Abstract
Background
The information provided by dense genomewide markers using high throughput technology is of considerable potential in human disease studies and livestock breeding programs. Genomewide association studies relate individual single nucleotide polymorphisms (SNP) from dense SNP panels to individual measurements of complex traits, with the underlying assumption being that any association is caused by linkage disequilibrium (LD) between SNP and quantitative trait loci (QTL) affecting the trait. Often SNP are in genomic regions of no trait variation. Whole genome Bayesian models are an effective way of incorporating this and other important prior information into modelling. However a full Bayesian analysis is often not feasible due to the large computational time involved.
Results
This article proposes an expectationmaximization (EM) algorithm called emBayesB which allows only a proportion of SNP to be in LD with QTL and incorporates prior information about the distribution of SNP effects. The posterior probability of being in LD with at least one QTL is calculated for each SNP along with estimates of the hyperparameters for the mixture prior. A simulated example of genomic selection from an international workshop is used to demonstrate the features of the EM algorithm. The accuracy of prediction is comparable to a full Bayesian analysis but the EM algorithm is considerably faster. The EM algorithm was accurate in locating QTL which explained more than 1% of the total genetic variation. A computational algorithm for very large SNP panels is described.
Conclusions
emBayesB is a fast and accurate EM algorithm for implementing genomic selection and predicting complex traits by mapping QTL in genomewide dense SNP marker data. Its accuracy is similar to Bayesian methods but it takes only a fraction of the time.
Background
Genomewide association (GWA) studies are being used more often for risk prediction in humans and trait prediction in livestock. Such studies associate individual single nucleotide polymorphisms (SNP) from a dense genomewide panel with betweenindividual variation in traits. The GWA provides measures of strength of association and estimates of the size of the effect of each SNP even though SNP identified as being of predictive value are unlikely to be causative. These GWA studies have had limited success as the individual effects of loci are often small and relatively few loci pass the very stringent statistical testing criteria imposed. The detected variants can be used to construct genetic profiles [1,2] but jointly the loci identified often explain less than 10% of the phenotypic variance [24]. This small fraction of variance explained is due in part to the stringent statistical thresholds required for identification in GWA studies [5]. Nevertheless the scope of the genomic information provided by high throughput technology using dense SNP panels remains of considerable potential.
Researchers in other fields, in particular animal and plant breeding, have developed methods of prediction of genetic value that use all available marker information simultaneously and do not apply such stringent tests of statistical significance [6,7]. Thus, instead of testing hundreds of thousands of separate hypotheses of 'is this single SNP associated with the trait' as in GWA, the problem is modified to 'what function of the entire SNP information provides the best predictor of the trait'. The outcome of these approaches is that many more loci are used in prediction. Although the set will now include false positive loci it also includes many more true positive effects and the overall predictive power is much improved [8]. This approach to genomewide prediction is called genomic selection and is being applied to livestock in practice [9].
Different statistical approaches to genomic selection have been attempted. One approach is to use the markers to construct the realised relationship matrix, rather than an expected one based upon pedigree, followed by use of this realised relationship matrix in established BLUP procedures [8]. When BLUP is used for genomic selection (hereafter called GSBLUP) the prior distribution of the marker effects is assumed normal, with the variance of the prior distribution being equal for each marker. But this "equal variance" assumption is biologically unrealistic as many markers will lie in regions that are not involved in trait determination and so contribute no trait variance. This was the finding in [6] where simulations of genomic selection found that GSBLUP was less accurate than Bayesian methods which allowed marker specific variances which cause differential shrinkage of marker effects.
