 Methodology
 Open access
 Published:
Principal component gene set enrichment (PCGSE)
BioData Mining volume 8, Article number: 25 (2015)
Abstract
Background
Although principal component analysis (PCA) is widely used for the dimensional reduction of biomedical data, interpretation of PCA results remains daunting. Most existing interpretation methods attempt to explain each principal component (PC) in terms of a small number of variables by generating approximate PCs with mainly zero loadings. Although useful when just a few variables dominate the population PCs, these methods can perform poorly on genomic data, where interesting biological features are frequently represented by the combined signal of functionally related sets of genes. While gene set testing methods have been widely used in supervised settings to quantify the association of groups of genes with clinical outcomes, these methods have seen only limited application for testing the enrichment of gene sets relative to sample PCs.
Results
We describe a novel approach, principal component gene set enrichment (PCGSE), for unsupervised gene set testing relative to the sample PCs of genomic data. The PCGSE method computes the statistical association between gene sets and individual PCs using a twostage competitive gene set test. To demonstrate the efficacy of the PCGSE method, we use simulated and real gene expression data to evaluate the performance of various gene set test statistics and significance tests.
Conclusions
Gene set testing is an effective approach for interpreting the PCs of highdimensional genomic data. As shown using both simulated and real datasets, the PCGSE method can generate biologically meaningful and computationally efficient results via a twostage, competitive parametric test that correctly accounts for intergene correlation.
Background
PCA is a well established statistical technique that performs a linear transformation of multivariate data into a new set of variables, the principal components (PCs), that are linear combinations of the original variables, are uncorrelated and have sequentially maximum variance [1–3]. The solution to PCA is given by the spectral decomposition of the covariance matrix with the variance of the PCs specified by the eigenvalues, arranged in decreasing order, and the PC directions specified by the associated eigenvectors.
In the biomedical domain, PCA has been extensively employed for the analysis of genomic data including measures of DNA variation, DNA methylation and RNA expression [4]. Features of these datasets that motivate PCA include the high dimensionality of the feature space, low sample size and significant collinearity. The most common uses of PCA with genomic data involve dimensionality reduction for visualization [5, 6] or clustering of the observations [7], with population genetics an important use case [8]. PCA has also been used as the basis for feature selection [9], gene clustering [10] and biclustering [11]. More recent applications include dimensionality reduction prior to gene set testing [12, 13] and highdimensional regression [14].
Although PCA is an effective tool for reducing the dimensionality of genomic data, application of the method remains limited by the challenge of biological interpretation [4, 15]. Because PCs are linear combinations of all original variables, which can number from the thousands to the millions for genomic datasets, they typically lack any clear biological meaning. While PCA may improve the performance of many statistical methods, e.g., better predictive accuracy in a regression context, the underlying model is often a black box.
Approaches for generating more interpretable PCs have evolved from component thresholding [3], simple components (i.e., PC loading vectors constrained to values from {−1,0,1}) [16] and rotation techniques (e.g., varimax) [17] to sparse PCA methods, which compute approximate PCs using cardinality [18] or LASSObased [15, 19] constraints on the component loadings. By generating approximate PCs with few nonzero loadings, all of these techniques improve interpretability by associating only a small number of variables with each PC. While such sparse PCA methods can be very effective when the true population PCs are associated with only a few variables, they will fail to accurately estimate the spectral structure of the data when the population PCs are defined by the coordinated action of large groups of variables with small marginal effects. For genomic data, the pathwaybased patterns that dominate the robust structure of genetic associations with clinical phenotypes [20], and are the motivation for traditional gene set testing methods [21], can be expected to also characterize the PCs of those datasets. The PCs of genomic data are therefore more likely to be quantitatively described, in a repeatable fashion, by collections of functionally related genes, e.g., gene sets from the Gene Ontology (GO) [22], than by individual genes.
To support interpretation of PCs in terms of a priori variable groups, rather than just individual variables, sparse PCA methods have recently been extended to include structured sparse penalties [23, 24], such as the group lasso [25]. Although structured sparse PCA techniques generate sparse PC loading vectors that reflect group structure, these methods cannot be easily used to compute the statistical association between variable groups and each PC in such a way that the variable groups can be ranked according to deviation from a specific null hypothesis, as is done in traditional gene set testing. Matrix correlation methods [3, 26] have also been used to quantify the association between groups of variables and one or more PCs. However, because such matrix correlation methods compute the association of each variable group independent of the variables that do not belong to the group, they can only be used for selfcontained gene set tests [27] (Q_{2} in the terminology of Tian et al. [28]) in a manner similar to Goeman and Buhmann’s globaltest [29] and not for competitive gene set testing (Q_{1} in the terminology of Tian et al.).
