ARTICLE
A simple Bayesian mixture model with a hybrid
procedure for genome-wide association studies
Yu-Chung Wei
1,2, Shu-Hui Wen
3, Pei-Chun Chen
1,4, Chih-Hao Wang
5and Chuhsing K Hsiao*
,1,4Genome-wide association studies often face the undesirable result of either failing to detect any influential markers at all because of a stringent level for testing error corrections or encountering difficulty in quantifying the importance of markers by their P-values. Advocates of estimation procedures prefer to estimate the proportion of association rather than test significance to avoid overinterpretation. Here, we adopt a Bayesian hierarchical mixture model to estimate directly the proportion of influential markers, and then proceed to a selection procedure based on the Bayes factor (BF). This mixture model is able to accommodate different sources of dependence in the data through only a few parameters. Specifically, we focus on a standardized risk measure of unit variance so that fewer parameters are involved in inference. The expected value of this measure follows a mixture distribution with a mixing probability of association, and it is robust to minor allele frequencies. Furthermore, to select promising markers, we use the magnitude of the BF to represent the strength of evidence in support of the association between markers and disease. We demonstrate this procedure both with simulations and with SNP data from studies on rheumatoid arthritis, coronary artery disease, and Crohn’s disease obtained from the Wellcome Trust Case–Control Consortium. This Bayesian procedure outperforms other existing methods in terms of accuracy, power, and computational efficiency. The R code that implements this method is available at http://homepage.ntu.edu.tw/~ckhsiao/Bmix/Bmix.htm. European Journal of Human Genetics (2010) 18, 942–947; doi:10.1038/ejhg.2010.51; published online 21 April 2010
Keywords: Bayesian inference; GWAS; mixture model; WTCCC
INTRODUCTION
Statistical analysis for testing simultaneously a large number of hypotheses in genome-wide association studies (GWAS) has received considerable attention. Several algorithms have focused on two- or multi-stage procedures, or on the selection of ordered P-values under a controlled error rate.1–3Other procedures have suggested a mixture
structure on P-values, or have adopted the null normal distribution with a dispersed density such as a mixture of beta distributions or a mixture of normals.4,5The underlying principle of all these methods is
to separate the markers into two groups on the basis of P-values, one showing association with the disease trait and the other not. It is thus intuitive to incorporate the mixture approach in genomic analysis, in which only a small proportion, say l, among the enormous number of markers shows association. Our rationale for using a mixture model, however, goes beyond the purpose of clustering, as it considers the estimation procedure primarily as a tool of inference and it accommodates the possible dependence among markers directly.
An estimation approach, rather than hypothesis testing, has earlier been advocated for large-scale association studies primarily so as to avoid overinterpretation of statistical significance.6 By adopting the
mixture model, we get an important advantage of incorporation of dependence among observed data, when several sources of depen-dence arise. For example, linkage disequilibrium (LD) among dense SNPs is one source of correlation. Genotyping data collected from a given individual may produce within-subject dependence. Markers
associating with the disease phenotype share the common feature of ‘showing association,’ and thus may induce yet another type of correlation within a gene set or pathway. Whereas the existence of such dependence in data, and thus among the statistics, may complicate the task of multiple hypotheses testing, a mixture structure on a properly defined statistic may in contrast provide an explanation of dependence in general. For these reasons, several GWAS have adopted the mixture model conceptualization.4,7
Bayesian statistical inference has attracted many researchers for its ability to deal with complex models and with uncertainty from different sources.8 For example, an exploration approach and an
empirical Bayes approach with hierarchical modeling have been considered.9–11Others have proposed a fully Bayesian approach with
a Dirichlet process mixture model for differential gene expression, or a hierarchical mixture model for normalized microarray data.12These