Differential shrinkage of marker effects across the genome can be performed by assuming the marker effects are normally distributed with variances which are independent random variables following a specific distribution. BayesA [6] assumes marker variances follow an inverted chisquare distribution while Bayesian LASSO (BayesL) [10] assumes an exponential distribution. Integrating out the variances it can be shown that the conditional distribution for the marker effects is a doubleexponential (DE) for BayesL and a tdistribution for BayesA. As the DE places more density at zero than a tdistribution this suggests that more shrinkage will occur for small effects with BayesL than with BayesA. In fact the original LASSO [11] can be interpreted as a Bayesian posterior mode when an independent DE prior is assigned to each marker effect as shown in equation (2) in [10]. However with dense marker data, many SNP will not contribute to predicting QTL genotypes through LD and the LASSO may not perform enough shrinkage of small marker effects to comply with this prior knowledge [12]. A somewhat similar conclusion was demonstrated in [6] for the tdistribution prior by comparing two Bayesian methods called BayesA and BayesB. BayesB used a prior mixture which assumed a BayesA prior for a small proportion of markers and allowed the rest of the marker effects to be precisely zero a priori. BayesB was shown to increase selection accuracy in simulated data when compared to BayesA. However this comparison has not been conducted in a full Bayesian analysis using a DE prior like in BayesL.
A major problem associated with a full Bayesian analysis is the computing time required to fit the model. The challenge is to fit hundreds of thousands of SNP to many thousands of individuals with genotypes. Markov Chain Monte Carlo (MCMC) techniques such as Gibbs sampling are tractable when the dimensionality and data size are small. However this is not the case with dense SNP data and thus has led to the development of fast algorithms for Bayesianlike marker selection models, involving either heuristic approximations to fit into standard BLUP models [9] or an iterated conditional expectation (ICE) approach [13] which iterates an analytical calculation of each SNP's conditional posterior mean. However it is unclear in what sense the solutions of these fast algorithms are optimal.
Expectation maximization (EM) algorithms can use the information in a prior distribution through the calculation of a maximum a posteriori (MAP) estimate [14] and are usually much faster than a full Bayesian approach. This result was demonstrated in an EM algorithm developed for implementing genomic selection [15]. In this paper we suggest a different formulation of the SNP prior mixture compared to the EM algorithm called wBST which was developed in [15]. This results in a number of advantages which will be discussed later. Hence this paper investigates a solution to the Bayesian SNP selection model through an EM algorithm which has a solid statistical foundation compared with the fast heuristic approaches. In the sections that follow (i) we develop an algorithm (called emBayesB) using standard EM theory, (ii) we propose an implementation to work with the dimensionality that is encountered in human data sets, (iii) we benchmark emBayesB by analysing a simulated workshop data set, and finally (iv) we explore the shrinkage features of emBayesB both analytically and graphically.
Methods
Data model for SNP effects
Each of the n individuals in the study is genotyped for m SNP markers and has a record for a continuous trait y. The trait is assumed to depend on alleles of unknown QTL which, either directly
or indirectly through LD, induce an association with the SNP markers. We assume that
SNP marker j has two alleles, 0 and 1, with 1 being the reference allele which has a frequency
p_{j }in the n individuals. The three possible genotypes '0_0', '0_1' and '1_1' for SNP marker j are coded 0, 1 and 2 respectively, and are standardised by subtracting the mean (2p_{j}) and dividing by the standard deviation
As each of the n individuals is genotyped for m SNP markers we can construct an n × m standardised frequency matrix B consisting of the m column vectors b_{j}. We assume a linear model for the 'SNP mediated' effects of the QTL, namely y = Bg + e where y is the n × 1 vector of phenotypic records, g is the m × 1 vector of SNP effects and e is an n × 1 vector of residuals which are assumed independent and identically distributed normal
random variables i.e.
Missing data and SNP prior distribution
We assume a priori that a proportion γ of the SNP markers are in LD with at least one QTL and that an unknown binary variable z_{j }(the missing data) indicates whether SNP j is in LD with QTL. That is, a priori
If z_{j }= 1 (i.e. SNP j is in LD with QTL), the SNP effect g_{j }is assumed to be from a DE distribution with parameter λ i.e.
Now the joint prior p(z_{j}, g_{j}) is as follows
Assuming independence of the m SNP effects, the joint prior for z and g is
Posterior distribution and EM algorithm
Apart from a normalising constant, the posterior distribution p(z, gy) is
where
To maximize the log posterior, we use z_{j }as missing data in an EM algorithm [14]. In the Estep we evaluate
Additional file 1. Appendix A. A pdf file giving the Estep of the EM algorithm: Derivation of
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This file can be viewed with: Microsoft Word Viewer
Estimators of g_{j}, γ, λ and
σ
e
2
for the Mstep
In Additional file 2, it is shown that the maximum a posteriori (MAP) estimate of g_{j }is
Additional file 2. Appendix B. A pdf file giving the derivation of the estimators
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where
as the DD_{mode }is always zero (i.e. DD_{mode }= 0) reflecting the posterior certainty due to the Dirac Delta prior certainty about the SNP effect.
It is also shown in equation (B9) of Additional file 2 that the ML estimators of γ, λ and
