.029  0.029  0.023 
OXSTRESS_BREASTCA_UP  24  0.027  0.046  0.044  0.047 
AGUIRRE_PANCREAS_CHR17  61  0.029  0.034  0.082  0.06 
ST_GAQ_PATHWAY  27  0.031  0.047  0.109  0.083 
HSA04340_HEDGEHOG_SIGNALING_PATHWAY  46  0.032  0.038  0.036  0.036 
NFATPATHWAY  47  0.034  0.041  0.065  0.053 
HYPOXIA_REVIEW  75  0.035  0.055  0.098  0.095 
HSA04614_RENIN_ANGIOTENSIN_SYSTEM  16  0.04  0.108  0.087  0.076 
CPR_NULL_LIVER_DN  16  0.041  0.047  0.038  0.036 
HSA00380_TRYPTOPHAN_METABOLISM  49  0.043  0.055  0.11  0.1 
HSA04630_JAK_STAT_SIGNALING_PATHWAY  135  0.045  0.065  0.173  0.167 
DIAB_NEPH_UP  58  0.046  0.051  0.196  0.194 
TRYPTOPHAN_METABOLISM  57  0.049  0.064  0.089  0.09 
INSULIN_SIGNALING  93  0.049  0.068  0.187  0.18 
PASSERINI_GROWTH  32  0.049  0.12  0.31  0.319 
TNFA_NFKB_DEP_UP  18  0.05  0.07  0.152  0.169 
FRUCTOSE_AND_MANNOSE_METABOLISM  24  0.055  0.02  0.039  0.041 
ANDROGEN_AND_ESTROGEN_METABOLISM  21  0.058  0.028  0.046  0.042 
POMEROY_DESMOPLASIC_VS_CLASSIC_MD_DN  38  0.091  0.048  0.13  0.116 
TCA  15  0.101  0.058  0.042  0.037 
HSA03050_PROTEASOME  21  0.103  0.113  0.051  0.048 
PROTEASOME_DEGRADATION  27  0.122  0.082  0.028  0.029 
Among the gene sets and pathways associated with LEP expression only by the LCT method, we highlight the insulin signaling candidate pathway (LCT pvalue = 0.049). A positive association between serum insulin levels and LEP expression has been reported in obese humans [26]. Furthermore, the association of circulating insulinlike growth factors with increased risk of prostate cancer has been reported in a metaanalysis [27]. Interestingly, the proteasome degradation candidate pathway was found to be significant by both Global Test (pvalue = 0.029) and SAMGS (pvalue = 0.028), but not by LCT (pvalue = 0.12). A small microarray study (N = 10) found that the genes in the proteasome degradation pathway were differentially expressed 72 hours after polyethylene glycolleptin injection [28]. Other gene sets and pathways found to be significantly associated with LEP expression but with less well elucidated roles are shown in Table 3 and may be worthy of future investigation.
Discussion
Many selfcontained GSA methods have been proposed. However, although many of these methods have the potential to be generalized to any design, they have only been illustrated for a binary or categorical outcome. Thorough extension of these methods to a continuous phenotype has rarely been reported, and studies into their implementation, simulation studies to check type I error and power, and their application to real datasets are lacking. Here, we describe the extension of a “selfcontained” GSA method from a binary to a continuous phenotype. The new GSA tests, LCT and LCT_{2}, address several important technical issues. First, they provide a rigorous and computationally efficient approach to extend the enrichment test of a given gene set against a continuous phenotype. This will be of great help in studying a variety of informative measurements that cannot always be easily or meaningfully reduced to binary or categorical phenotypes. Second, because a pathway often consists of genes that are together involved in a biological mechanism or disease, gene expression levels within a pathway are expected to be correlated. Yet most traditional GSA methods fail to accommodate this important characteristic feature of gene expression datasets. While permutation methods using a valid test statistic can result in appropriate Type I error, the incorporation of a covariance matrix estimator into the test statistic is highly desirable because it often results in better power. Furthermore, we noted that when the gene set to be tested is larger than the sample size, the covariance matrix is illconditioned. To address this problem, a shrinkage method for covariance matrix estimation can provide a useful solution; however, shrinkage methods are rarely used in GSA, in spite of their implementation as an Rpackage which is free for download [7]. The computational cost of including a shrinkage covariance matrix estimator, especially for permutationbased hypothesis testing, can be very high. Notably in our LCT algorithm, we overcame this difficulty by using an orthogonal transformation of the gene expression matrix. In the LCT algorithm, therefore, the eigenvalue decomposition of the shrinkage covariance matrix is performed only for the original data, and not for the permuted versions.