models are useful when the information about an evolutionary region or linkage is available, though they may be computationally intensive because of a large number of parameters. Other proposals have included use of an asymptotic Bayes factor (BF) as a means of incorporating the information of minor allele frequency (MAF) and sample size,13and using a hierarchical mixture model on the loga-rithm of odds ratios as was briefly mentioned earlier.14Nevertheless, a
specific mixture distribution accommodating different sample sizes and variances among various markers has not been previously suggested.14The large number of parameters remains a challenging
Received 13 November 2009; revised 9 March 2010; accepted 10 March 2010; published online 21 April 2010
1Department of Public Health, Institute of Epidemiology and Research Center for Gene, Environment, and Human Health, National Taiwan University, Taipei, Taiwan, ROC; 2Institute of Statistics, National Chiao Tung University, Hsinchu, Taiwan, ROC;3Department of Public Health, College of Medicine, Tzu-Chi University, Hualien, Taiwan, ROC; 4Bioinformatics and Biostatistics Core, Research Center for Medical Excellence, National Taiwan University, Taipei, Taiwan, ROC;5Department of Cardiology, Cardinal Tien
Hospital, College of Medicine, Fu-Jen Catholic University, Taiwan, ROC
*Correspondence: Dr CK Hsiao, Department of Public Health, Institute of Epidemiology and Research Center for Gene, Environment, and Human Health, National Taiwan University, No. 17, Xu-Zhou Road, Room 523, Taipei 100, Taiwan, ROC. Tel: +886 2 3366 8032; Fax: +886 2 2341 8562; E-mail: ckhsiao@ntu.edu.tw
task. Other proposals have included the false positive report prob-ability, a combined measure of frequentist significance and Bayesian subjective probability.15Criticism arises, however, with its Bayesian
interpretation.16Therefore, for GWAS, a fully Bayesian approach with
a simpler model may provide a better tool for inference and for wider applicability.
We propose here a hybrid procedure based on a Bayesian hier-archical mixture model for biallelic markers. This Bayesian mixture model considers not only the grouping of markers, but also the general dependence among data. This model is able to capture the existing dependence with only a few parameters, and thus is simpler than other Bayesian models. There are two important features of our proposed model of particular note. First, we consider a statistic yi,
standardized to be of unit variance, and use a mixture structure on its parameter gi. In other words, this yi represents the ‘standardized’
difference in allele frequencies between case and control groups, and is drawn from a distribution with a meanOnigiand variance 1, where
niis the sample size and the index i ranges from 1 to M markers. The
use of such yi has a major function in our procedure and its
contribution will be explained in later sections. The mixture prior on giis an important element to account for the general dependence
among statistics. In contrast to other mixture structures on P-values or on their transformations, such as test statistics, the adopted model is more intuitive and simpler to implement. Second, the information contained in the mean parameter gi is not affected by the sample
size. Without worrying about the influence from the sample size, a unit information prior for the i-th marker becomes easier to formulate. In addition, most procedures consider the small proportion of association markers, l, a nuisance parameter and often fail to provide a stable estimate for it, not to mention failing to use its estimate at the stage of selecting significant components. We will show here that the posterior inference of l improves greatly the performance of the association test. After estimation of l, a test with the BF at the marker-specific level is carried out to rank and identify susceptible genes.17 Several GWAS from the Wellcome Trust Case–Control
Consortium (WTCCC) as well as simulations are used to evaluate performance of the model.
MATERIALS AND METHODS Bayesian mixture structure
Suppose there are M markers under study, and let l denote the small proportion of association markers among them. The mixture setting of Ml and M(1l) markers within a Bayesian hierarchical model then explains the dependence among markers through their corresponding ‘standardized’ statis-tics yi’s as follows. For the i-th marker, yi is the standardized difference in genetic measurements, say gene expression levels, between two types of tissues or experimental conditions:
yi¼
^ gcs;i ^gcn;i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varð^gcs;iÞ+Varð^gcn;iÞ
p ¼^gcs;iffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^gcn;i ^ s2 cs ncs+ ^ s2 cn ncn q
where gˆcs,iand gˆcn,iare the observed average levels for the case group and for the
control group, respectively; ncs and ncnare the numbers of tissues in each
group; and ^s2
csand ^s2cnare estimates of variance in each group.