To date, competitive gene set testing relative to PCs has been limited to methods, such as Fisher’s Exact Test, that are based on a 2×2 contingency table representing the association between gene set membership and a discretization of the ranked list of PC loading values [30]. Such contingency table tests have two key flaws: they rely on an arbitrary threshold of the genelevel test statistic, which reduces statistical power and, more importantly, they are based on the incorrect assumption of independence among the genelevel test statistics, causing them to generate high type I error rates [27, 31, 32]. Because of the anticonservative nature of contingency tablebased tests, and other approaches that assume independence among genelevel test statistics under the null, the use of these methods for standard gene set testing has been strongly discouraged in favor of techniques that preserve intergene correlation, usually via permutation of the sample labels [27]. Competitive gene set testing methods that correctly account for correlation among genelevel test statistics, either through sample permutation, parametric approximation of the sample permutation distribution or correlation adjustment of parametric test statistics, include SAFE [31, 33, 34], GSEA [35], GSA [36] and CAMERA [32]. All of these methods, however, are designed for use in a supervised context to measure the statistical significance of the association between sets of genomic variables and a phenotype variable.
Although biologically meaningful and repeatable interpretation of the PCs of genomic data requires approaches based on functional gene sets, researchers currently lack methods that competitively test the association between gene sets and PCs with correct handling of intergene correlation. To address this gap, we have developed principal component gene set enrichment (PCGSE), an approach for interpreting the PCs of genomic data via twostage competitive gene set testing in which the correlation between each gene and each PC is used as a genelevel statistic with flexible choice of both the gene set test statistic and the method used to compute the null distribution of the gene set statistic. Although described in the context of functional gene sets and genomic data, the PCGSE method can be used to compute the statistical association between any collection of variable groups and the PCs of an empirical dataset. To support use of the PCGSE method by other researchers, we implemented the PCGSE R package, which is available from the CRAN repository. Using simulated data with simulated gene sets and real gene expression data with curated gene sets, we demonstrate that biologically meaningful and computationally efficient results can be obtained from a simple parametric version of the PCGSE technique, based on the CAMERA method [32], that performs a correlationadjusted twosample ttest between the genelevel test statistics for gene set members and genes not in the set.
Methods
PCGSE inputs
The PCGSE method takes as input both an n×p genomic data matrix X quantifying p genomic variables under n experimental conditions and an f×p binary annotation matrix A that specifies the association between the p genomic variables and f functional categories.
The genomic data held in X, e.g., mRNA expression levels, will be modeled as a sample of n independent observations from a pdimensional random vector x. Although PCGSE does not have specific distributional requirements, sources of genomic data, especially gene expression data, are often well approximated by a multivariate normal distribution after appropriate transformations. It is assumed that any desired data transformations have been performed and that missing values have been imputed or removed.
The rows of the annotation matrix A represent f distinct biological functions, e.g., GO categories, and the elements a_{i,j} hold indicator variables whose value depends on whether an annotation exists between the function i and genomic variable j.
PCGSE algorithm
Enrichment of the gene sets defined by A relative to one of the PCs of X is performed using the following sequence of steps. This workflow is graphically illustrated in Fig. 1 and each step is explained in more detail in the Sections “PCA for PCGSE” thru “Gene set statistical significance” below. Note that steps 2 thru 5 have close parallels to modules in Ackermann and Strimmer’s general modular framework for gene set enrichment analysis [37].

1.
Perform PCA on a standardized version of X.

2.
Compute genelevel statistics, z_{ j },j=1,…,p, for all p genomic variables that quantify the association between the genomic variable and the PC.

3.
(Optional) Transform the genelevel statistics.

4.
Compute gene set statistics, S_{ k },k=1,…,f, for all f gene sets defined by A using the genelevel statistics, z_{ j }.

5.
Determine the statistical significance of the gene set statistics according to a competitive null hypothesis.
Output: For each of the f gene sets, the observed value of the gene set test statistic, S_{ k }, and a pvalue representing the probability of encountering a gene set statistic as or more extreme than then the observed S_{ k } under the appropriate competitive null hypothesis.
PCA for PCGSE
Because PCs are not invariant under scaling of the data [3], PCA is performed on a mean centered and standardized version of X, \(\tilde {\mathbf {X}}\). The PC loading vectors and variances of \(\tilde {\mathbf {X}}\) are thus the eigenvectors and eigenvalues of the sample correlation matrix, \(\mathbf {S} = 1/(n1) \tilde {\mathbf {X}}^{T} \tilde {\mathbf {X}}\), rather than the sample covariance matrix. For computational efficiency, the PCA solution is realized via the singular value decomposition (SVD) of \(\tilde {\mathbf {X}}\), \(\tilde {\mathbf {X}} = \mathbf {U} \boldsymbol {\Sigma } \mathbf {V}^{T}\), where the columns of V represent the PC loading vectors, the entries in the diagonal matrix Σ are proportional to the square roots of the PC variances and the columns of UΣ are the PCs.