where γ^{k }is the vector of posterior probabilities at iteration k.
emBayesB using GaussSeidel iteration
The steps in the EM algorithm using GaussSeidel (GS) iteration are as follows:
Step 1. Start with an initial set of values for
Step 2. For SNP j (j = 1,…,m), calculate
Step 3. Use the current estimates of
Step 4. Repeat Steps 2 and 3 until convergence which is assessed at iteration k using the criterion
If needed, fixed effects can be fitted in the model simultaneously with the SNP effects as explained in [13].
emBayesB for large SNP panels
In Step 2 of the EM algorithm using GS we calculate all possible combinations of b'_{j}b_{l }(i.e. B'B) each iteration. It is more computationally efficient to store the symmetric matrix
B'B at the start. However this matrix is of order m × m which will be huge for large SNP panels. To avoid the calculation of B'B we use GaussSeidel iteration with residual update (GSRU) as described in [16] where it was used to avoid the calculation of B'B in a heuristic BLUP approach to genomic selection. Basically GSRU avoids the calculation
of B'B by using the identity
Step 2^{GSRU}. For SNP j (j = 1,…, m), calculate
Step 3^{GSRU}. Now update the ML estimates of γ, λ and
As mentioned in [16], e should be recalculated periodically (e.g. at each iteration) using
Simulation example
To benchmark the capabilities of emBayesB the SNP data distributed to participants of the QTLMAS XII workshop was analysed. A summary of the data simulation is given here, with full details available in [17]. An initial population of 50 male and 50 female founder individuals was created. For the next 50 generations, 50 males and 50 females were produced by random sampling parents each generation. For the last six generations, 15 males and 150 females were selected randomly for a hierarchical mating, with each male mated randomly to 10 females who produce 10 progeny each, giving a total of 1500 pedigreed progeny per generation. The 1200 individuals in the validation data set consisted of a random sample of 400 progeny from each of the last three generations. The 4665 individuals in the training data set were progeny from the preceding four generations; three generations of 1500 progeny plus the initial 15 males and 150 females. The training data set contained both SNP genotypes and phenotypic records, while the validation data contained only SNP genotypes.
There were 6000 biallelic marker loci on six 100 cM chromosomes with a 0.1 cM spacing between marker loci which gave 1000 markers per chromosome. Marker alleles were sampled with equal probability in the founders. QTL effects were sampled from a gamma distribution. The genomic location and allele substitution effects of the 48 simulated biallelic and additive QTL are shown in Figure 1. More detail about the QTL effects is available in [18]. The number of QTL which explain more than 0.1, 1, 5 and 10% of the total genetic variation in the training data set were 28, 15, 6 and 4 respectively. The true breeding value (TBV) of an individual was calculated as the sum of its QTL effects. Phenotypic records were calculated for the training data set by adding a normally distributed residual error term to each individual's TBV. The variance of the normally distributed residual error term was chosen to produce a heritability of 0.3 for the trait.
Statistical analysis
The prediction equation
GEBV were also calculated for GSBLUP, LASSO and the ICE algorithm. The estimated
SNP effects for GSBLUP were solutions to
Additional file 3. emBayesB.zip. The zip file "emBayesB.zip" contains the Fortran 90 source code "emBayesB_gs.f90" and the Windows executable "emBayesB_gs.exe" for the emBayesB program. A readme file gives instructions on using the program and the input/output files. The two input data files "emBayesB_input*.txt" are also in the zip file.
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The emBayesB algorithm had difficulty with estimation of λ for some heritabilties. This is probably a reflection of the flat likelihood surface for estimating λ particularly when combined with estimating γ. Hence an upper bound was placed on λ in each analysis with the upper bound being the corresponding λ used as the initial value for the LASSO. If the bound was reached then the current estimate of λ was reset to its initial value. This procedure seemed to produce a good searching algorithm for parameter estimation with emBayesB given the complexity of the likelihood surface.
Results
Comparison of methods using simulated data
The emBayesB algorithm, and indeed all methods in Table 1, took only a few minutes to converge on a 2 GHz laptop PC for the 6 k SNP panel simulated. This was considerably faster than a full Bayesian analysis similar to [6] which took approximately 2 days (R. PongWong, pers. comm.). A similar difference in computer time was reported in [13] where ICE was compared with a full Bayesian analysis (an MCMC implementation of the BayesB algorithm).
Table 1. Correlation and regression coefficient of TBV on GEBV for various generations of the validation data.
