Abstract
Background
Genomewide association studies (GWAS) have identified hundreds of genetic variants
associated with complex human diseases, clinical conditions and traits. Genetic mapping
of expression quantitative trait loci (eQTLs) is providing us with novel functional
effects of thousands of single nucleotide polymorphisms (SNPs). In a classical quantitative
trail loci (QTL) mapping problem multiple tests are done to assess whether one trait
is associated with a number of loci. In contrast to QTL studies, thousands of traits
are measured alongwith thousands of gene expressions in an eQTL study. For such a
study, a huge number of tests have to be performed (
Results
The results show that PEB has an edge over NPEB. The proposed methodology has been applied to human liver cohort (LHC) data. Our method enables to discover more significant SNPs with FDR<10% compared to the previous study done by Yang et al. (Genome Research, 2010).
Conclusions
In contrast to previously available methods based on pvalues, the empirical Bayes method uses local false discovery rate (lfdr) as the threshold. This method controls false positive rate.
Introduction
Genomewide association studies (GWASs) have done a remarkable progress in searching for susceptibility genes. In GWAS, instead of one gene at a time, variation across the entire genome is tested for association with disease risk. GWASs exploit the linkage disequilibrium (LD) relationships among single nucleotide polymorphisms (SNPs), making it possible to assay genome by testing a finite number of SNPs. Till date, the signals that can be discovered through GWAS has not been reported exhaustively. It is important to annotate SNPs information on expression for the better understanding of the genes and mechanisms driving the association. In many situations, there are more common variants truly associated with disease. These variants are highly likely to be expression quantitative trait loci (eQTLs). eQTLs are derived from polymorphisms in the genome that result in differential measurable transcript levels. Microarrays are used to measure gene expression levels across genetic mapping populations. For at least a subset of complex disorders, gene expression levels could be used as a surrogate/biomarker for classical phenotypes. The gene underlying the eQTL is considered to be an excellent candidate for phenotypic QTL.
eQTL mapping is a statistical technique to locate genomic intervals, that are likely
to regulate the expression of each transcript, by correlating quantitative measurements
of mRNA expression with genetic polymorphisms segregating in a population. In a GWAS,
millions of SNPs are tested at once. Associations that initially appear to be significant
must be statistically adjusted to account for the large number of tests being performed.
A large number of false positives will result in if this correction is ignored. The
multipletesting correction, however, sets a very high threshold for genomewide significance,
on the order of
Two closely related inferential procedures for multiple testing have been discussed in this workafrequentist approach based on Benjamini and Hochberg's ([2]) false discovery rate procedure, and an empirical Bayes methodology developed in Efron et al. [3,4]. These two methods are not only very closely related, they can be used to support each other. In a classic twosample problem in a microarray experiment, these approaches have been discussed by Efron and Tibshirani[5]. However, they have considered nonparametric empirical Bayes (NPEB) model. Parametric Bayesian modeling has been considered by Newton et al. [6], Lee et al. [7], Kendziroski et al. [810], Gelfond et al. [11]. Hierarchical models like gammagamma [6] or lognormalnormal [8] are used quite often in PEB procedures. These models suffer from a serious drawback that the variation is constant among genes. An extension has been done to these models by considering gene specific variations[12]. The application of empirical Bayes has been somehow not very common in literature. The obvious reason is that, experimenters have not brought us many data sets having the parallel structure necessary for empirical Bayes to do its stuff. Because of the recent surge in highthroughput ([13]) technologies and genome projects, many genome studies are now underway. These studies have become a major data generator in the postgenomics era. Empirical Bayes procedures seem to be particularly wellsuited for combining information in expression data.
One of the fundamental statistical problems in microarray gene expression analysis is the need to reduce dimensionality of the transcripts. This can be achieved by identifying differentially expressed (DE) genes under different conditions or groups. Regulatory network can be obtained by associating differential expressions with the genotype of molecular markers. It is possible to have a large number of DE genes that influences a certain phenotype while their relative proportion is very small. It is very important to identify these DE genes from among the number of recorded genes [6,7,9,14,15]. Empirical Bayes methods provide a natural approach to reduce the dimensionality significantly [16,17]. Following the empirical Bayes approach DE genes are identified using the posterior probability for differential expression. EB approaches detect a DE gene by sharing information across the whole genome.