Genelevel statistics
The PCGSE method supports the following genelevel statistics, represented using the notation z_{ j },j=1,…,p, for quantifying the association between genomic variable j and the target PC.

PC loading. For genomic variable j and target PC m, the genelevel statistic is element v_{j,m} of matrix V from the SVD of \(\tilde {\mathbf {X}}\).

Pearson correlation coefficient. Where the correlation is computed between each genomic variable and the target PC.

Fishertransformed Pearson correlation coefficient. This creates a statistic whose distribution is approximately \(\mathcal {N}(0,1)\).
Because the Pearson correlation coefficients between genomic variables and PCs of the sample correlation matrix are proportional to the PC loadings (see (1) below), all of these genelevel statistics provide a measure of the correlation between genomic variables and PCs. Specifically,
where U,Σ and V are from the SVD of \(\tilde {\mathbf {X}}\), \(\mathbf {P}_{\tilde {\mathbf {X}}, \mathbf {U} \boldsymbol {\Sigma }}\) is the matrix of Pearson correlation coefficients between the standardized genomic variables held in \(\tilde {\mathbf {X}}\) and the PCs of \(\tilde {\mathbf {X}}\), with element i,j of \(\mathbf {P}_{\tilde {\mathbf {X}}, \mathbf {U} \boldsymbol {\Sigma }}\) set to the Pearson correlation coefficient between column i of \(\tilde {\mathbf {X}}\) and column j of UΣ, and \(\mathbf {S}_{\tilde {\mathbf {X}}, \mathbf {U} \boldsymbol {\Sigma } \sqrt {1n} \boldsymbol {\Sigma }^{1}}\) is the matrix of sample covariances between the columns of \(\tilde {\mathbf {X}}\) and the columns of \(\mathbf {U} \boldsymbol {\Sigma } \sqrt {1n} \boldsymbol {\Sigma }^{1}\).
The choice between the different genelevel statistics will be guided by the gene set statistic and significance testing method employed for PCGSE as well as computational constraints. For example, the added computational expense to generate zstatistics from correlation coefficients is motivated by parametric tests of the mean difference statistic, whereas, for rank sum tests, the PC loadings are sufficient.
Transformation of genelevel statistics
An absolute value transformation can optionally be applied to the genelevel statistics, i.e., \(\tilde {z}_{j} = z_{j}\). Such a transformation gives the PCGSE method increased power to detect scale alternatives, i.e. gene sets that contain both significantly enriched and significantly repressed genomic variables, whereas the use of untransformed genelevel statistics provides better power against shift in location alternatives, i.e., gene sets containing genomic variables with a common direction of association [36].
Gene set statistics
The PCGSE method supports two competitive gene set statistics, represented using the notation S_{ k },k=1,…,f, for quantifying the association between gene set k and a target PC.
Mean difference statistic
This statistic is computed as the standardized difference between the mean of the z_{ j } for genomic variables in the gene set and genomic variables not in the set and corresponds to U_{ D } in the notation of Barry et al. [31]. Benefits of the mean difference statistic include its parametric null distribution and excellent power, relative to other gene set test statistics, for shift in location alternatives when using untransformed z_{ j } [36]. For gene set k, this statistic is defined as:
where m_{ k } is the number of genes in set k, \(\bar {z}_{k}\) is the mean of the z_{ j } for members of gene set k, \(\bar {z}_{k^{c}}\) is the mean of the z_{ j } for genes not in set k and σ_{ p } is the pooled standard deviation of the z_{ j }.
Rank sum statistic
This statistic is computed as the standardized Wilcoxon rank sum statistic given the ranks of the z_{ j } for genomic variables in the set and genomic variables not in the set and corresponds to U_{ W } in the notation of Barry et al. [31]. Benefits of the rank sum statistic include lack of distributional assumptions and robustness to outliers. For gene set k, the Wilcoxon rank sum statistic is defined as the sum of the ranks of the genelevel statistic for all genomic variables belonging to gene set k minus the minimum possible value for this sum of ranks: \(W_{k} = \sum _{j =1}^{p} a_{k,j} \text {Rank}(z_{j})  \frac {m_{k} (m_{k} +1)}{2}\), where \(m_{k} = \sum _{j =1}^{p} a_{k,j}\), the size of gene set k. A version of this statistic that has an asymptotic \(\mathcal {N}(0,1)\) distribution under the null can be generated as:
where \(\mu _{W_{k}} = (m_{k}(pm_{k}))/2\) and \(\sigma ^{2}_{W_{k}} = (m_{k}(pm_{k})(m_{k}+1))/12\).
Gene set statistical significance
To compute the statistical significance of the association between gene set k and a target PC, the distribution of the gene set statistic S_{ k } must be calculated under the appropriate null hypothesis. The PCGSE approach supports three different methods (parametric, correlationadjusted parametric and permutation) for computing the competitive null distributions of the standardized mean difference statistic, \({S^{D}_{k}}\), and standardized rank sum statistic, \({S^{W}_{k}}\), defined in (2) and (3) respectively.