emBayesB was the most accurate method of predicting TBV in the validation data over all heritabilities (Table 1). The emBayesB correlation of 0.88 between GEBV and TBV for all 1200 individuals was similar to correlations of 0.84 to 0.87 for Bayesian MCMC methods performed on the same data, but larger than correlations of 0.5 to 0.77 for various BLUP models [17]. GSBLUP produced correlations of 0.75, 0.71 and 0.66 for heritabilities 0.3, 0.5 and 0.7 respectively (Table 1). Using the top 10% of individuals ranked on TBV in the validation data, the calculated Pearson correlation was 0.51 for emBayesB, while the rank correlation between GEBV and TBV was 0.41 when initial heritability was 0.5. This rank correlation was lower than the rank correlations of 0.46 to 0.58 for Bayesian MCMC methods applied to the same data [17] but larger than the GSBLUP rank correlation of 0.27.
ICE with γ = 0.01 produced a correlation of 0.87 when heritability was 0.1 (Table 1). However the correlations for ICE decreased as initial heritability increased, whereas
for emBayesB the correlations remained constant due to the ability of the EM algorithm
to estimate the unknown parameters. If the emBayesB parameters γ, λ and
The LASSO produced similar correlations to emBayesB when heritability was 0.3 and 0.5, but smaller correlations when heritability was 0.1, 0.7 and 0.9 (Table 1). Heritabilities of 0.1, 0.3, 0.5, 0.7 and 0.9 correspond to λ values of 161, 93, 72, 61 and 54 respectively for the LASSO. As λ decreases the LASSO performs less shrinkage such that the number of nonzero LASSO estimates of SNP effects increases and was 20, 57, 132, 233 and 1029 as heritability increased in Table 1. In practice the λ value would usually be determined by cross validation for the LASSO. When heritability was 0.3 or 0.5, the LASSO correlations decreased very little across the three generations similar to emBayesB (Table 1). Using γ = 1 the ICE algorithm was not able to match the performance of the LASSO which used a fixed γ = 1 in the emBayesB algorithm with all other parameters fixed (Table 1). The reason for this result is illustrated graphically later.
The regression of TBV on GEBV was biased for GSBLUP and ICE for all heritabilities in Table 1. For emBayesB and the LASSO the regression of TBV on GEBV was only unbiased when heritability was 0.5 although emBayesB displayed the least bias for each heritability.
For an initial heritability of 0.5, the final parameter estimates were
Figure 1. 48 QTL effects and 5726 SNP posterior probabilities of being in LD with QTL. Average effect of an allelic substitution in the training data set (▲) plotted against genomic location for each of the 48 QTL. Also the SNP posterior probability (+) of being in LD with at least one QTL plotted against genomic location for each of the 5726 SNP. The QTL effects are in absolute values.
SNP results for emBayesB when h^{2 }= 0.5
The SNP results that follow were obtained using emBayesB with an initial heritability of 0.5. As expected most SNP have a small posterior probability of being in LD with at least one QTL (Figure 1). In fact 5660 SNP have posterior probabilities less than 0.1, while only 27 SNP have posterior probabilities greater than 0.5. emBayesB detected all QTL with allele substitution effects greater than 0.18 by calculating posterior probabilities of 0.98 or more for nearby SNP (Figure 1). On chromosome 6 all SNP have posterior probabilities less than 0.22 which was in accord with the absence of QTL simulated on this chromosome.
Of the 48 QTL simulated, there were 15 QTL which, individually explained more than 1% of the total additive genetic variation, and in total, explained over 95% of the additive genetic variance. emBayesB detected each of these 15 QTL by calculating posterior probabilities of 0.99 or more for nearby SNP (Figure 2). The distance from each of these 15 QTL to the nearest high probability SNP averaged 0.7 cM, with the largest distance being 1.7 cM. Three QTL each explained more than 12% of the genetic variation and this large variation resulted in multiple nearby SNP having posterior probabilities of 1 (Figure 2).
Figure 2. Genetic variation explained by each of the 48 QTL and 5726 SNP posterior probabilities. Percentage of the total genetic variance in the training data set explained by each QTL (▲) plotted against genomic location for each of the 48 QTL. Also the SNP posterior probability (+) of being in LD with at least one QTL plotted against genomic location for each of the 5726 SNP.
There were 25 SNP with posterior probabilities greater than 0.9 and the distance averaged 0.85 cM from each of these 25 SNP to the nearest QTL explaining more than 1% of the total genetic variation. As the genetic variation explained by a QTL dropped below 1%, the posterior probability of nearby SNP decreased toward zero. Hence in this simulation it was found that the SNP posterior probabilities could be used to accurately locate QTL explaining more than 1% of the total genetic variation.
In general the SNP used for prediction were different for emBayesB and the LASSO. For example with an initial heritability of 0.5, the number of estimated SNP effects greater than 0, 0.01 and 0.1 was 2841, 15 and 10 for emBayesB compared with 132, 72 and 6 for the LASSO. However the LASSO did use SNP which emBayesB estimated as having a nonzero posterior probability of being in LD with QTL. For example, the LASSO used 57 and 132 nonzero estimates of SNP effects for heritabilities of 0.3 and 0.5 respectively, and these SNP had average posterior probabilities of 0.31 and 0.16 of being in LD with QTL as estimated by emBayesB.
Analytical emBayesB shrinkage
