Fast empirical Bayesian LASSO for multiple quantitative trait locus mapping
© Cai et al; licensee BioMed Central Ltd. 2011
Received: 3 February 2011
Accepted: 26 May 2011
Published: 26 May 2011
The Bayesian shrinkage technique has been applied to multiple quantitative trait loci (QTLs) mapping to estimate the genetic effects of QTLs on quantitative traits from a very large set of possible effects including the main and epistatic effects of QTLs. Although the recently developed empirical Bayes (EB) method significantly reduced computation comparing with the fully Bayesian approach, its speed and accuracy are limited by the fact that numerical optimization is required to estimate the variance components in the QTL model.
We developed a fast empirical Bayesian LASSO (EBLASSO) method for multiple QTL mapping. The fact that the EBLASSO can estimate the variance components in a closed form along with other algorithmic techniques render the EBLASSO method more efficient and accurate. Comparing with the EB method, our simulation study demonstrated that the EBLASSO method could substantially improve the computational speed and detect more QTL effects without increasing the false positive rate. Particularly, the EBLASSO algorithm running on a personal computer could easily handle a linear QTL model with more than 100,000 variables in our simulation study. Real data analysis also demonstrated that the EBLASSO method detected more reasonable effects than the EB method. Comparing with the LASSO, our simulation showed that the current version of the EBLASSO implemented in Matlab had similar speed as the LASSO implemented in Fortran, and that the EBLASSO detected the same number of true effects as the LASSO but a much smaller number of false positive effects.
The EBLASSO method can handle a large number of effects possibly including both the main and epistatic QTL effects, environmental effects and the effects of gene-environment interactions. It will be a very useful tool for multiple QTL mapping.
Quantitative traits are usually controlled by multiple quantitative trait loci (QTLs) and environmental factors. Interactions among QTLs or between genes and environmental factors make a substantial contribution to variation in complex traits . The goal of QTL mapping is to infer genomic loci that are associated with the trait and to estimate the genetic effects of these loci including their main effects and gene-gene (epistasis) and gene-environment (G × E) interactions. Due to the physical linkage of and/or epistatic interactions among multiple QTLs, it is highly desirable to analyze a large number of loci simultaneously. Since hundreds or thousands of genomic loci or markers are usually genotyped and involved in QTL mapping studies, including all these markers and their possible interactions in a model leads to a huge number of model variables, typically much larger than the sample size. Two general techniques often employed to handle such oversaturated models are variable selection and shrinkage.
Variable selection attempts to identify a subset of all possible genetic effects that best explain the phenotypic variation, typically using a stepwise search procedure in conjunction with a selection criterion such as the Bayesian information criterion (BIC) . On the other hand, a shrinkage method includes all variables in the model but uses a penalty function of the variables or appropriate prior distributions on the variables to shrink most non-effect variables toward zero. Early shrinkage methods include ridge regression  and the least absolute shrinkage and selection operator (LASSO) . More recently, Bayesian shrinkage method  has received considerable attention and been applied to multiple QTL mapping [6–10]. All these works employ Markov chain Monte Carlo (MCMC) simulation to fit the Bayesian model and provide comprehensive information about the model drawing from the posterior distribution of the model variables. Despite the advances in the development of the MCMC simulation algorithms , MCMC simulations are computationally intensive and time consuming.
In order to reduce the computational burden of the fully Bayesian approach relying on MCMC simulation, one of the authors of this paper developed an empirical Bayes (EB) method  that uses a properly chosen prior distribution for the model variables to shrink variables toward zero. It was demonstrated that the EB method can handle a large number of model variables simultaneously. More recently, the EB method has been extended to handle classification predictor variables . Although the EB method  requires much less computation comparing to the fully Bayesian approach, its efficiency is limited by the fact that a numerical optimization algorithm such as the simplex algorithm  is needed to estimate the variance components. On the other hand, a very efficient EB method, named relevance vector machine (RVM), for learning a linear model was developed in the machine learning community [15, 16]. The RVM can estimate the variance components in a closed form, which along with other algorithmic techniques make it a very fast algorithm. The RVM assumes a uniform prior distribution for the variance components. Although this choice of the prior distribution gets rid of any hyperparameters to be pre-specified, it lacks the flexibility of adjusting the degree of shrinkage needed for analyzing a specific data set. Particularly, its uniform prior distribution may not provide enough shrinkage in multiple QTL mapping that includes a very large number of possible effects, often resulting in a large number of false effects .
