Genetic and Nongenetic Variation Revealed for the Principal Components of Human Gene Expression

Abstract

Principal components analysis has been employed in gene expression studies to correct for population substructure and batch and environmental effects. This method typically involves the removal of variation contained in as many as 50 principal components (PCs), which can constitute a large proportion of total variation present in the data. Each PC, however, can detect many sources of variation, including gene expression networks and genetic variation influencing transcript levels. We demonstrate that PCs generated from gene expression data can simultaneously contain both genetic and nongenetic factors. From heritability estimates we show that all PCs contain a considerable portion of genetic variation while nongenetic artifacts such as batch effects were associated to varying degrees with the first 60 PCs. These PCs demonstrate an enrichment of biological pathways, including core immune function and metabolic pathways. The use of PC correction in two independent data sets resulted in a reduction in the number of cis- and trans-expression QTL detected. Comparisons of PC and linear model correction revealed that PC correction was not as efficient at removing known batch effects and had a higher penalty on genetic variation. Therefore, this study highlights the danger of eliminating biologically relevant data when employing PC correction in gene expression data.

GENE expression profiling has become a very popular technique used to quantify regulatory changes in messenger (m)RNA expression associated with disease and environmental factors. Gene expression acts as an intermediate phenotype between genotypes and complex traits and is known to act as a modifier to disease susceptibility (Nica and Dermitzakis 2008; Li et al. 2012). Genetic variation underlying gene expression levels has been well established and reported within the literature, with the transcript levels for the majority of genes being heritable to some degree (Price et al. 2011; Grundberg et al. 2012; Powell et al. 2012b).

Microarray technology can simultaneously capture the expression of thousands of transcripts within an individual. However, these arrays are sensitive to environmental or experimental perturbations, for example due to different laboratory technicians and reagents (Churchill 2002; Irizarry et al. 2005), microarray chip and chip position (Luo et al. 2010), temperature (Scherer 2009), and even ozone levels (Thomas et al. 2003). These effects can constitute a substantial proportion of variance within a data set (Leek et al. 2010).

Normalization strategies have become standard in gene expression studies to correct for nonnormal distributions and inconsistencies between arrays (Allison et al. 2006). However, normalization techniques do not control for batch effects caused by technical artifacts. These batch effects require additional correction techniques (Scherer 2009) and failure to do so has led to spurious associations (Spielman and Cheung 2007; Baggerly et al. 2008).

Many different correction and normalization techniques are currently used in gene expression studies (for review see Chen et al. 2011; Qin et al. 2012). Principal components analysis (PCA) is one method that has been used for the correction of widespread batch effects (Leek and Storey 2007; Pickrell et al. 2010; Fehrmann et al. 2011; Qin et al. 2012). PCA determines linear combinations of variables and projects them into orthogonal vectors that are ranked on variance explained (Jackson and Wiley 1991). PCA is often used for dimensionality reduction (Holter et al. 2000), identifying gene correlations (Fukushima et al. 2008); determining coexpressed networks (Hirai et al. 2007); classifying gene expression modules (Fukushima et al. 2008); analyzing time-series data (Raychaudhuri et al. 2000); and determining differentially expressed genes in different tissues (Misra et al. 2002), environments, and various conditions (Ma and Kosorok 2009).

Clearly PCA is a powerful tool to analyze and understand high-dimension gene expression data. However, while PCA is commonly used to decompose variation into common axes, the components of variation contributing to each principal component (PC) are often unknown. Information on the components of variation captured by PCs is important if the correct inferences and interpretation of PCA results are to be made. A clear example is in the use of PCs to correct for batch effects, where variation in the first 1–50 PCs is removed from the data set by fitting these PCs as covariates in a linear model and using the residuals as a corrected phenotype (Leek and Storey 2007; Stegle et al. 2008; Leek et al. 2010; Fehrmann et al. 2011; Fu et al. 2012; Qin et al. 2012; Stranger et al. 2012). Although this method has been demonstrated to increase the power to detect eQTL by eliminating confounding batch and environmental effects (Fehrmann et al. 2011), there is a risk of the indirect removal of biologically relevant information due to the large amount of variance held in the first few PCs (Qin et al. 2012).

In this study we investigated the sources of variation driving PCs generated on a gene expression data set produced from whole blood. This data set is composed of 860 individuals from 314 families from the Brisbane Systems Genetics Study (BSGS) (Powell et al. 2012a). We combine the power of pedigree and SNP-based designs to quantify and dissect the total genetic variance of PCs. From these results we demonstrate that the PC correction methods used in published literature (for example, in Fehrmann et al. 2011; Fu et al. 2012) simultaneously remove both genetic variation and batch effects. We also show that with careful analysis, the information contained within PCs can be leveraged to provide an understanding of gene expression pathways underlying complex disease. We use these results to emphasize the importance of using other correction methods when applicable instead of removing PCs to correct for batch effects.