In this section we graphically explore features of emBayesB in order to assist with
understanding how the algorithm works. Figure 3 shows the shape of the conditional posterior distribution of g_{j }given in equation (A2) of Additional file 1. The graphs assume
Figure 3. Graphical illustration of how a posterior probability is calculated for a SNP. Graphs of the mixture prior p(g_{j}), conditional likelihood h(g_{j}G_{j}, σ^{2}) and conditional posterior distribution p(g_{j }g_{j,}y) as given in equations (A1) and (A2) of Additional file 1 for
When the cML estimate (G_{j}) of g_{j }is far from 0 the conditional posterior resembles the conditional likelihood, but
is slightly shifted (or shrunk) toward 0. The mode of the shrunk likelihood is G_{j}λσ^{2}(=G_{j}0.02) when G_{j }is much greater than 0. This is the LASSO estimate as the Spike has little influence
when G_{j }is far from 0. However as G_{j }approaches 0, the conditional posterior becomes bimodal, with the height of the mode
at 0 increasing the closer G_{j }is to 0 (Figure 3). This reflects the fact that, the closer G_{j }is to 0, the higher is the probability that the true g_{j }is 0. Using numerical integration in equation (A3) it can be shown that the area under
the DE part of the conditional posterior is 0.99, 0.67 and 0.18 for G_{j }values of 0.19, 0.15 and 0.11 as shown in Figure 3. In the EM algorithm this DE area is
Using numerical integration it can also be shown that the mean of the conditional
posterior is 0.1677, 0.0868 and 0.0165 for G_{j }values of 0.19, 0.15 and 0.11 respectively, while the MAP estimates of g_{j }(calculated using equation (7)) are 0.1677, 0.0868 and 0.0163 for the same values
of G_{j}. So the MAP estimate of g_{j }is an accurate estimate of the conditional posterior mean. Hence at convergence, it
is reasonable to expect that the MAP estimate will be an accurate estimate of the
marginal posterior mean of g_{j}. Bayesian MCMC methods use the marginal posterior mean of each SNP in the prediction
equation
The analytical relationship between the conditional posterior mean E(g_{j } g_{j},y) and the MAP estimate of g_{j }is explored further in Figure 4. The analytical derivation of E(g_{j } g_{j},y) is given in Appendix 1 of [13], while the MAP estimate of g_{j }is calculated using equation (7) with γ_{j }given by equation (A4). Plots of the E(g_{j } g_{j},y) versus G_{j }are given in Figures 4A and 4C, while plots of the MAP estimate
Figure 4. Shrinkage of the cML estimate using the posterior mean or the MAP estimate. Plots of the analytical formulae for the conditional posterior mean E(g_{j}g_{j},y) (Figures 4A and 4C) as given in Appendix 1 of [13] and the MAP estimate of g_{j }(Figures 4B and 4D) as given by equation (7) with the posterior probability γ_{j }given by equation (A4). All plots have G_{j }(the conditional ML estimate of g_{j}) on the horizontal axis. G_{j }is in σ units as all plots use σ = 1.
When γ = 1 in Figure 4B, the MAP curve resembles a broken stick which is absolutely flat around the origin.
This is the LASSO estimate which is a broken stick for all values of λ. The LASSO's
When γ = 0.1, the MAP curves show strong shrinkage to 0 for any G_{j }values between 2σ and 2σ for all values of λ (Figure 4D). Even more shrinkage of small G_{j }values occurs when γ = 0.01 as the G_{j }interval increases to (3σ, 3σ) as shown in Figure 4B. As λ increases in Figure 4D, the variance of the DE distribution gets smaller which results in a smaller total genetic variance for fixed m and γ. Hence we need more shrinkage (±λσ^{2}) of large G_{j} values as shown by the different asymptotes in Figure 4D.
Discussion
This study has developed a fast EM algorithm for genome wide prediction in which there is a joint prediction of breeding value from accumulated SNP data. The benefits of the algorithm are its fast performance, its verity in relation to the proposed model, and the optimality properties it brings from application of the EM algorithm. The time advantage of emBayesB over a full Bayesian analysis is expected to be even greater with dense 500 k SNP panels currently being used in GWA studies. A disadvantage of emBayesB is that no standard errors are routinely available from an EM algorithm. However there are methods of obtaining standard errors with an EM algorithm [14] and even bootstrapping is a possibility given the fast performance.
The predictive power of emBayesB comes from the use of the informationrich prior mixture distribution which is of particular value when the number of QTL is small relative to the number of independent genomic segments [19]. In fact it is expected that there will be no advantage in using emBayesB over GSBLUP if the simulated QTL more closely fit an infinitesimal model. As with other recent studies [10,11] emBayesB uses a DE prior distribution for QTL effects which has some experimental justification [8]. In addition emBayesB incorporates a priori that an unknown proportion of SNP will not be in LD with QTL through the use of the Dirac Delta function in the prior mixture distribution for the SNP effects. This SNP prior mixture is quite different to that used in the EM algorithm wBSR in [15] where the Dirac Delta function was not used to model the absence of LD. The wBSR algorithm derived in [15] is only an approximate EM algorithm due to the approximation used to include the missing data variable γ_{l }(the SNP weight) into the EM modelling process. Using a Dirac Delta function in the prior mixture seems a more theoretically attractive way of modelling the LD between SNP and QTL and produces some appealing analytical results like the posterior probability formula in equation (A4) and the result that the best estimate of a SNP effect can be viewed as a regressed DE mode as shown in equation (7).