The development of the empirical Bayes methodologies that improve the power to detect DE genes essentially reduces to the choice of whether genespecific effects should be modeled as fixed or random [18]. Both mean and error variance can be of either of these two: fixed or random. Fixed mean and random error variance has been considered by Wright and Simon [19] and Cui et al. [20] whereas Lonnstedt et al. [21], Tai and Speed [22], Lonnstedt and Speed [23] have considered both the parameters to be random. Random mean effect with homogeneous fixed error variance has been considered by Newton et al. [6,24], Kendziroski et al. [9] and Kendziroski et al. [10]. However an extension to this fixed error variance has been considered by Gelfond et al. [11]. They have considered discrete uniform prior for the variance component.
The paper is organized as follows. In the Methods section we introduce the necessary notations for our additive genetic model along with the notions of false discovery rate (fdr). In this section we have tried to establish the relationship between fdr and empirical Bayes. Methods section also describes, the proposed Expectation/Conditional Maximization Either (ECME) (Liu and Rubin [25]) in details. This algorithm generalizes the ExpectationMaximization algorithm with better convergence rate. A simulation study has been performed and described in the Results section. We show that proposed parametric empirical Bayes performs better compared to nonparametric empirical Bayes in terms of controlled fdr. In the Results section, as an application, we have applied the proposed methodology to the Liver Cohort (LHC) dataset. We conclude the article the Discussion section.
Methods
In a microarray experiment, we obtain several thousand expression values, one or many
for each gene. These studies offer an unprecedented ability to do largescale studies
of gene expression. Let us define G_{i}i = 1.....l as the genomic marker(i.e. SNP), and T_{j}(j = 1......J) as the transcripts. The identified eQTLs refer to the significant Gs that are associated
with Ts. These associations can be found using a test statistics based on all
where
When expression measurements between two groups are compared for any transcript, the
observations are partitioned into two user defined groups of sizes
where
where
For any transcript and any SNP there may be three possible relations: no association, positive association and negative association. Extending the idea of two component mixture model, the distribution of the test statistics is modeled by the following mixture model:
Where
with
Full Bayesian analysis of (4) will require prior specifications of
Empirical Bayes, false discovery rates (fdr) and local false discovery rate (lfdr)
False discovery rate (fdr) is defined as the expected proportion of errors committed
by falsely rejecting null hypotheses. Benjamini and Hochberg's [2]fdr criterion has very close relation with the empirical Bayes analysis. This relation
improved the connection between Bayesian and frequentist testing theory. The close
connection between fdr and the empirical Bayes methodology follows directly from Bayes
theorem and this has been established by the "Equivalence theorem"[28]. Tail area rejection regions like
The empirical Bayes approach suggests a local version of the fdr called local false
discovery rate (
Analytically,
For the above set up in (3),
and hence
All other parameters will be estimated by EM algorithm assuming
Nonparametric empirical Bayes (NPEB)
The main difference between parametric empirical Bayes (PEB) and nonparametric empirical
Bayes (NPEB) is the way in which
ECME algorithm
To fit a mixture model, EM algorithm is widely used. In case of t distribution the mean parameter
For the
where
and
McLachlan and Krishnan [32] have already discussed the application of the EM algorithm for ML estimation in case
of single component t distribution. In ECME algorithm, this result has been extended to cover the present
set up of a 3component mixture of t distribution. For the sake of brevity, in this section we omit the suffix ij for all the variables. To define t distribution with mean
then marginally,
Following the above definition, the complete data likelihood
where
and
EStep
To compute the Estep of the proposed algorithm, at (t+1)th step we need to calculate
where
and
which is the posterior probability that
Similarly,
Where
CMstep
In usual Mstep parameters
and
To get an efficient algorithm, let us partition
CMStep 1. Keeping
CMStep 2. Now fix
Furthermore to make the algorithm more efficient, after the first CMstep, we replace
the Estep with
Simulation study
To assess the proposed methodology, a small sample simulation study has been performed. This gives an idea whether or not the parameters are well estimated and most importantly, they provide information of false discovery rates.
First we simulated a dominant model with 10,000 transcripts and 10 SNPs. The equivalently
expressed (EE) transcripts are generated from N(0,1) after logtransformation. We
have simulated the data under three choices of proportions of differentially expressed
(DE) transcripts (
Figure 1. A part of the simulated data for
The impact of minor allele frequency (MAF) on the distributions under null has also been studied. Under null, for a tdistribution, the only parameter to be estimated is its degrees of freedom. The comparison has been made by computing different quantiles for six choices of MAFs. For the lower quantiles, they almost overlapped with each other. Very small deviations are observed for upper quantiles (Figure 2).