Parametric tests
Under the competitive H_{0} that the z_{ j } are i.i.d, \({S^{D}_{k}}\) has a tdistribution with p−2 df and a twosided ttest can therefore be used to determine statistical significance. For \({S^{W}_{k}}\), the asymptotic standard normal distribution under this H_{0} can be used as the basis for a twosided ztest. Both of these parametric tests fall into the class 1 test category as outlined in Barry et al. [31] and are similar to the Q_{1} test defined by Tian et al. [28].
While it is often safe to assume a normal distribution for the z_{ j }, especially after transformation, the z_{ j } will not be independent. Indeed, because the z_{ j } used with PCGSE are proportional to the PC loadings, they have an asymptotic multivariate normal distribution [38], assuming multivariate normality for the underlying genomic data, with significant correlation present between the loadings associated with the genes that have high pairwise correlations [3]. Because both the ttest for \({S^{D}_{k}}\) and the ztest for \({S^{W}_{k}}\) ignore this correlation between the z_{ j }, they will generate inflated type I error rates. These tests are therefore only supported by the PCGSE method for the purpose of comparative evaluation.
Correlationadjusted parametric tests
A computationally efficient approach for addressing correlation among the z_{ j } involves the use of correlationadjusted parametric tests. Correlationadjusted versions of \({S^{D}_{k}}\) and \({S^{W}_{k}}\) were first discussed in the context of gene set testing by Barry et al. [31]. Simplified versions of these correlationadjusted statistics were later developed into the CAMERA method by Wu et al. [32]. Specifically, the approach taken by CAMERA assumes that correlation among the z_{ j } can be approximated by the correlation among the genomic variables (this is supported by results in Barry et al. [31]), ignores all intergene correlation except the correlation among the members of the tested gene set and estimates a single average pairwise correlation for gene set members using residuals from a linear regression.
The PCGSE method makes similar simplifying assumptions as those made by CAMERA, i.e., correlation between the z_{ j } can be approximated by correlation among the genomic variables, only gene set members have nonzero intergene correlation and all pairwise correlations between gene set members are the same. An important difference between PCGSE and CAMERA is that PCGSE estimates the average intergene correlation directly from the sample correlation matrix. The correlationadjusted standardized mean difference statistic used by PCGSE is:
where VIF (variance inflation factor) \( = 1 + (m_{k}  1) \bar {\rho }_{k}\) and \(\bar {\rho }_{k}\) is the average Pearson correlation coefficient between members of gene set k. Following Wu et al. [32], this correlationadjusted statistic has a tdistribution with n−2 df under H_{0}. Likewise the correlationadjusted standardized rank sum statistic is computed as:
where \(\sigma ^{2}_{\text {VIF},W_{k}} = (m_{k}(pm_{k}))/(2\pi) (sin^{1} (1) + (pm_{k}  1) sin^{1} (.5) + (m_{k}1) (pm_{k}1)sin^{1}(\bar {\rho }_{k}/2) + (m_{k}1)sin^{1}((\bar {\rho }_{k}+1)/2))\), as derived in Wu et al. [32] based on the formula in Barry et al. [31].
Permutation test
The most common approach in the gene set testing literature for addressing correlation between the z_{ j } has been sample permutation. This approach, which corresponds to the class 2 test in Barry et al. [31], generates the null distribution of the S_{ k } via permutation of the outcome variable. For each permutation of the outcome variable, all z_{ j } are recomputed to generate permutation statistics \(z^{*}_{j}\) and then permutation gene set statistics \(S^{*}_{k}\) are calculated using the \(z^{*}_{j}\). The statistical significance for a given gene set k is based on the proportion of all permutation \(S^{*}_{k}\) more extreme than the observed S_{ k }. In standard gene set testing, permutation is applied to a clinical outcome variable, e.g., a case/control label.
For PCGSE, permutation is applied to the elements of the target PC, i.e., the elements of one of the columns of UΣ. Because permutation is applied to the PC elements, this test can only be used with Pearson correlation coefficients or Fishertransformed Pearson correlation coefficients as genelevel statistics since only these genelevel statistics can be recomputed after permutation of the PC elements (the PC loadings are fixed). A key assumption of the permutation null distribution is that the permuted values are i.i.d. Assuming the original n observations of the pdimensional random vector x are i.i.d, the elements of each PC will also be i.i.d., since each PC is a linear function of the original x. Permutation of the PC elements therefore generates a valid permutation distribution for both \({S^{D}_{k}}\) and \({S^{W}_{k}}\).