emBayesB is an EM algorithm which has similarities with the fast heuristic algorithm
called ICE [13]. ICE uses the same formulation of the data model and the SNP prior distribution but
iterates on the mean of each SNP effect conditional on all the other SNP mean effects,
the y data and assuming fixed values for γ, λ and
The simulated example of [13] used an 8010 SNP panel with 1000 individuals in the training and validation data sets. The speed advantage of ICE was large; ICE converged in 2 to 5 minutes compared to 47 hours for the Bayesian MCMC analysis. The computational speed advantage of ICE comes from the analytical calculation of the conditional posterior mean; emBayesB uses a similar analytical calculation for the conditional posterior probability. As ICE and emBayesB took similar amounts of CPU time in Table 1, the results for ICE in [13] provide further evidence of the computational speed advantage of emBayesB over a full Bayesian analysis.
emBayesB is similar to the empirical Bayes method suggested by [20] where Bayesian hyperparameters are estimated by marginal and conditional maximum likelihood methods. Taking an empirical Bayes approach in a wavelet regression application, [21] used marginal maximum likelihood with various prior mixtures involving the Dirac Delta function (including the DE as in emBayesB) to evaluate shrinkage of wavelet noise. They compared the posterior mean and posterior median as shrinkage methods and showed that the posterior median, unlike the posterior mean, produces a threshold rule for estimation in that estimated wavelet coefficients below a calculated threshold were set to exactly zero. The emBayesB estimate of a SNP effect is also calculated using a thresholding rule (see equation (7) and Figure 4). As with emBayesB, the empirical Bayes methods of [21] combine fast computation with good theoretical properties.
The simulated example used in this paper did not show any advantage for emBayesB over the LASSO. However in a simulated example of wavelet denoising, [22] demonstrated an advantage over the LASSO of both a Bayesian sigmoid model and the empirical Bayes method of [21] which uses a DD+DE mixture prior like in emBayesB. In fact various methods for shrinking coefficients in regression models were compared in [22] including the Bayesian sigmoid model which has a single hyperparameter to tune the shrinkage. The bimodal nature of the marginal posterior for a regression coefficient in the Bayesian sigmoid model (Figure 2 in [22]) is very similar to the bimodal nature of the conditional posterior distribution of a SNP effect as shown in Figure 3. The shape of the shrinkage graph for the Bayesian sigmoid model (Figure 4 in [22]) is also similar to the emBayesB shrinkage graph when γ is small and λ is small (see Figure 4D with γ = 0.1, λ = 0.1). However emBayesB will estimate values for γ and λ, and so, unlike the Bayesian sigmoid model, emBayesB can adapt its shrinkage such that it is appropriate for the prevailing nature of the data like in Figure 4.
Conclusions
This paper reports an EM algorithm called emBayesB for genome wide prediction in which there is a joint prediction of breeding value from dense SNP marker data. A formulation of the emBayesB algorithm using GSRU is developed to handle large SNP panels. Using a simulated and widely available dataset it was found that the accuracy of emBayesB was similar to Bayesian approaches, but emBayesB took only a fraction of the computational time. Using emBayesB may be a promising solution to the problem found in GWA studies with the use of stringent statistical thresholds. The emBayesB calculation of posterior probabilities of SNP being in LD with QTL may also be useful in the area of SNP subset selection. Due to the fast computational speed, opportunities exist with emBayesB to explore fitting innovative models which could include nonadditive genetic variation or even simultaneous fitting of multiple traits. More research is needed to explore the opportunities which emBayesB offers and to benchmark its capabilities.
Availability and requirements
The simulated data analysed in the paper is available on the 12^{th }QTLMAS workshop website http://www.computationalgenetics.se/QTLMAS08/QTLMAS/DATA.html webcite. The program emBayesB is available both as Fortran 90 source code and as a Windows executable in Additional file 3.
Authors' contributions
RKS developed the emBayesB theory, wrote the emBayesB software and the paper. JAW developed aspects of the emBayesB theory and wrote sections of the paper. THEM formulated the basic Gauss Seidel algorithm, and helped formulate the research and writing the paper. All authors read and approved the final version of the paper.
Acknowledgements
This paper reports collaborative research instigated while RKS was on sabbatical at the Roslin Institute with support from CQUniversity and the Roslin Institute.
References