In this paper, capitalizing on the idea of RVM, we developed a fast empirical Bayesian LASSO (EBLASSO) algorithm based on the Bayesian LASSO model [10, 17] with an exponential prior distribution for the variance components in contrast to the inverse chi-square distribution for the variance components used by the EB method . Simulation studies demonstrate that our EBLASSO method can provide a speed up to orders of magnitude faster than the EB method and can detect more true QTL effects without increasing the false positive rate. Real data analysis also show that the EBLASSO method is able to detect some effects when the EB method fails.
Linear model of multiple QTLs
where μ is the population mean, vectors β E and β G represent the environmental effects and the main effects of all markers, respectively, vectors β GG and β GE capture the epistatic effects and the G × E interactions, respectively, X E, X G, X GG and X GE are the corresponding design matrices of different effects, and e is the residual error that follows a normal distribution with zero-mean and covariance . Throughout the paper we use I to denote an identity matrix whose size can be clearly identified from the context.
The design matrix X G depends on a specific genetic model. We adopt the widely used Cockerham genetic model as also used by  in their generalized linear model for multiple QTL mapping. For a back-cross design, the Cockerham model defines the values of the main effect of a marker as -0.5 and 0.5 for two genotypes at the marker. For an intercross (F2) design, there are two possible main effects named additive and dominance effects. The Cockerham model defines the values of the additive effect as -1, 0 and 1 for the three genotypes and the values of the dominance effect as -0.5 and 0.5 for homozygotes and heterozygotes, respectively. The columns of the design matrix X GG are obtained as the element-wise product of any two different columns of X G , and similarly the columns of X GE are constructed as the element-wise product of any pair of columns from X E and X G .
Suppose that there are p environmental covariates and q markers whose main effects are additive, then the size of matrix X is n × k where k = p + q(q + 1)/2 + pq. Typically, we have k ≫ n. If dominance effects of the markers are considered, k is even larger. Our goal is to estimate all possible environmental and genetical effects on y manifested in the regression coefficients β, which is a challenging problem because k ≫ n. However, we would expect that most elements of β are zeros and thus we have a sparse linear model. Taking into account this sparsity, we will adopt the Bayesian LASSO model  where appropriate prior distributions are assigned to the elements of β as described in the next section.
Prior and posterior distributions
The unknown parameters in model (2) are, μ and β. While our main concern is β, parameters μ and need to be estimated so that we can infer β. To this end, we assign a noninformative uniform prior μ to and , i.e., p(μ) ∝ 1 and p( ) ∝ 1. Following the Bayesian LASSO model , we assume a three-level hierarchical model for β. Let us denote the elements of β as β i , i = 1, 2, ⋯, k. At the first level, β i , i = 1, 2, ⋯, k follow independent normal distributions with mean zero and unknown variance . At the second level, , follow independent exponential distribution with a common parameter . For a given λ, the distribution of β i is found to be the Laplace distribution: , which is known to encourage the shrinkage of β i toward zero . However, the degree of shrinkage strongly depends on the value of λ. To alleviate the problem of choosing an inappropriate value for λ, we add another level to the hierarchical model at which we assign a conjugate Gamma prior Gamma(a, b) with a shape parameter a > 0 and an inverse scale parameter b > 0 to the parameter λ. As discussed in , we can pre-specify appropriate values for a and b so that the Gamma prior for λ is essentially noninformative.