Because permutation tests handle correlation among the z_{ j } without attempting to estimate this correlation or make simplifying assumptions about the correlation structure, they are likely the most accurate of the statistical tests supported by PCGSE and are therefore used to evaluate the performance of the parametric and correlationadjusted parametric tests. The exact permutation test was also used as a “goldstandard” in Zhou et al. [34]. Although they provide superior handling of intergene correlation, permutation tests do suffer from two important disadvantages relative to parametric tests: computational complexity and lower power to detect gene sets whose members all have a small common association with the outcome. Because of these disadvantages, correlationadjusted parametric tests are preferred for most PCGSE applications.
Another alternative to sample permutation testing that addresses the key challenge of computational complexity is the parametric approximation of the sample permutation distribution of genelevel score statistics developed by Zhou et al. [34]. Although the Zhou et al. beta distributionbased parametric approximations may be a useful option for the PCGSE method, it is not currently supported due to the lack of a parametric approximation for a directional, competitive gene set test statistic that is equivalent to \({S^{D}_{k}}\) or \({S^{W}_{k}}\) using untransformed z_{ j }. In Zhou et al. [34], parametric approximations are only detailed for two selfcontained gene set test statistics (sum of the score statistics and sum of the squares of the score statistics) and one nondirectional competitive test statistic (a weighted sum of the squares of local score statistics).
PCGSE evaluation
Benchmark PC gene set testing method
Because contingency tablebased tests represent the current stateoftheart for competitive gene set testing relative to the sample PCs of genomic data [30], it is important to compare the performance of the PCSGSE method, specifically the tests based on the S^{D,adj} and S^{W,adj} gene set statistics, against tests based on a 2×2 contingency table populated via a discretization of the ranked list of PC loading values. Since tests based on a discretization of the genelevel test statistics, e.g., Pearson’s difference in proportions test [31], are simply a special case of the unadjusted mean difference statistic, S^{D}, the PCGSE method using the unadjusted ttest can be used as a proxy for existing contingency table methods in both the simulation and real data examples. Furthermore, because a twosample ttest based on S^{D} is more powerful than the corresponding contingency table test based on a discretization of the genewise test statistics, this comparison is conservative, i.e., the difference in performance between the PCGSE method using S^{D,adj} or S^{W,adj} and contingency table tests should be greater than the difference between PCGSE using S^{D,adj} or S^{W,adj} and PCGSE using S^{D}.
Evaluation using simulated gene sets and simulated data
As a simple example, the PCGSE method was used to compute the statistical association between 20 disjoint gene sets, each of size 10, against the PCs of 1,000 simulated gene expression datasets each comprised by 75 independent observations of a 200dimensional random vector simulated according to a multivariate normal distribution ∼MVN(μ,Σ). The population covariance matrix was generated as: \(\boldsymbol {\Sigma } = \lambda _{1} \boldsymbol {\alpha }_{1} \boldsymbol {\alpha }_{1}^{T} + \lambda _{2} \boldsymbol {\alpha }_{2} \boldsymbol {\alpha }_{2}^{T} + \lambda _{d} \boldsymbol {I}\), where λ_{1}= 2, λ_{2}= 1, λ_{ d }= 0.1, α_{1} is a 200dimensional vector with all elements equal to 0 except for the first 10 which were set to \(\sqrt {.1}\), α_{2} is a 200dimensional vector with all elements equal to 0 except for the second 10 which were set to \(\sqrt {.1}\). Figure 2 graphically illustrates the variance and loadings of the population and sample PCs simulated according to this model. The PCGSE method was executed using the Fishertransformed Pearson correlation coefficient between each variable and each PC as the z_{ j } with \({S^{D}_{k}}\), as defined in (2), as the gene set test statistic. The statistical significance of the association between each of the 20 simulated gene sets and each PC was computed using all supported tests described in Section “Gene set statistical significance” with 10,000 permutations for the permutation tests. Because the true association was known between simulated gene sets and the PCs of the simulated data, it was possible to compute contingency table statistics. In this case, the type I error rates for the different statistical testing methods were computed for gene set 2 relative to PC 1 and for gene set 1 relative to PC 2, both cases with no true association.
Evaluation using Spellman et al. α factorsynchronized yeast gene expression data and yeast cell cycle gene sets
The PCGSE method was used to compute the statistical association of the yeast cell cycle gene sets defined by Spellman et al. [39] relative to the first three PCs of a specially processed version of the α factorsynchronized yeast gene expression data collected by Spellman et al. and reexamined by Alter et al. [5]. Both the α factorsynchronized data and yeast cell cycle gene sets were downloaded from the Additional file 1 website for Alter et al. To support comparison against the results reported in Alter et al., PCA was performed on a version of the gene expression data that was specially processed according to the steps outlined in Alter et al. so that the first three PCs were identical to the first three socalled eigengenes. A reproduction of Fig. 5 from Alter et al. is included as Fig. 2 in Additional file 1 with the value of the first three PCs of the specially processed gene 350 expression data shown relative to the 22 α factor arrays. The PCGSE method was executed on the Spellman et al. data and gene sets using the Fishertransformed Pearson correlation coefficient between each gene and each PC as the z_{ j } with \({S^{D}_{k}}\) as the gene set statistic. The statistical significance of the gene set statistic was computed using all supported tests described in Section “Gene set statistical significance” with 10,000 permutations for the permutation tests.