Each yi is of unit variance and represents a standardized effect size. For
biallelic markers such as SNPs, yi is the ‘standardized’ difference in allele
frequencies between two groups, yi¼ ^ pcs;i ^pcn;i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ pcs;ið1^pcs;iÞ ncs + ^ pcn;ið1^pcn;iÞ ncn q
where pˆcs,iand pˆcn,iare the observed minor allele frequencies (MAFs) of the i-th marker for the case and control groups, respectively. Note that for biallelic
markers, this standardization uses the empirical variance estimates without loss of degrees of freedom in estimation. Therefore, this procedure is able to avoid the formulation of a prior distribution on nuisance variance parameters. This model is more flexible than one that assumes equal variance for all markers,9
and is easier to work with than a highly parameterized model with unequal variances for all markers.
To satisfy the purpose of association studies, we specify the mixture structure on the mean standardized risk difference gi. For ease of notation, let ni¼n c-s,i¼ncn,i; then the mean of yican be factored as giOni, where giis a function of
the MAFs. This gifollows a mixture distribution with a mixing parameter l,
gijl ð1lÞf0ðgiÞ+lf1ðgijt; uÞ
That is, with probability 1l, gi follows a distribution f0 of no association,
whereas with probability l, gifollows f1, a uniform distribution over (u,t) and
(t,u) of association. The mixing weight (1l) denotes the majority of non-associated markers, and f0(gi) can be a conservative indicator function, I(0),
degenerating at the zero mass, or a distribution over a pre-specified interval for ‘no association.’ Under f1, giranges in the set (u,t),(t,u) (see Supplementary
Materials for discussion about their values and the complete model specification).
Effect size and rare MAF
The standardized risk difference gihere can be considered as a measure of effect
size. As compared with the odds ratios, it is much less sensitive to rare MAFs. The influence from rare MAFs is reduced to a minimum because yihas been
standardized by its SD. In Figure 1a and b, the solid lines show the behavior of giand the odds ratio, respectively, versus the rare MAFs of the control group. It
is apparent that giremains stable in the range of rare MAFs, while the odds
ratio, even divided by 10 (OR/10), contains much more variation. The same pattern can be observed when the difference in allele frequencies between two groups is fixed (dashed lines in Figure 1). Therefore, a mixture structure on giis
more suitable in terms of variability than an odds ratio.
Formulation of the priors
For the two components f0and f1in the mixture distribution of gi, we consider
an indicator function for f0(gi) and a uniform distribution for f1. The latter
implies that all values are equally likely over the intervals (t,u) and (u,t). When the case and control groups are of the same size (ie, ncs¼ncn), the range
can be derived analytically as (t¼0.071, u¼0.825) for biallelic markers with MAFs in (0.05, 0.50). When sample size differs, the range depends on the ratio of ncs and ncn (see Supplementary Material). The above finite values are recommended from a conservative viewpoint, because a larger u would imply stronger evidence against the null a priori, and may lead to an improper posterior for inference.
Hybrid method of the global estimate and the BF
With the likelihood, mixture structure, and prior distribution established, the posterior distribution of l can then be derived (Supplementary Material). Next, based on this posterior distribution f(l|y1,y,yM), one can estimate l, the
proportion of influential markers, and proceed to the statistical inference. We term the estimated posterior mode, ^l, the global estimate of the association proportion. In other words, M^l is the estimated posterior mode of Ml, the number of association markers. This estimate borrows strength from all marker information, and, as the results will show later, it is accurate and stable. If the estimated proportion ^l is non-zero, then a total of M^l markers are considered candidates and will be selected by the rankings of the BFs17for
the M markers. The smaller the BF, the stronger the evidence supporting the association with the disease under study.
To compute the BF for the i-th marker, we test li¼0 indicating no
association versus li¼1 indicating an association. Next, we select the leading
M^l markers with the largest values of 1/BF, the inverse of BF. The magnitude indeed implies directly the strength of evidence for association.17In general,
when BF is less than 1/100, the strength of evidence against the null is considered decisive;17 that is, the data provide strong evidence supporting
the association. Using the threshold 1/100 for BF usually results in more signals than using the global estimate, and is considered more conservative.