We focused here on selfcontained approaches and because competitive and selfcontained methods test different hypotheses, it is important for the user to make an informed choice based on the hypothesis of interest and their understanding of the limitations of the two approaches (see reviews by Nam and Kim [2] and Dinu et. al.[4]). An important limitation of the selfcontained approaches is that only a few genes can drive the association between the gene set and the phenotype. In such cases, posthoc analysis can be used to reduce the gene set to a core subset associated with the phenotype. Such an analysis has been reported in simulations and in a real example for a binary phenotype [4].
The improvements that we have incorporated into our new GSA tests have given these tests a variety of advantages over the existing methods. We hope that they will be used for the rigorous testing of associations between different molecular pathways and gene signatures. At least of the measured clinicpathological phenotypes are continuous. They include tissue features such as tumor size, staining based readouts; cellular characteristics such as the amount of lymphocytic infiltration in a tumor environment; and subjectspecific measurements such as diagnostic or prognostic marker protein or metabolite concentrations. The LCT algorithm can adjust for continuous or categorical covariates following a regression framework. The LEP levels in the prostate tumor example that we considered may also have been influenced by patientspecific covariates such as body mass index (BMI), age, and/or smoking status. Smoking status did not show a significant association with LEP expression levels (pvalue = 0.36), and BMI and age data were not available for our analysis.
To check the linearity assumption, exploratory data analysis should be used prior to running a formal inference. However, we noted that the small sample sizes that are common in microarray studies, would limit a thorough check for nonlinearities. We also noted that the LCT method could be extended to accommodate nonlinearities; however, a larger sample size would be needed. The simulations and real microarray studies which we conducted indicated that the LCT and LCT_{2} methods both performed very well for small sample sizes. The question of how small is small is debatable and depends largely on the study design. In the case of a binary/categorical phenotype, at least five samples per group are desirable. In the case of a continuous phenotype, assessing significance based on less than 10 samples is dangerous; an alternative would be to rely upon representations that are more descriptive/exploratory in nature. While LCT tests only linear associations between sets of genes and a continuous phenotype, SAMGS and Global Test have been extended in a generalized linear model (GLM) framework and can accommodate multiclass, continuous, count, rate, and censored survival phenotypes. SAMGS uses the sum of squares of the Wald statistic for individual genes constituting the pathway as the test statistic. Wald statistics are calculated as the ratio between the regression coefficient for an individual gene and its corresponding standard error. Global Test reduces the GLM to a random effects model, assuming the regression coefficients corresponding to the genes constituting the set are sampled from a common distribution with mean zero and constant variance. A score test statistic is used to test the null hypothesis of no association between the set and the phenotype. The SAMGS and Global Test algorithms can both adjust for covariates, an attractive feature when accounting for other known prognostic factors in the screening of associations between gene sets and a phenotype.
Conclusions
Our proposed LCT method for gene set analysis efficiently incorporates the gene expression covariance matrix into the test statistic. This approach has resulted in a powerful and computationally attractive method for testing the association of a given gene set with a continuous phenotype. Additional file 1.
Availability and requirements
Project name: Linear Combination Test for GeneSet Analysis of a Continuous Phenotype
Project home page: http://www.ualberta.ca/~yyasui/homepage.html
Operating system: Microsoft Windows XP.
Programming language: R 2.10.1.
Abbreviations
 GSA:

Gene set analysis
 LCT:

Linear combination test
 SAMGS:

Significance analysis of microarray for gene sets.
Declarations
Acknowledgements
LEK is supported by Canadian Institutes of Health Research New Investigator award (MSH87734). SP is supported by Ramalingswami Fellowship from DBT, India and grants from MoS&PI and DST, India. We thank the reviewers for their constructive comments which have improved the exposition of this paper substantially.
Authors’ Affiliations
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