Materials and Methods

Samples

A total of 335 unrelated individuals were selected from the BSGS, a cohort comprising 860 individuals from 314 families (Powell et al. 2012a). Individuals were selected on two criteria: the first was to obtain the maximum number of unrelated individuals by selecting parents (n = 165) and a single individual from families with no parents (n = 170). Unrelated individuals were selected from this data set to exclude family effects from being present within the PCs. All individuals were genotyped on an Illumina 610-Quad Beadchip (Illumina, San Diego) by the Scientific Services Division at deCODE Genetics, Reykjavik, Iceland. A total of 488,462 SNPs were present after appropriate quality control (see Genome-wide association study on PCs below).

RNA was collected from whole-blood samples that were collected in a PAXgene tube (QIAGEN, Valencia, CA), analyzed for purity on an Agilent Bioanalyzer, converted to cDNA, amplified, and purified using the Ambion Illumina TotalPrep RNA Amplification Kit (Ambion). The expression levels were quantified on an Illumina HumanHT-12 v4.0 Beadchip. Samples were randomized across the chip to minimize batch effects due to families, sex, and generation. Quality control methods for selecting highly expressed probes in the samples are described in detail in Powell et al. (2012a). After appropriate quality control, the probes were further filtered for expression in 100% of samples. Probe names starting with HS, KIAA, and LOC were removed from the data set, as they did not map to characterized ref-seq genes. After filtering, a total of 9086 probes remained.

Replication sample

The Center for Health Discovery and Well Being (CDHWB) study is a population-based cohort consisting of 139 individuals collected in Atlanta (Nath et al. 2012). Gene expression profiles were generated with Illumina HT-12 V3.0 arrays from whole blood collected with Tempus tubes that preserve RNA. Whole-genome genotypes were measured using Illumina Omni Quad arrays.

Normalization and batch effect correction

The gene expression levels of the 9086 probes were normalized using the quantile, log2, and z-score transformation. Correction methods were then applied to create four different data sets that were used for further analysis. These are (a) standard normalization (log, quantile, and z-score transformation), (b) standard normalization with linear model correction for batch effects, (c) standard normalization with correction for 1–25 PCs, and (d) standard normalization with correction for 1–50 PCs. These data sets/correction procedures are referred throughout this article as data sets a–d.

Batch effects arise from various sources during the generation of the data and processing date records can provide a useful estimate of differences between subsets of groups (Qin et al. 2012). In previous work (Powell et al. 2012a), we showed that chip identification (ID) and chip position comprehensively account for batch effects. Data set b was created using the residuals from a linear model with the batch effects of chip ID and chip position as well as sex and age as covariates. This model was fitted as

$$y=X\beta +e,$$ (1)

where

$$y=\left(\begin{array}{c}{y}_1\\ ⋮\\ y_n\end{array}\right)$$

is a vector of probe values from data set a with n = 335 individual values.

$$\text{X}=\left(\begin{array}{cccc}1& x_1& \cdots & x_{1c}\\ ⋮& ⋮& \ddots & ⋮\\ 1& x_n& \dots & x_{nc}\end{array}\right)$$

is a matrix with c = 86 covariates: chip position (11 levels), chip ID (73 levels), and sex and generation, which were coded as dichotomous variables.

$$\beta =\left(\begin{array}{c}\mu \\ {\beta }_1\\ ⋮\\ {\beta }_c\end{array}\right)$$

is a vector of parameters for the model. The parameter μ represents the mean expression level across all the individuals. The vector

$$e=\left(\begin{array}{c}{e}_1\\ ⋮\\ e_n\end{array}\right)$$

holds the model residuals. The parameters in β are estimated by

$$\beta ={\left({\mathbf{X}}^{\mathbf{T}}X\right)}^{-1}{\mathbf{X}}^{\mathbf{T}}Y.$$ (2)

Data sets c and d were created from the residuals obtained after correcting for the first 25 and 50 PCs using Equation 1, where y was the normalized data set (a) and the PC score values (Equation 3, see below) obtained from data set a were fitted as covariates in X. The first 50 PCs were selected to follow the procedure used in previously published gene expression articles (Fehrmann et al. 2011; Fu et al. 2012).