RESULTS
WTCCC association studies
We consider genotyping data from the WTCCC18 to evaluate the
performance of the hybrid Bayesian procedure and to compare it with other existing methods. From the WTCCC archive, we obtained genotyping data originally from 1999 rheumatoid arthritis (RA) patients, 1988 patients with coronary artery disease (CAD), 2004 with Crohn’s disease (CD), and 3004 common controls. Exclusion criteria for SNPs were (1) MAF o0.05, (2) call rate o95%, and (3) failure to meet the quality control criteria of the WTCCC.18
Rheumatoid arthritis
After passing the quality control filters of the WTCCC, a total of 366 037 SNPs were selected for 1860 RA patients and 2938 controls for our analysis. Earlier studies have established an association region, the major histocompatibility complex, also called the HLA complex, on 6p21, and a gene PTPN22 on 1p13.2.17–21Here, we examine whether
these findings can be replicated by the proposed method.
Table 1 lists the estimated number of association markers based on the Bayesian hybrid procedure (Bayes), the Bonferroni procedure (Bon), q-values, the Benjamini and Hochberg procedure (BH), and a false discovery control with a P-value weighting scheme (wBH).22 Among the 232 SNPs identified by the Bayesian procedure, 9 are near PTPN22 on 1p13.2, and 200 are in 6p21.32–6p21.33. Both regions generated strong signals. In fact, 217 of the SNPs are on either chromosome 1 or 6. For comparison, the number of SNPs identified by all other procedures (except Bon) is 41.5 times the number (232) obtained with our procedure, and may result in more false positive regions, while the Bon provides the smallest number 200. Figure 2a displays the distributions of these 232 SNPs, and Figure 2b shows the Manhattan plot of the negative log10 transformation of BFs. Supplementary Table A.1 lists their chromosome locations. For those SNPs whose individual BFs imply decisive evidence, details are in Supplementary Table A.2. The Bayesian inference further provides a measure of association strength for each marker (ie, BFo 1/100). The identified SNPs merit further functional or pathway analysis, and can serve as the next candidate regions for further analysis (Supplementary Table A.2).
Coronary artery disease
On the basis of the quality control criteria, a total of 365 984 SNPs of 1926 subjects with CAD were derived. The Bayes identified 26 SNPs, the Bon selected 29 SNPs, and the others 42. Among the 26 SNPs, 16 are in 9p21.3. This region contains two cyclin-dependent kinase inhibitors, CDKN2A and CDKN2B, both of which have been reported to associate with CAD.23 The locations of identified SNPs
are displayed in Figure 3 and the details are in Supplementary Table A.1. It is worth noting that, although the earlier identified APOE gene on 19q13 did not show a signal in the WTCCC reports and was not selected by ^l either, the Bayesian test did provide decisive evidence of association (ie, BFo1/100, Supplementary Table A.3). This again shows the advantage of the hybrid procedure in providing candidate markers based on the strength of evidence.
Crohn’s disease
A total of 366 251 SNPs and 1748 subjects with CD passed the quality control filters, and were included in the analysis. Seventy-four SNPs were identified by the Bayesian global estimate, 72 were determined by the Bon, 178 by q-values, and 167 by both BH and wBH. The locations of identified SNPs are presented in Supplementary Figure A.1 with details in Supplementary Table A.1.
Among the identified 74 SNPs, 22 were in 5p13.1, 15 in 1p31.3, 12 in 16q12.1, 7 in 10q24.2, and 5 in 2q37.1, respectively. These regions Table 1 Numbers of influential markers for three studies (RA, CAD, CD) from WTCCC under different methods
Bayes
P-value
thresholda Bon q-value BH wBH
Rheumatoid arthritis (1860 cases; 366 037 SNPs)
M^l 232 234 200 372 372 372
Coronary artery disease (1926 cases; 365 984 SNPs)
M^l 26 31 29 42 42 42
Crohn’s disease (1748 cases; 366 251 SNPs)
M^l 74 86 72 178 167 167
aNumber of significant SNPs identified by the P-value threshold of 5107used by WTCCC.