Evaluation using MSigDB C2 v4.0 gene sets and Armstrong et al. leukemia gene expression data
The PCGSE method was also used to compute the statistical association between the MSigDB C2 v4.0 gene sets and the first 3 PCs of the leukemia gene expression data [40] used in the 2005 GSEA paper [35]. The MSigDB C2 v4.0 cancer modules and collapsed leukemia gene expression data were both downloaded from the MSigDB repository. With a minimum gene set size of 15 and maximum gene set size of 200, 3,076 gene sets out of the original 4,722 were used in the analysis. The PCGSE method was executed using the Fishertransformed Pearson correlation coefficient between each genomic variable and each PC as the z_{ j } and \({S^{D}_{k}}\) as the gene set test statistic. The statistical significance of the association between each of the MSigDB C2 gene sets and each of the first 3 PCs of the standardized leukemia gene expression data was computed using all supported tests described in Section “Gene set statistical significance” with 10,000 permutations for the permutation tests. The enrichment of the MSigDB C2 gene sets was also computed relative to the acute myeloid leukemia (AML) versus acute lymphoblastic leukemia (ALL) phenotype using the GSA method [36] with the restandardized mean statistic and 10,000 permutations. For each of the first three PCs and each of the PCGSE methods for computing statistical significance of the standardized mean difference gene set statistic, the Spearman correlation coefficient was computed between PC gene set enrichment pvalues and phenotype enrichment pvalues. For PC 2, for which the PC and phenotype gene set enrichment pvalues were highly correlated, contingency table statistics were computed measuring how well PCGSE was able to identify MSigDB C2 gene sets significantly associated with the AML/ALL phenotype.
Results and discussion
Simulation example
According to the population covariance matrix, Σ, used to simulate the 1,000 datasets, only the first gene set should be significantly enriched on the first PC and only the second gene set should be significantly enriched on the second PC. This relationship can be seen easily in the loading values for population PCs 1 and 2 as shown in Fig. 2, plots ( c ) and ( e ). The significant loading of gene set 2 on PC 2, however, will result in a high pairwise correlation between the PC loadings for gene set 2 members on PC 1. The fact that high loadings on one PC result in correlation among the PC loadings on other PCs follows from the formula for the asymptotic distribution of the PC loadings for MVN data [38]: \(\mathbf {v}_{j} \sim \mathcal {N}(\boldsymbol {\alpha }_{j}, \mathbf {T}_{j}), \mathbf {T}_{j} = \frac {\lambda _{j}}{n1} \sum _{k=1, k \neq j}^{p} \frac {\lambda _{k} \boldsymbol {\alpha _{k} \alpha _{k}}^{T}}{(\lambda _{k}\lambda _{j})}\), where j=1,…,p, p is fixed and n→∞, λ_{ j } is an eigenvalue of the population covariance matrix, λ_{1}>λ_{2}>…>λ_{ p }, and α_{ j } is an eigenvector of the population covariance matrix.
The genelevel test statistics computed for gene set 2 on PC 1 and for gene set 1 on PC 2 will therefore have a nonzero average pairwise correlation. The impact of this correlation between the genelevel test statistics can be seen in the PCGSE results shown in Fig. 3. The unadjusted ttest uses an incorrectly small variance for the \({S^{D}_{k}}\) gene set statistic and, as expected, generates the high type I error rate of 0.382 given a nominal α of 0.05 for gene set 2 relative to PC 1 and 0.257 for gene set 1 relative to PC 2. The correlationadjusted twosided ttest and the twosided permutation test are much more successful at controlling the type I error rate. For PC 1 and gene set 2, the type I error rate was 0.057 for the correlationadjusted ttest and 0.05 for the permutation test. For PC 2 and gene set 1, the type I error rate was 0.016 for the correlationadjusted ttest and 0.014 for the permutation test. For this example, all gene set testing methods were able to correctly reject the null hypothesis for almost all cases where the gene set had a true association with the PC, e.g., gene set 1 relative to PC 1 and gene set 2 relative to PC 2. When this simulation is performed with smaller sample sizes, both the type I and type II error rates increase due to increased uncertainty in the estimation of the sample covariance matrix and sample PCs (results not shown). The lower type I error rates for gene set 1 relative to PC 2 versus gene set 2 relative to PC 1 can be explained by the larger uncertainty in the estimation of the eigenvector for the second PC relative to the first PC and the consequent overestimation of correlation between genelevel test statistics. PCGSE results computed for this simulation example using \({S^{W}_{k}}\) as the gene set statistic can be found in the Additional file 1.