Let us define vector . The joint posterior distribution of all the parameters (μ, , β, σ2, λ) can be easily found . In principle, MCMC simulation can be employed to draw samples from the posterior distribution for each parameter. However, since the number of parameters 2k + 3 in our model can be very large, the fully Bayesian approach based on MCMC sampling requires a prohibitive computational cost. To avoid this problem, Xu developed an empirical Bayes method for inferring β. Our goal here is to develop a much faster and more accurate empirical Bayes method that can easily handle tens of thousands of variables.
Maximum a posteriori estimation of variance components
Note that α* in (8) and (9) depends on other unknown parameters through s i and q i , and thus, α i will be estimated iteratively as detailed in the EBLASSO algorithm described in the next section. Comparing with the EB method , our method finds each α i (and equivalently ) in a closed form, whereas the EB method generally needs to employ a numerical optimization algorithm to find each . Therefore, our method not only saves much computation but also gives more accurate estimate of each α i . Moreover, exploiting the similar techniques used in the RVM , we can further increase computational efficiency as described in the ensuing section.
Fast Empirical Bayesian LASSO algorithm
After we get estimates of μ, and α, we finalize the model (10), where the prior distribution of is now . For those x i s not in the model, we can declare that they do not affect the quantitative trait because their regression coefficient is zero. For those x i s in the matrix , the posterior distribution of their regression coefficients is Gaussian with covariance ∑ in (13) and mean u in (17) . We can then use the z-score or more conservative t-statistics to test if at certain significance level. In this paper, the posterior mean u j of the j th effect is reported as the empirical Bayes estimate of β j , denoted by , and the posterior standard deviation, , is used as the standard error of .
We now summarize our fast EBLASSO algorithm as follows.
Algorithm 1 (EBLASSO algorithm)
2. Initialize the model: Find , and calculate α j from (9), set all other α i s to be ∞ and .
3. Calculate ∑ from (13), S i and Q i , ∀i, from (14).
4. Update the model
while the local convergence criterion is not satisfied
Calculate q i and s i from (11), ∀i.
if x j is in the model, delete it and update ∑ , S i and Q i , ∀i.
and Q i , ∀i.
6. Calculate ∑ from (13) and C-1from (12).
8. Calculate S i and Q i from (14).
9. If the global convergence criterion is not satisfied, go to step 4.
The parameters a and b can be set to be a small number (e.g., a = b = 0.01) so that the Gamma prior distribution is essentially noninformative . Alternatively, we can use the predicted error (PE) obtained from cross-validation  to choose the values of a and b. As done in , the initial value of is chosen in step 1 to be a small number and the initial model is selected in step 2 to have a single effect that corresponds to the maximum L(α i ) with a = -1. The outer iteration loop consists of steps 4-9, while the inner iteration loop is step 4, where we use the greedy method described earlier to update the model. In step 4, we use the method given in the Appendix of  to efficiently update ∑, S i and Q i after a x j is added to or deleted from the model or after α j is updated. The local convergence criterion can be defined as the simultaneous satisfaction of the following three conditions: 1) no effect can be added to or deleted from the model, 2) the change of L(θ) between two consecutive inner iterations is smaller than a pre-specified small value, and 3) the change of between two consecutive inner iterations is less than a pre-specified value. The global convergence criterion can be defined as the simultaneous satisfaction of the following two conditions: the change of L(θ) between two consecutive outer iterations is smaller than a pre-specified small value, and 2) the total change in μ, and between two consecutive outer iterations is less than a pre-specified value. A Matlab program has been developed to implement the algorithm; and a more efficient C++ program is under development.
True and estimated QTL effects for the simulated data with main and epistatic effects.