0.00 0.02 0.04 0.06 0.08 0.10 −0.2 0.0 0.2 0.4 freq(control), MAF gamma p_cs=0.01 p_cs=0.02 p_cs=0.03 p_cs=0.04 p_cs=0.05 p_cs−p_cn=0.02 p_cs−p_cn=0.01 p_cs−p_cn=0 0.00 0.02 0.04 0.06 0.08 0.10 −0.2 0.0 0.2 0.4 freq(control), MAF OR/10 p_cs=0.01 p_cs=0.02 p_cs=0.03 p_cs=0.04 p_cs=0.05 p_cs−p_cn=0.02 p_cs−p_cn=0.01 p_cs−p_cn=0
Figure 1 The solid lines are values of (a) gi(OR/10 in (b)) versus pcn(small
MAFs of the control group) at different fixed values of pcs. The dashed lines
are values of (a)gi (OR/10 in (b)) versus pcn at fixed values of pcspcn
have all been identified in earlier research as containing genes associated with CD: the gene PTGER4 on 5p13, IL23R on 1p31, NOD2 on 16q12, NKX2-3 on 10q24.2, and ATG16L1 on 2q37.1. Three other SNPs on 5q23.1, 10q21.2, and 18p11.21 were in different regions. For the remaining 10 SNPs, the values of BFs also imply remarkably strong association (Supplementary Table A.4). Further replication studies or pathway analysis would be necessary for confirmation.
Simulation study
Here, we show the performance of this proposed procedure with simulated data, especially with correlated markers. We consider the independent and correlated settings separately with l between 0.001 and 0.1. The dependence setting uses a spatial structure for correlation for LD, where the correlation between any two markers depends on the physical distance,
rij corrðmarkeri;markerjÞ ¼
1dij
y; if dij y
0; if dij4y
(
where dijis the distance between the i-th and j-th marker, and y is a
pre-specified tolerance distance for correlation. The dij follows a
Poisson distribution and y depends on the maximum correlation in the generated data. The procedures for generation of dijand
determi-nation of y are documented in Supplementary Material. Note that
when M is larger than 1000, the correlation between markers at two ends becomes extremely small and thus markers become almost independent. Therefore, it suffices to set M at 1000 in all 100 replications. The global estimation of l is derived, and other existing methods are also performed for comparison.
The results are displayed in Figure 4 and Supplementary Table A.5. For the point estimate of l under independent markers, all procedures provide good estimates, except the conservative Bon. However, when data are simulated with the dependence structure, the Bayesian global estimate performs the best (Figure 4b). Its performance is consis-tently stable and is not affected by the data structure. The correspond-ing standard error is small as well (o0.1% under the independence model and o1% under the dependence model). The q-value (Storey_q) overestimates to a greater degree, confirming the findings in the WTCCC analysis that it tends to overestimate, whereas BH, wBH, and Bon all underestimate (details in Supplementary Materials). We have carried out other simulation studies with settings such as M¼100 000 and l¼0.0001, and the results are as good as the case reported here.
DISCUSSION
On the basis of a simple Bayesian hierarchical mixture model, the hybrid procedure we have used in this article to detect susceptible genes for GWAS has proven useful and robust for extremely small l. The major contributions of our approach are the use of the standar-dized statistic yi, and a further decomposition of its mean to the
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 X Chromosome 0 10000 20000 30000 40000 a b 10 2 1 1 0 207 0 1 0 1 0 1 1 0 0 0 2 2 1 0 0 0 2 80 60 40 1 22
Chromosome and position
– log10(1/100) – log10(1/1665.25) X 20 – log 10 (Ba y e s F actor) 0
Figure 2 (a) Grey bars represent the numbers of SNPs passing the quality control filters for the corresponding chromosomes. The values at the bottom of the bars indicate the numbers of SNPs selected by ^l for RA. (b) Plot oflog10(BF) by chromosome locations.