Figure 2. Effect of minor allele frequency (MAF) on the null distribution. Only upper quantiles (from 80%) have been considered as lower quantiles showing almost no difference.
For the 10 SNPs, we fitted the null distribution using permutation method in a balanced way. From each group, randomly selecterd 35 samples are shifted from one group to the other and the value of the statistic is noted. This process is repeated 40 times and histograms are plotted. From the histograms, the degrees of freedom corresponding to the null distribution for eack SNP is estimates. To get an idea about the goodnessoffit, QQ plots are done (Figure 3). These plots show that the null distribution is well approximated by the standardized tdistribution with appropriate degrees of freedom.
Figure 3. QQplot for eight SNPs.
Parameters related to the mixture model (4) are estimated using proposed ECME algorithm after estimating the null distribution using permutation method. Then FDR is computed under both proposed parametric empirical Bayes and nonparmetic empirical Bayes setup and the result is given in Table 1.
Table 1. The True FDR Performance of Controlled FDR in EB Models
It is evident from the above table that the nonparmateric empirical Bayes is much conservative compared to its parametric alternative. For parametric set up, the true FDR is very much close to the controlled one, whereas, for nonparametric empirical Bayes these values are not so close as the true fraction of DE transcripts increases.
HLC data analysis
We applied the empirical Bayes model to analyze a sequencing data publicly available. In the current study, we have started with liver tissue data of 213 Caucasian samples from apreviously described human liver cohort (LHC) (Yang et al. [33]). To get the genotypes and gene expression profiles, DNA and RNA have been isolated. Illumina platform is used to get the expressions. After putting some filtration (MAF>5%, HWE<10^{5},) we are left with 173 samples, 472,000 SNPs and 30,000 expressions.
The distribution of minor allele frequency (MAF) over SNPs is given in the histogram
(Figure 4). For all possible SNPtranscript combinations, test statistic,
Figure 4. Minor allele frequency (MAF) distribution. X axis corresponds to minor allele frequency 25% to 50%.
Conclusion
To compare our result with [33], we focus on 18 of the 54 P450 genes used in the study. These are CYP3A5, CYP2D6, CYP4F12, CYP2E1, CYP2U1, CYP1B1, CYP2C18, CYP4F11, CYP4V2, CYP2F1, CYP39A1, CYP26C1, CYP2C19, CYP2C9, CYP2S1, CYP46A1, CYP4A11 and CYP4X1.However our method fails to identify a single SNP with FDR<10% for CYP2R1 and that gene symbol has been excluded from the table (Table 2). It can be seen from the table (Table 2) that for a threshold of 10% FDR number of significant eQTL pairsis4916.Since we have considered only top SNPs, this may be an overestimate. SNPs which are within <1Mb distance from gene location are defined as cisSNPs. It is interesting to note that, among these 18 genes, the first five (CYP3A5, CYP2D6, CYP4F12, CYP2E1 and CYP2U1) having more than 40 cisSNPs. In all cases FDR based analysis results in identifying more cisSNPs for these 18 genes compared to that of Yang et al. (2010) [33].
Table 2. Number of eQTL pairs after crossing the threshold of FDR
Discussion
In contrast to previously available methods based on pvalues, the empirical Bayes method uses local false discovery rate (lfdr) as the threshold. This method controls false positive rate. For a particular SNP, the lfdr is computed for the sitespecific evidence whereas the FDR averages over other sites with stronger evidence. There are some limitations of using FDR which may result in misleading inferences in genome studies. In such a situation, it is better to use lfdr which is a bit difficult to estimate compared to FDR.However there is still one computational problem which needs much attention. Due to the high dimensionality in the data, sometimes existing algorithms fail. This necessitates the need to find some more efficient algorithms. The choice of threshold FDR value is an important deciding factor in such studies. It would be interesting to see, how number of cisSNPs vary with the change in FDR threshold. In this way FDR criterion can be used to estimate number of SNPs that we may need to consider.
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
This work is supported by the U.S. National Institutes of Health grants R01 GM74217 (Lang Li) and AHRQ Grant R01HS01981801 (MalazBoustani)
Declarations
The publication costs were funded by the authors through P50 CA113001 (Huang, T.M.), R01 GM088076 (Skaar, T.), R01 HS019818 (Dexter).
This article has been published as part of BMC Genomics Volume 14 Supplement 8, 2013: Selected articles from the International Conference on Intelligent Biology and Medicine (ICIBM 2013): Genomics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcgenomics/supplements/14/S8.
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