Principal components

PCs were calculated on data set a to facilitate PC correction, which was used to generate data sets c and d. PCs were also calculated on all four data sets a–d to analyze the extent of batch effect associations. To follow the methods used in previous gene expression studies (Leek and Storey 2007; Fehrmann et al. 2011; Fu et al. 2012), the principal components used in this study were generated using singular value decomposition (SVD). SVD decomposes the high-dimensional data set

$$M=\left(\begin{array}{ccc}{m}_{11}& \cdots & m_{1p}\\ ⋮& \ddots & ⋮\\ m_{n1}& \cdots & m_{np}\end{array}\right),$$

with n = 335 (samples) and p = 9,085 (probes), into a set of uncorrelated, orthogonal vectors

$$M=U\Sigma V^{\mathbf{T}},$$ (3)

where

$$U=\left(\begin{array}{ccc}{u}_{11}& \cdots & u_{1n}\\ ⋮& \ddots & ⋮\\ u_{n1}& \cdots & u_{nn}\end{array}\right)$$

is a matrix with score values for each PC (columns), which gives the correlation values between the samples (rows).

$$\Sigma =\left(\begin{array}{c}{e}_1\\ ⋮\\ e_n\end{array}\right)$$

is a diagonal matrix containing the eigenvalues, which represent the amount of variance each PC explains.

$$V=\left(\begin{array}{ccc}{v}_{11}& \cdots & v_{1n}\\ ⋮& \ddots & ⋮\\ v_{p1}& \cdots & v_{pn}\end{array}\right)$$

are the eigenvectors that hold the correlation values for each probe (rows) to the PC (columns) (Golub and Reinsch 1970).

Data set a demonstrates a homogenous variance structure within the score plots (Supporting Information, Figure S1) with no clear population stratification or substructure comprising independent clusters of individuals. As this data set contains the same information as b–d, this indicates the uniform nature of the samples within the BSGS data set.

The eigenvector values in V (Equation 3) represent the correlation between the probes and the principal component. As multiple probes are correlated with each PC, the selection was based on the eigenvector values for each probe. The minimum eigenvector demonstrates the largest negative correction, while the maximum eigenvector demonstrates the largest positive correlation. Across all PCs the maximum eigenvector values ranged between 0.02 and 0.08 and the minimum values ranged between −0.02 and −0.08 (Figure S2A). The PCs that demonstrated the smallest maximum and minimum values were the first 1–20 PCs. As the number of probes selected for each PC can affect later enrichment analysis, we wanted to select a consistent number of probes between all PCs. Therefore a cutoff eigenvector value was used to select probes as this produced the most consistent results and provided a standard approach to use for all PCs. This cutoff value, selected to be < −0.02 or >0.02, incorporated the maximum and minimum values for all PCs to ensure that only the most highly correlated probes for each PC were selected. This eigenvector cutoff value chose a consistent number of probes (∼550) across all the PCs (Figure S2B). The eigenvalues for the first 10 PCs were confirmed via linear regression with the relationship tested using an F-test and P-values corrected for multiple testing, using a Bonferroni adjustment.

Association with batch effects

Principal variance components analysis (PVCA) was used to quantify the extent of batch, age, and sex effects within data sets a–d. This method has been described fully previously (see Li et al. 2009). PCs are generated by an eigenvalue decomposition of a covariance matrix, and the batch effects are quantified with a linear mixed model, using the batch effect terms as covariates. The variance components in each model are estimated using restricted maximum likelihood and are scaled by the eigenvalues obtained for each PC. The variance attributed to each factor is divided by the variance (determined from the eigenvalue) of the corresponding PC and then standardized across all factors.

A similar method was used to quantify the extent of batch effects on the principal components obtained from SVD. Score values obtained from the SVD of data sets a–d were tested for batch effect association, using a multiple linear regression model (Equations 1 and 2), where y is a vector of score values for one PC, with n = 335 individual score values. X is a matrix with c = 86 covariates: chip position (11 levels), chip ID (73 levels), and sex and generation, which were coded as dichotomous variables.

The R² values for the linear regression are calculated as the sum of squares explained by the regression (SSL) divided by the total sum of squares (SST), which gives the percentage of variance in each PC that is associated with batch effects,

$$R^2=\frac{1-\text{SSL}}{\text{SST}},$$ (4)

and are adjusted for multiple covariates using

$$R_{\text{adj}}^2=1-\frac{(1-R^2)(n-1)}{(n-c-1)}.$$ (5)

P-values were obtained from an F-test of the total variance explained by the model as opposed to total variance. Multiple testing was accounted for by Bonferroni correction.