( i, j )
β ( h 2 )
where var(x j ) is the variance of X j . In the simulated data, the proportion of contribution from an individual QTL varied from 0.30% to 9.75%, whereas the proportion of contribution from a pair of QTLs ranged from 0.26% to 7.25%, as shown in Table 1. Some of the markers had only main or epistatic effect, while the others had both main and epistatic effects. The QTL model contained a total of possible effects, a number about 115 times of the sample size.
The data were analyzed in Matlab on a personal computer (PC) using the EBLASSO algorithm, the EB method, the RVM and the LASSO. The Matlab program SPARSEBAYES for the RVM was downloaded from http://www.miketipping.com. We translated the original SAS program for the EB method  into Matlab, and slightly modified the program to avoid possible memory overflow due to the large number of possible effects. We also got the program glmnet  that is a very efficient program implementing the LASSO and other related algorithms. The PC version of glmnet uses Matlab to initialize and call the core LASSO algorithm that is implemented efficiently with Fortran code.
Summary of results for the simulated data with main and epistatic effects
PE ± STE*
Number of effects†‡
CPU time (mins)
16.49 ± 0.8908
15.95 ± 0.7477
15.89 ± 0.7498
15.81 ± 0.8359
15.86 ± 0.7717
16.07 ± 0.7203
16.14 ± 0.8557
15.92 ± 1.0161
The EBLASSO took about 3.4 minutes of CPU time for each set of values of a and b listed in Table 2, whereas the EB took 249 hours of CPU time for τ = -1 and ω = 0.001 and about 46, 69, 46 hours for τ = -1 and ω = 0.0001, 0.0005, 0.01, respectively. This simulation study showed that the EBLASSO method not only can detect more effects, but also offers a huge advantage in terms of computational time. Note that all simulations were done in Matlab. It is expected that the EBLASSO algorithm will be even faster, after its implementation in C++ is completed.
where λ is a positive constant that can be determined with cross-validation . We tried to run the program glmnet  with the simulated data. However, glmnet could not handle the big design matrix X of 1, 000 × 115, 922 in our QTL model, and we did not get any results from glmnet.
Summary of results for the simulated data with only main effects
PE ± STE*
Number of effects†‡
CPU time (sec)
11.52 ± 0.5677
11.52 ± 0.578
11.36 ± 0.6088
11.23 ± 0.5571
11.32 ± 0.5937
11.4 ± 0.5929
11.57 ± 0.5593
10.87 ± 0.5599
10.78 ± 0.5646
11.09 ± 0.5045
17.73 ± 2.0244
15.81 ± 2.5732
12.21 ± 1.7635
10.69 ± 0.9903
11.63 ± 0.5743
10.77 ± 0.4583
10.52 ± 0.4442
10.50 ± 0.5248
10.52 ± 0.4382
10.59 ± 0.4434
The optimal values for the parameters of the EB method were τ = -1 and ω = 0.01, since they gave the smallest PE in cross-validation as listed in Table 3. With τ = -1 and ω = 0.01, the EB method detected 19 true effects and 4 false positive effects. The RVM detected all 20 true effects as the EBLASSO did, but it also output a large number of 42 false positive effects. This result is consistent the observation  that the uniform prior distribution used in the RVM usually yields many false positive effects. To choose the optimal value of λ for the LASSO, we ran ten-fold cross validation starting from λ = 4.9725 (which gave only one nonzero effect) and then decreasing λ to 0.0025 with a step size of 0.0768 on the logarithmic scale (Δ ln(λ) = 0.0768). The smallest PE was achieved at λ = 0.0675. We then used this value to run glmnet on the whole data set, which yielded 97 nonzero effects. For each of these nonzero effects, we calculated their standard error using equation (7) in , and then calculated the p-value of each nonzero effect. This gave 20 true effects and 48 false positive effects with a p-value less than 0.05. Comparing the number of effects detected by the EBLASSO, EB, RVM and LASSO, the EBLASSO offered the best performance because it detected all true effects and a very small number of false positive effects.