Although based on a simple twofactor MVN model, this simulation example demonstrates the importance of controlling for correlation between genelevel test statistics. Tests which assume independence among the statistics that quantify the association between genes and PCs, such as a twosample ttest, Fisher’s exact test or a gene permutation test, will underestimate the variance of the gene set test statistic and will therefore reject too many H_{0}. This example also shows that the correlationadjusted ttest can achieve enrichment sensitivity and specificity comparable to a sample permutation test with a lower computational burden.
Yeast cell cycle gene expression example
The Spellman et al. [39] α factorsynchronized gene expression data was selected for PCGSE analysis because it is easily accessible, has been widely reanalyzed and has a spectra with a published biological interpretation. In particular, the reanalysis by Alter et al. [5] was one of the first to illustrate that the spectra of gene expression data can represent important biological features, in this case phases of the yeast cell cycle. In Alter et al., the authors provided a qualitative interpretation of the first two eigengenes in terms of the yeast cell cycle by examining the correlation between the eigengenes and genes known to be active during different cell cycle phases, as defined by Spellman et al. Alter et al. concluded that the first eigengene was correlated with genes that peak late in cell cycle phase G_{1} and early in phase S and was anticorrelated with genes that peak late in cell cycle phase G_{2}/M and early in phase M/G_{1}. Alter et al. also concluded that the second eigengene was correlated with genes that peak late in cell cycle phase M/G_{1} and early in phase G_{1} and was anticorrelated with genes that peak late in phase S and early in phase S/G_{2}.
Table 1 contains pvalues representing the statistical significance of the association between each of the Spellman et al. [39] yeast cell cycle gene sets and the first two PCs of a specially processed version of the Spellman et al. gene expression data. As described in Section “Evaluation using Spellman et al. α factorsynchronized yeast gene expression data and yeast cell cycle gene sets”, this special processing ensured that the PCs were identical to the eigengenes analyzed in Alter et al. [5]. When a twosided ttest was used to determine the statistical significance of the \({S^{D}_{k}}\) gene set statistic, the gene sets corresponding to cell cycles G_{1}, S and G_{2}/M were highly significantly associated with PC 1 and the gene sets corresponding to M/ G_{1}, G_{1}, S and S/G_{2} were significantly associated with PC 2. However, when either a twosided ttest was used to compute the significance of \(S^{D,adj}_{k}\) or a twosided permutation test was used to determine the statistical significance of \({S^{D}_{k}}\), PC 1 only had a significant association with the gene set corresponding to phase G_{1} (with a marginally significant association with phase G_{2}/M) and none of the cell cycle gene sets were significant for PC 2.
Comparing the output from PCGSE with the analysis in Alter et al. [5], the results from the twosided ttest align closely with the qualitative conclusions of Alter et al. The output from the correlationadjusted ttest and permutation test, although generally in agreement for PC 1, are in direct contrast with Alter et al. regarding PC 2, finding no cell cycle association. The agreement between Alter et al. and the ttest results is expected since the authors had based their analysis simply on a qualitative inspection of the genelevel correlations without a more formal test of a gene set test statistic. The fact that the PCGSE methods which account for intergene correlation failed to find an association between PC 2 and the cell cycle gene sets indicates that the published association in Alter et al. may well have been a false positive due to either the high intergene correlation present among the members of these sets or the selective examination by Alter et al. on a subset of the genes in each of the cell cycle gene set with a common direction of association with the eigengene. In the later case, it is likely that a gene set statistic such as the maxmean [36] would identify significant cell cycle enrichment for the second eigengene.
This example highlights the importance of using formal statistical methods for gene set testing when attempting to interpret the PCs of genomic data in terms of gene sets. Such gene set testing methods must specifically account for the correlation between genelevel test statistics.
Leukemia gene expression example
The classic Armstrong et al. [40] leukemia gene expression dataset is another excellent example of a case where the genomic patterns associated with an interesting phenotype have a clear representation in the spectral structure of the data. For the Armstrong et al. data, the second PC of the gene expression data is strongly associated with the AML versus ALL status of the subjects. Use of the Armstrong et al. gene expression data and MSigDB C2 v4.0 gene sets for evaluation of PCGSE was also motivated by the extensive use of this dataset and gene set collection in the gene set enrichment literature (e.g., Subramanian et al. [35]) and easy accessibility from the MSigDB repository, factors that will facilitate interpretation and replication of the reported PCGSE results by other researchers.
Figure 4 shows the association between phenotype and PC gene set enrichment pvalues for the MSigDB C2 v4.0 gene sets, the AML versus ALL phenotype and the first three PCs of the Armstrong et al. leukemia gene expression data. Each of the columns in the multiplot corresponds to results for one PC and each row corresponds to one of the three different statistical tests supported by PCGSE on the \({S^{D}_{k}}\) gene set statistic (i.e., ttest, correlationadjusted ttest and permutation test). The association between PC 2 and the AML versus ALL phenotype can be clearly seen in Fig. 4 plots (b), (e) and (h). For all three PCGSE methods, the PC enrichment pvalues for the MSigDB C2 v4.0 gene sets are highly correlated with the enrichment pvalues computed for these gene sets relative to the AML versus ALL phenotype.