Estimation of heritability

To assess the extent of genetic variability held within the PCs that are removed for PC correction in data sets c and d, estimates for heritability were calculated for all PCs generated from data set a. To compute heritability for the PCs the score values for related individuals had to be estimated. This was done to minimize the presence of family trends within the PCs and to retain the same PCs between expression (e)QTL and heritability analyses. The estimated scores were calculated by multiplying the probe values for those individuals by the eigenvectors calculated from the PCA decomposition,

$$U_{est}=XV,$$ (6)

where

$$U_{est}=\left(\begin{array}{ccc}{u}_{11}& \cdots & u_{1n}\\ ⋮& \ddots & ⋮\\ u_{n1}& \cdots & u_{nn}\end{array}\right)$$

is the resulting matrix of estimated PC score values with n = 425 individuals.

$$X=\left(\begin{array}{ccc}{x}_{11}& \cdots & x_{1p}\\ ⋮& \ddots & ⋮\\ x_{n1}& \cdots & x_{np}\end{array}\right)$$

is a matrix of probe values with P = 9085 probes.

$$V=\left(\begin{array}{ccc}{v}_{11}& \cdots & v_{1n}\\ ⋮& \ddots & ⋮\\ v_{p1}& \cdots & v_{pn}\end{array}\right)$$

is a matrix of eigenvectors, previously derived from the analysis of data from the 335 unrelated individuals.

U_est was combined with U from the original PCA to create an ${\mathrm{ℝ}}^{860\times 335}$score matrix containing values of all individuals for the 335 PCs.

Heritabilities were estimated for the PCs generated from data set a, using Quantitative Trait Disequilibrium Test (QTDT), which partitions variance components attributed to additive genetic, environmental, or common family variance (Abecasis et al. 2000). This method utilizes the complete pedigree structure within the data. Two models were compared in this analysis: an AE model, which includes an additive genetic component and a unique environment component, and a CE model, which includes common and unique environment components. Additive genetic, common, and unique environment variance estimates were divided by the total phenotypic variance to determine the proportion contributed by each factor.

Heritability estimates for all probes were calculated using QTDT on the full data set of 860 individuals. The full data set was corrected using the same four methods used for data sets a–d. The distributions of probe heritability estimates obtained from the four different correction methods were compared. QTDT heritability estimates are constrained to have a minimum value of 0.

Genome-wide association study on PCs

A genome-wide association analysis for each PC was performed using PLINK software (Purcell et al. 2007) on data set a to provide an independent assessment of genetic effects present for PCs. SNPs were filtered based on a minor allele frequency >0.05, missingness >0.10, and Hardy–Weinberg equilibrium P-value <1e-6. After filtering, 488,462 SNPs remained for analysis. Significance was determined both at a family-wise error rate, using Bonferroni correction at an α-level of 0.05, and with an empirical P-value estimation for each PC, using 1000 permutations in PLINK (Purcell et al. 2007).

eQTL analysis

To determine the impact of each of the different correction methods employed in data sets a–d, an eQTL analysis was performed for each probe within these four data sets, using the same procedure as described above. To replicate the effect on eQTL detection we performed the sample analysis for an independent sample comprising 139 unrelated individuals (CHDWB). Cis- and trans-eQTL at multiple FDR levels (Benjamini and Hochberg 1995) were extracted and compared between the data sets. Cis-eQTL associations are defined as being within ±1 Mb of the gene tested.

Biological pathway analysis

The pathway analysis was performed on the online Database for Annotation, Visualization and Integrated Discovery (DAVID) Bioinformatics Resources 6.7 (Da Wei Huang and Lempicki 2008). Probes associated to the PCs of interest were submitted as a list of Illumina probe IDs and Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway enrichment was performed using the Functional Annotation Chart implemented by DAVID (Sherman et al. 2007). Significance of pathway enrichment was determined from a modified Fisher’s test, which calculates the chance that a set of genes of related terms is presented at a certain percentage in the list. Multitesting was accounted for, using a Benjamini–Hochberg false discovery rate (FDR) of 0.05 (Benjamini and Hochberg 1995).

Results

Nongenetic contributions to PCs

PVCA was used to estimate the extent of batch, sex, and age effects within the whole data set (Boedigheimer et al. 2008; Li et al. 2009). This method involves first decomposing the data set into a series of PCs and estimating the effects of each covariate on these components. The total variance of each covariance is then summed across all PCs and scaled by the respective eigenvalue. PVCA was applied to data sets a–d to compare the effectiveness of these methods at removing batch effects from the data.