van Hoek M, Dehghan A, Wittentan JCM, van Duiin CM, Uitterlinden AG, Oostra BA, Hofman A, Sijbrands EJG, Janssens ACJW: Predicting Type 2 diabetes based on polymorphisms from genomewide association studies. A populationbased study.
Diabetes 2008, 57:31223128. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Weedon MN, Lango H, Lindgren CM, Wallace C, Evans DM, Mangino M, Freathy RM, Perry JR, Stevens S, Hall AS, Samani NJ, Shields B, Prokopenko I, Farrall M, Dominiczak A, Diabetes Genetics Initiative; Wellcome Trust Case Control Consortium, Johnson T, Bergmann S, Beckmann JS, Vollenweider P, Waterworth DM, Mooser V, Palmer CN, Morris AD, Ouwehand WH, Cambridge GEM Consortium, Zhao JH, Li S, Loos RJ, et al.: Genomewide association analysis identifies 20 loci that influence adult height.
Nature Genetics 2008, 40:575583. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Barrett JC, Hansoul S, Nicolae DL, Cho JH, Duerr RH, Rioux JD, Brant SR, Silverberg MS, Taylor KD, Barmada MM, Bitton A, Dassopoulos T, Datta LW, Green T, Griffiths AM, Kistner EO, Murtha MT, Regueiro MD, Rotter JI, Schumm LP, Steinhart AH, Targan SR, Xavier R, the NIDDK IBD Genetics Consortium, Libioulle C, Sandor C, Lathrop M, Belaiche J, Dewit O, Gut I, Heath S, Laukens D, Mni M, Rutgeerts P, Van Gossum A, Zelenika D, Franchimont D, Hugot JP, de Vos M, Vermeire S, Louis E, the BelgianFrench IBD consortium, the Wellcome Trust Case Control Consortium, Cardon LR, Anderson CA, Drummond H, Nimmo E, Ahmad T, Prescott NJ, Onnie CM, Fisher SA, Marchini J, Ghori J, Bumpstead S, Gwilliam R, Tremelling M, Deloukas P, Mansfield J, Jewell D, Satsangi J, Mathew CG, Parkes M, Georges M, Daly MJ: Genomewide association defines more than 30 distinct susceptibility loci for Crohn's disease.
Nature Genetics 2008, 40:955962. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Prokopenko I, McCarthy MI, Lindgren CM: Type 2 diabetes: new genes, new understanding.
Trends in Genetics 2008, 24:613621. PubMed Abstract  Publisher Full Text