Similar to the PCGSE results outlined in previous sections on simulated data and yeast gene expression data, the unadjusted twosided ttest on \({S^{D}_{k}}\) generates PC gene set enrichment pvalues that are substantially lower than the enrichment pvalues output by either the ttest on \(S^{D,adj}_{k}\) or the permutation test on \({S^{D}_{k}}\). Although the true enrichment status of the MSigDB C2 v4.0 gene sets relative to the PCs of the Armstrong et al. [40] gene expression data is unknown, the phenotype enrichment results can be used as a proxy for the true gene set association with PC 2 under the assumption that this PC captures the AML versus ALL signal. If gene sets with a phenotype enrichment significance at or below 0.1 are considered AML/ALL markers, the PCGSE method is able to correctly identify these gene sets via enrichment relative to PC 2 with an area under the receiver operator characteristic curve (AUC) of 0.82 for the ttest results displayed in plot (b), an AUC of 0.88 for the correlationadjusted ttest results displayed in plot (e) and an AUC of 0.83 for the permutation test results displayed in plot (h). Considering identification of AML/ALLassociated gene sets via PC enrichment using just α= 0.1, the PCGSE method has a positive predictive value of 0.4 for the ttest results displayed in plot (b), 0.85 for the correlationadjusted ttest results displayed in plot (e) and 0.51 for the permutation test results displayed in plot (h).
PCGSE analysis of the MSigDB C2 v4.0 gene sets and Armstrong et al. leukemia gene expression data illustrates the biological motivation for PC gene set enrichment and demonstrates the superior performance of the computationally efficient correlationadjusted ttest relative to either an unadjusted ttest or permutation test.
Conclusion
Although PCA is widely used for the dimensional reduction of biomedical data, with applications in visualization, clustering and regression, interpretation of PCAbased models remains challenging. While rotation methods and sparse PCA techniques can generate approximate PCs with few nonzero loadings that support interpretation in terms of individual variables, these approaches will perform poorly on genomic data in which important biological signals are defined by the collective action of groups of functionally related genes. Although gene set testing methods have been widely applied in supervised settings to analyze the association between gene sets and clinical phenotypes, such variable group testing methods have seen little application in unsupervised contexts to test the association between gene sets and the spectra of genomic data. To address the challenge of gene setbased interpretation of the PCs of genomic data, we have developed the principal component gene set enrichment (PCGSE) method, available as an R package from CRAN. PCGSE performs a twostage competitive gene set test with the correlation between each gene and each PC as the genelevel test statistic and with the flexible choice of both the gene set test statistic and the method used to compute the null distribution of the gene set statistic. On both simulated gene sets with simulated data and on curated gene sets with real gene expression data, a computationally efficient version of the PCGSE method based on a correlationadjusted ttest has been shown to accurately compute the statistical association between gene sets and the PCs of genomic data. Methods for combining the results from PCGSE tests on multiple PCs will be explored in future work.
Availability of supporting data
The MSigDB C2 v4.0 gene sets can be downloaded from http://www.broadinstitute.org/gsea/msigdb/collections.jsp. The Armstrong et al. [40] leukemia gene expression data can be downloaded from http://www.broadinstitute.org/gsea/datasets.jsp. The Spellman et al. [39] yeast cell cycle data can be downloaded from http://genomewww.stanford.edu/SVD/htmls/pnas.html. An implementation of the PCGSE algorithm is available in the PCGSE R package (http://cran.rproject.org/web/packages/PCGSE/index.html). Due to the dependency on the Bioconductor package safe, it is recommended that PCGSE be installed using the biocLite() function. At the R prompt, enter: source(“http://bioconductor.org/biocLite.R”) biocLite(“PCGSE”)
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Authors’ contributions
HRF developed the PCGSE methodology, implemented the algorithm, designed the simulation study, performed the real data analysis and drafted the manuscript. ZL participated in the development of the PCGSE methodology, assisted with the design of the simulation study and real data analysis and helped draft the manuscript. JHM participated in the development of the PCGSE methodology, assisted with the design of the simulation study and real data analysis and helped draft the manuscript. All authors read and approved the final manuscript.
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National Institutes of Health R01 grants LM010098, LM011360, EY022300, GM103506 and GM103534.
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Additional file 1
Supplementary figures for simulation and cell cycle examples. (PDF 698 kb)
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Frost, H.R., Li, Z. & Moore, J.H. Principal component gene set enrichment (PCGSE). BioData Mining 8, 25 (2015). https://doi.org/10.1186/s130400150059z
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DOI: https://doi.org/10.1186/s130400150059z