For data set a, batch effects explained 29.2% of the total variance, with chip ID having the largest effect (Figure 1A). This proportion is far less than the 75.5% cumulative variance explained by the first 50 PCs (Figure S3A), which were removed in previous studies to correct for batch effects (Fehrmann et al. 2011; Fu et al. 2012). The results from linear model correction (Figure 1B) and PC correction (Figure 1, C and D) show that the majority of batch effects have been removed using these methods.

Figure 1.

(A–D) Principal variance components analysis (PVCA) of gene expression data sets. PVCA calculates the proportion of variance in the entire data set that is attributed to certain batch, sex, and age effects. Altogether batch, sex, and age effects account for 29% of the variance in the noncorrected data set (A) and are fully removed in the corrected data set (B). Batch effects present in PC corrected dataset are represented in (C) PC25 and (D) PC50. The residuals represent the remaining variance in the data set not attributed to these batch effects.

To analyze how batch effects are distributed across PCs, each of the four data sets a–d was decomposed using SVD into 335 PCs. Multiple regression of each PC on chip ID, chip position, sex, and age was used to analyze the distribution of these effects across all PCs. The majority of variance attributed to batch effects in the normalized data set a was held within the first 58 PCs of the data set (Figure 2A), with the proportion of variance explained by batch effects ranging from 24% to 74%. This indicates that unknown sources of variance, not associated to batch, sex, or age effects, are also present within these PCs. As the initial PCs pick up the majority of variance within the data set (Figure S3A), significant associations to these PCs indicate a much larger impact of batch, sex, and age effects within the data set. Therefore signification association to the later PCs as in Figure 2B demonstrates a negligible effect (as quantified for each data set in Figure 1). Batch effect associations to the first few PCs are expected to occur more frequently than in later PCs, because the initial PCs can capture the majority of correlation structure within the data set (Jackson and Wiley 1991).

Figure 2.

Batch effect associations to PCs. Significant values are highlighted in red and are obtained from an F-test on the linear regression models. (A) R² from a regression analysis of each PC on batch, sex, and age effects in a data set that is not corrected for any of those factors. (B) R² from a regression analysis of each PC on the batch, sex, and age effects in a data set that is corrected for these factors. There is no significant association to batch effects present. (C) Association to batch effects in a data set corrected for the first 25 PCs. (D) Association to batch effects in a data set corrected for the first 50 PCs.

Overall it is clear that batch effects have a large impact on gene expression variation. However, fitting these as covariates in a linear model during the correction procedure removes the presence of these batch effects (Figure 1B). This correction procedure (data set b), which fits all batch effects as individual factors in a linear model (see Materials and Methods), removes the vast majority of associations with batch effects in the principal components generated on the residuals (Figure 2B). There was only one significant association with PC330 present in this corrected data set and that accounted ∼0% of the variance, which explains why the variance attributed to batch effects was calculated to be zero with PVCA (Figure 1B). Batch effects were also efficiently removed in data sets c and d, which were corrected for the first 25 PCs (68.4% of the variance) and 50 PCs (75.5% of the variance) (Figure 1, C and D, respectively). However, there are still 6% and 2% associations to batch effects in data sets c and d, respectively, and analyses of PCs generated on these two data sets show many components capturing these residual batch effects (Figure 2, C and D, respectively). These results indicate that PC correction does not remove known batch effects as effectively as correcting for chip ID, chip position, sex, and generation with linear models.

Traditionally the last few PCs, which contain only a small fraction of the variance, are removed as they are attributed to noise or experimental error (Gauch 1982). However, the correlation structure present in the data set can drastically change which PCs are attributed to noise or batch effects (Peres-Neto et al. 2005). In gene expression data, it has been demonstrated that the final PCs usually hold genetically relevant information (Yeung and Ruzzo 2001; Ma and Kosorok 2009). Due to the large amount of variance that has been attributed to batch effects within gene expression data sets (Leek et al. 2010), removal of PCs has recently focused on the initial PCs (up to 50) that explain the majority of the variance within the data sets (Stegle et al. 2008; Fehrmann et al. 2011; Fu et al. 2012; Qin et al. 2012). We demonstrated that the initial PCs did have the highest association to these batch effects. However, the variability in which PCs are associated with the batch effects (Figure 2A) makes it difficult to just select an arbitrary number of components to correct for. Removing an uninformed number of PCs could lead to inefficient correction as opposed to using linear models with recorded batch effects. Other factors including biological information could be present alongside these artifacts in these PCs, although we acknowledge that additional batch effects, not recorded, may also be present.