Yang J, Benyamin B, McEvoy BP, Gordon S, Henders AJ, Nyholt DR, Madden PA, Heath AC, Martin NG, Montgomery GW, Goddard ME, Visscher PM: Common SNPs explain a large proportion of the heritability for human height.
Nature Genetics 2010, 42:565569. PubMed Abstract  Publisher Full Text

Meuwissen THE, Hayes BJ, Goddard ME: Prediction of total genetic value using genome wide dense marker maps.
Genetics 2001, 157:18191829. PubMed Abstract  PubMed Central Full Text

Xu S: Estimating polygenic effects using markers of the entire genome.
Genetics 2003, 163:789801. PubMed Abstract  PubMed Central Full Text

Goddard ME: Genomic selection: prediction of accuracy and maximisation of long term response.
Genetica 2009, 136:245257. PubMed Abstract  Publisher Full Text

van Raden PM: Efficient methods to compute genomic predictions.
J Dairy Sci 2008, 91:44144423. PubMed Abstract  Publisher Full Text

de los Campos G, Naya H, Gianola D, Crossa J, Legarra A, et al.: Predicting quantitative traits with regression models for dense molecular markers and pedigree.
Genetics 2009, 182:375385. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Yi N, Xu S: Bayesian LASSO for quantitative trait loci mapping.
Genetics 2008, 179:10451055. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Goddard ME, Hayes BJ: Genomic selection.
J Anim Breed Genet 2007, 124:323330. PubMed Abstract  Publisher Full Text

Meuwissen THE, Solberg TR, Shepherd R, Woolliams JA: A fast algorithm for BayesB type of prediction of genomewide estimates of genetic value.
Genet Sel Evol 2009, 41:2. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

McLachlan GJ, Krishnan T: The EM Algorithm and Extensions. Second edition. Hoboken, NJ; Wiley; 2008.

Hayashi T, Iwata H: EM algorithm for Bayesian estimation of genomic breeding values.
BMC Genetics 2010, 11:3. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

Legarra A, Misztal I: Technical Note: Computing strategies in genomewide selection.
J Dairy Sci 2008, 91:360366. PubMed Abstract  Publisher Full Text

Lund MS, Sahana G, de Konig DJ, Su G, Carlborg O: Comparison of analyses of the QTLMAS XII common dataset. I: Genomic selection.
BMC Proceedings 2009, 3(Suppl 1):S1. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

Crooks L, Sahana G, de Konig DJ, Lund MS, Carlborg O: Comparison of analyses of the QTLMAS XII common dataset. II: Genomewide association and fine mapping.
BMC Proceedings 2009, 3(Suppl 1):S2. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

Daetwyler HD, PongWong R, Villanueva B, Woolliams JA: The impact of genetic architecture on genomewide evaluation methods.
Genetics 2010, 185:10211031. PubMed Abstract  Publisher Full Text

George EI, Foster DP: Calibration and empirical Bayes variable selection.
Biometrika 2000, 87:731748. Publisher Full Text

Johnstone IM, Silverman BW: Empirical Bayes selection of wavelet thresholds.
Ann Statist 2005, 33:17001752. Publisher Full Text

ter Braak CJF: Bayesian sigmoid shrinkage with improper variance priors and an application to wavelet denoising.