product of giandOni. This novel design and factorization are essential
within this setting as they make inference of the proportion of association feasible with a fewer number of parameters – this is a case different from and not explicitly considered by others.14 The
standardized risk difference gi, in addition to the direct interpretation
it affords and the easier formulation of priors, is robust to small MAFs and small differences in MAFs, as opposed to when OR or log(OR) is the parameter of interest. The global Bayesian estimate performs well in terms of point accuracy, true positive rate, false positive rate, and
proportions of false positive, for both independent and dependent markers (Supplementary Figures A.2–A.4). Furthermore, the second step of the BF evaluation quantifies the strength of evidence for association, and offers a way to measure the ‘importance’ of markers, thereby making possible allocation of appropriate research resources to the different markers. This quantitative comparison is an advantage that the procedures based on P-values cannot provide.
There are several issues meriting discussion with regard to extension of the model. First, the statistic yi illustrated here is based on the
-log 10 (Ba y es f a ctors) -log10 (1/14822.56) -log10 (1/100) 1 22 X
Chromosome and position 20 15 10 5 0 -5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 X Chromosome 0 40000 30000 20000 10000 0 0 1 2 2 0 0 0 16 1 0 0 0 0 0 0 1 1 0 0 0 0 2
Figure 3 (a) Grey bars represent the numbers of SNPs passing the quality control filters for the corresponding chromosomes. The values at the bottom of the bars indicate the numbers of SNPs selected by ^l for CAD. (b) Plot oflog10(BF) by chromosome locations.
Bmix Bmix Storey_q BH wBH Bon Storey_q BH wBH Bon 0.15 0.10 0.05 0.00 0.001 0.025 0.050 0.100 0.001 0.025 0.050 0.100 λ λ λ -hat 0.15 0.10 0.05 0.00 λ -hat
Figure 4 Estimated proportion ^l versus true l for (a) independent and (b) dependent markers. Bmix is the global estimate, Storey_q for q-values, BH for Benjamini and Hochberg procedure, wBH is false discovery control with P-value weighting, and Bon for Bonferroni procedure.
standardized difference in allele frequencies. This can be extended to data of genotype counts or microarray data. For instance, for any risk allele under the dominant or recessive model, the difference in frequencies of genotypes under study can be computed and standar-dized to derive the global estimate. However, one should be cautious in estimating variance. For biallelic markers as discussed here, estima-tion of the variance does not lose extra degrees of freedom, whereas for other data types, this may not hold true. Second, with microarray data, the range of giin the reference prior will differ from the values
considered for (t,u) in simulations. In our implementations, we suggest the 10th percentile of yi for t. In fact, our investigations
show that any other smaller value will not make any difference in the results. As for u, it has to be greater than all yi’s, and hence the
maximum of yiis recommended. Other larger numbers are certainly
allowable. Third, for gene expression profiling data, this procedure can be compared with current procedures such as the maxT test. Fourth, for the prior specification on gi, other choices with informative priors
are possible. The analytical derivation of the global estimate should be straightforward, or numerical implementations can be adopted. Finally, as already mentioned, this Bayesian hierarchical model is used here to detect association markers based on the hybrid proce-dure, particularly when the proportion of influential markers is very small, or when quantification of the degree of association is of interest. Further applications include meta-analysis of various GWAS to measure small effects and to assess the degree of heterogeneity among studies. The need to synthesize results across multiple studies indeed opens the era for methodological research and collaborations among different disciplines.
CONFLICT OF INTEREST
The authors declare no conflict of interest.
ACKNOWLEDGEMENTS
This study makes use of data generated by the Wellcome Trust Case–Control Consortium. A full list of the investigators who contributed to the generation of the data is available from http://www.wtccc.org.uk. Funding for the project was provided by the Wellcome Trust under award 076113. We are grateful to WTCCC for granting the access, and to Dr Lee H Chen for help with plots. We also thank reviewers and the associate editor for their suggestions, which improved this manuscript greatly (NSC 97-2314-B-002-040-MY3).
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