Genetic contribution to PCs

To determine the total genetic component of the PCs that would be removed when using PC correction (as in data sets c and d), heritabilities were estimated for all 335 PCs generated from the normalized data set a (see Materials and Methods). The heritability estimates showed that there was a considerable genetic component to nearly all 335 PCs generated (Figure 3). The mean of the heritability across all PCs was 0.429 (SD 0.1), with the first 50 PCs having an average heritability of 0.39 (SD 0.13). This demonstrates that a high genetic component is still captured in the first 50 PCs despite the strong association to batch effects. Comparisons between genetic and common family environment models indicate that there is a slight confounding between the two (Figure S4). There is a significant association between the estimates between the two models with an R² = 0.08 (P = 8.57e-08). This indicates that the heritability estimates could be slightly biased upward due to confounding with common family effects (Lynch and Walsh 1998).

Figure 3.

Heritability estimates. Narrow-sense heritability estimates of each PC obtained from QTDT. These results indicate that nearly all PCs hold genetic information.

To assess the impact of linear model and PC correction on genetic variation, we estimated the heritabilities of the 9086 probes in the entire data set of 860 individuals, using the same four correction methods a–d (see Materials and Methods). Mean heritabilities for the 9086 probes from the four normalization strategies were 0.32, 0.23, 0.21, and 0.18, respectively (Figure 4). The high mean heritability from strategy a is likely due to inflation from batch effects such as the date of RNA extraction, which was performed in family groups and therefore could not be corrected without removing heritable variation. With PC corrections c and d, there was a much higher proportion of zero heritability probes as opposed to those in strategy b. The lower mean estimate and the higher proportion of zero heritability probes when correcting for 1–50 PCs suggest that PC correction has a much higher penalty of genetic variation than linear model correction.

Figure 4.

Distribution of heritability estimates for 9086 probes from four different correction methods. Black, standard normalized; red, standard normalization with linear model correction for batch effects; blue, corrected for batch effects using the first 25 PCs; purple, corrected for batch effects using the first 50 PCs. There is a drop in heritability using the correction methods; however this is more pronounced in the PC corrected data set. The heritability estimates are constrained to zero due to the nature of genetic variance component estimating in QTDT.

To investigate the impact of PC correction on eQTL detection, we ran a series of eQTL analyses for each probe in the unrelated BSGS data sets a–d. Associated cis and trans variants were extracted at multiple FDR thresholds (Table 1). The results demonstrated a much higher number of both cis- and trans-eQTL detected within the data set corrected with linear models (b) as opposed to PC-corrected data sets (c and d). PC correction, however, enhanced eQTL detection when compared to noncorrected data sets (a), likely reflecting a removal of false positives caused by batch effects. The improved removal of batch effects in the PC50 (d) vs. PC25 (c) corrected data set (Figure 1 and Figure 2) is also reflected by an increased number of cis- and trans-eQTL in data set d. These same trends were also observed in our replication sample (CHDWB) (Table 1). The results from these eQTL studies indicate that the PC correction method negatively affects the number of eQTL that can be detected within gene expression data sets.

Table 1. eQTL results for datasets a–d.

	BSGS
	No correction, data set a		PC 25 correction, dataset c		PC 50 correction, dataset d		Linear model correction, dataset b
FDR	Cis	Trans	Cis	Trans	Cis	Trans	Cis	trans
0.2	1586	8056	1676	9085	1707	9085	2824	9085
0.1	840	2144	929	2530	1149	2564	1914	3025
0.05	596	170	650	254	806	301	1746	490
0.01	449	99	502	137	737	169	1199	264
0.001	349	75	402	104	640	114	1005	218
0.0001	273	56	316	88	510	91	848	183
	CHDWB
0.2	743	8102	815	8234	909	8695	1164	8841
0.1	513	1249	597	1471	684	1538	822	2076
0.05	345	90	373	98	457	152	627	388
0.01	189	51	252	46	366	120	594	215
0.001	143	22	207	31	295	86	470	136
0.0001	101	9	168	15	183	27	385	141

Cis regions were defined as ±1 Mb either side of the transcription start site. Trans was defined as elsewhere in the genome. The numbers of probes with a cis or trans association significant at various study-wide FDR thresholds is provided for each of the four correction methods

To investigate loci driving genetic variation captured within PCs generated from data set a, we performed a genome-wide association study for each PC. The PCs were tested for association with 488,462 genotyped SNPs, using linear models implemented in PLINK software (Purcell et al. 2007). At a study-wide significance level determined by Bonferroni correction (0.05/(488,462 SNPs × 335 PCs) = 3e⁻¹⁰), no significant SNPs were found. We next examined the top SNPs that were significant after Bonferroni correction on each PC (0.05/448,462 = 1.0e⁻⁷). There were 23 SNPs that were significant at this threshold and these were confirmed with 1000 permutations (Figure S5 and Table S1). The lack of genome-wide significant associations could be attributed to the study not having enough power due to a small sample size (Park et al. 2010). Another explanation is that the genetic variance attributed to PCs is highly polygenic. As multiple probes are driving each PC (Figure S2B), the magnitude of different signals can prevent the detection of individual SNP effects.

Biological pathway analysis

We next sought to evaluate whether the first 50 PCs generated from data set a contained linear combinations of genes involved in an expression pathway. Pathway analysis was performed using the DAVID Bioinformatics Resources 6.7, Functional Annotation Tool (Da Wei Huang and Lempicki 2008) on the first 50 PCs (Figure S6 and Table S2). For numerous PCs we are able to demonstrate significant enrichment for multiple different KEGG pathways (FDR = 0.05).

Immune function was the most common process with PC3, PC12, PC24, PC25, and PC26 all showing significant enrichment for immune functional pathways. PC3 showed enrichment for Fc-gamma R-mediated phagocytosis (P = 5.4e⁻³). PC12 showed enrichment for hematopoietic cell lineage (P = 2e⁻³), B-cell receptor signaling (P = 3.8e⁻²), graft-vs.-host disease (P = 4.4e⁻²), non-small-cell lung cancer (P = 4e⁻²), and asthma (P = 4.4e⁻²). PC24 and PC25 were also enriched for B-cell receptor signaling (P = 2.7e⁻⁵ and P = 1.6e⁻², respectively) and PC26 showed enrichment for primary immunodeficiency (P = 1.2e⁻²).

Metabolic processes were enriched in PC8, PC13, and PC32. Enrichment for oxidative phosphorylation was found in PC8, PC13, and PC32 (P = 4e⁻⁵, P = 5.6e⁻³, and P = 3.7e⁻²). PC8 also showed enrichment for genes known to be involved in susceptibility to Parkinson’s (P = 3e⁻³), Alzheimer’s (P = 3e⁻³), and Huntington’s disease (P = 9e⁻⁵) and proteasomal components involved in peptide processing (P = 4e⁻⁵). These brain conditions have been linked to oxidative metabolic dysfunction, due to oxidative damage to neurons (Browne et al. 1997; Mecocci et al. 2004; Rhein et al. 2009) and neuronal energy deficiency (Hoyer 1992) and also improper peptide processing that leads to the buildup of amyloid plaques (Jonsson et al. 2012).

Enrichments for ribosomal components were found in PC1, PC13, and PC18 (P = 4.5e⁻⁵, P = 2.4e⁻², and P = 1e⁻⁴). PC13 also showed enrichment for RNA degradation processes (P = 1.8e⁻³). PC7 was enriched for porphyrin metabolism (P = 2e⁻²) involved in heme biosynthesis.

Of the 11 PCs that showed a significant enrichment (1, 3, 7, 8, 12, 13, 18, 24, 25, 26, and 32), most of them also demonstrated relatively high heritability estimates (0.4, 0.4, 0.58, 0.49, 0.4, 0.45, 0.37, 0.39, 0.57, 0.37, and 0.49, respectively), with an average heritability of 0.45. These results together demonstrate that biologically relevant and interesting probe enrichments are present within the first 50 PCs despite the high association with batch effects within the data set.

Discussion

The removal of genetic variation alongside batch effects when using PC correction has been alluded to but never formally investigated in several articles (Stegle et al. 2010; Fehrmann et al. 2011; Brown et al. 2012; Fu et al. 2012). Due to the unique study design of the BSGS, which contains both unrelated and family information, we are able to quantify the genetic components driving each PC with two independent approaches: SNP association analysis and heritability estimates. From this, we have demonstrated that all PCs obtained from the decomposition of a gene expression data set contain relevant genetic information. Most importantly, we show that the first 50 PCs, which have been removed in previously published articles to correct for batch effects (Leek and Storey 2007; Stegle et al. 2008; Leek et al. 2010; Fehrmann et al. 2011; Fu et al. 2012; Qin et al. 2012; Stranger et al. 2012), contain both a considerable proportion of genetic variation influencing gene expression (Figure 3) and an enrichment for biological networks. The considerable genetic variation found within the initial PCs cautions against the removal of such components in the data set due to the potential loss of genetic information.

We also show that batch effects are distributed across the first 59 PCs with varying effect sizes. As these initial PCs contain a combination of both genetic and batch effects, there appears to be a trade-off between removing batch effects and removing biologically relevant data when employing PC correction. Removing a higher number of PCs improves batch effect correction while at the same time increasing the amount of genetic variation that is removed (Figure 4 and Table 1) and the removal of a smaller number of PCs (as in some studies, e.g., Pickrell et al. 2010) can lead to the incomplete removal of batch effects. While the exact proportions of genetic and batch effect variance observed in the PCs here are unique to this data set, similar patterns of variance distributions are expected to be present in other high-throughput expression data sets.

PC correction became prevalent in gene expression studies after being used to correct for expression heterogeneity by a method called surrogate variable analysis (SVA) (Leek and Storey 2007). This method corrects for noise in the data set that is not accounted for by the primary variable of interest, which may be different experimental conditions or genes. It has been demonstrated as an effective means of enhancing the genetic signals of interest and minimizing false discovery rates. One key difference between SVA and PCA correction is that the principal components were generated on the residuals of the data that were corrected for the primary variable of interest. These principal components contain residual “noise” within the data set and their subsequent removal enhanced the power to detect signals associated with the primary variable. Later studies removed principal components from the whole data set without former corrections for variables of interest (Stegle et al. 2008; Pickrell et al. 2010; Fehrmann et al. 2011; Fu et al. 2012). We have demonstrated here that this can remove genetically driven variation within the gene expression data set that could be of interest to the researcher, due to the ability of PCs to pick up multiple sources of variation (Stegle et al. 2010; Qin et al. 2012).

As the PCs pick up linear combinations of factors in the data set, they also have the power to detect gene regulatory networks and coexpressed modules (Holter et al. 2000). Gene regulatory networks have a considerable impact on disease as opposed to single-gene changes (Chen et al. 2008), as groups of genes are known to interact and respond to environmental perturbations together (Ihmels et al. 2003). These networks are made of coexpressed gene modules that are robust to environmental changes and even different microarray platforms (Chaussabel et al. 2008). Principal components have been used to quantify regulatory SNPs governing metabolic networks in both yeast (Biswas et al. 2008) and human data (Abo et al. 2012). We observed in our data set that there was enrichment for biological networks within the initial PCs. These were dominated by immune function processes, which have been detected previously with clustering analysis of gene expression data (Chaussabel et al. 2008) as well as of metabolic, ribosomal, protein, and heme biosynthesis pathways. As these PCs also demonstrate a high heritability, this indicates that a considerable proportion of the variance in these PCs is being explained by biological factors. The removal of such PCs to compensate for batch effects would have led to the removal of this interesting information. It has also been demonstrated previously that PC correction also removes a large proportion of covariance in a data set, which could constitute these gene networks and interactions (Qin et al. 2012).

Our results show that PC correction negatively affects both average probe heritability (Figure 4) and the number of eQTL hits detected (Table 1). While the removal of a larger number of PCs improved batch effect correction (Figure 1) and increased the number of eQTL detected in the data set (Table 1), it had a much larger impact on mean probe heritability. This genetic variation could be composed of additional factors such as genetic covariance contributing to gene networks that may not necessarily be found in an eQTL study. Linear model correction, on the other hand, demonstrated superior eQTL detection and retained a much higher probe average heritability. Therefore, if recorded processing dates are present, these values can be used to correct for batch effects in the data by incorporating them as covariates in a linear model (Li and Rabinovic 2007). This not only increases eQTL detection by increasing power and reducing false positives (Stegle et al. 2008) but also ensures that, unlike PC correction, large amounts genetic variance are not removed (Figure 4). Strong associations with PCs explaining large proportions of variance in a data set can be used to select for appropriate batch effects to model within a data set if a large number of different factors have been recorded (Parts et al. 2012). We demonstrate here that linear model correction also has the advantage over PC correction in that it is more effective in correcting for known batch effects (Figure 2).

We demonstrate that the removal of PCs from gene expression data sets to correct for batch and environmental effects should be treated with caution, as many different sources of variation can be present within them. This comes from the ability of PCs to detect many linear combinations of trait values (Yeung and Ruzzo 2001). As it can be difficult to distinguish between which factors are driving the PCs without prior knowledge of the batch and environmental trends in the data set, removal of PCs runs the risk of removing biologically interesting data. However, removal of the first 50 PCs can sometimes provide a good method to account for technical artifacts that can lead to spurious associations when no batch effects have been recorded. Our results here show that it is clearly preferable to record such information during the generation of the data and correct for it using standard linear approaches (Scherer 2009).

The data used in this study are available to potential researchers through the Consortium for Genetic Architecture of Gene Expression (CAGE). Should individuals wish to obtain a copy of the data they can apply with a research proposal for membership to CAGE. Applications can be made by e-mailing Peter Visscher (peter.visscher@uq.edu.au) or Joseph Powell (joseph.powell@uq.edu.au).