It is seen from Table 3 that the EBLASSO and the LASSO took much less time than the EB method and the RVM on analyzing this data set. It is expected that the EBLASSO is much faster than the EB method because as we discussed earlier the EB needs a numerical optimization procedure. The RVM and EBLASSO generally should have similar speed because two algorithms use the similar technique to estimate effects. However, when applying to the same data set, the RVM often yields a model with much more nonzero effects than the EBLASSO as is the case here, because the RVM does not provide sufficient degree of shrinkage. Due to this reason, the RVM algorithm requires more time than the EBLASSO. The LASSO took slightly less CPU time than the EBLASSO in this example. However, we would emphasize that the LASSO was implemented with Fortran but our EBLASSO was implemented with Matlab. The speed of EBLASSO is expected to increase significantly once it is implemented in C/C++.
Real data analysis
This dataset was obtained from . This dataset consists of n = 150 double haploids (DH) derived from the cross of two spring barley varieties Morex and Steptoe. The total number of markers was q = 495 distributed along seven pairs of chromosomes of the barley genome, covering 206 cM of the barley genome. The phenotype was the spot blotch resistance measured as the lesion size on the leaves of barley seedlings. Note that spot blotch is a fungus named Cochliobolus sativus. This dataset was used as an example for the application of the EBLASSO method. Genotype of the markers were encoded as +1 for genotype A (the Morex parent), -1 for genotype B (the Steptoe parent), and 0 for missing genotype. Ideally, the missing genotypes should be imputed from known genotypes of neighboring markers. For simplicity, we replaced the missing genotypes with 0 in order to use the phenotypes of the individuals with missing genotypes. The total missing genotypes only account for about 4.2% of all the genotypes. Including the population mean, the main and the pair-wise epistatic effects, the total number of model effects was , about 818 times as large as the sample size.
Summary of the results of the EBLASSO algorithm for the real data
Eight effects estimated with the EBLASSO algorithm for the real data.
Xu  compared several methods for multiple QTL mapping including the EB , LASSO , penalized likelihood (PENAL)  and stochastic search variable selection (SSVS) [22, 23] methods. The SSVS method is much slower than the EB method; whereas LASSO and PENAL methods are faster than the EB method. Although we did not directly compare the speed of our EBLASSO with that of the PENAL method, based on all comparisons with the EB method in , we observed that the EBLASSO method is faster than PENAL methods. Direct comparison between the EBLASSO and LASSO showed that the LASSO is slightly faster than the current version of EBLASSO, however, this may not be the case when the EBLASSO is implemented in C/C++ instead of Matlab. Although EB, LASSO, PENAL and SSVS methods all produced satisfactory results in a simulation study , the EB method outperformed the other three methods in terms of the mean-squared error. Moreover, when being applied to a real data set, the EB and LASSO detected some effects, whereas the PENAL and SSVS failed to generate any meaningful results . In our simulation studies, we observed that the EBLASSO method detected more true effects than the EB method with almost the same false positive rate, and the same number of true effects as the LASSO but with a much smaller number of false positive effects. When analyzing a real data set, we found that the EBLASSO method detected a reasonable number of effects, but the EB method detected one or zero effect depending on values of the hyperparameters used. These observations in both simulation study and real data analysis demonstrated that the EBLASSO method outperforms the EB method and the LASSO.
The EBLASSO method was built upon the idea of the RVM in machine learning. The EBLASSO and EB methods, as well as the RVM, all are based on a Bayesian hierarchical linear regression model and all estimate the variances of the regression coefficients. The difference of the three methods in the regression model is the different prior distributions for the hyperparameters. The EB method and the RVM employ inverse chi-square and uniform distributions, respectively, for the variances of the regression coefficients, while the EBLASSO assigns exponential distributions to the variance components and uses a Gamma distribution for the parameter of the exponential distribution, which leads to the prior distribution in (3) for the variance components. The uniform prior distribution used by the RVM may not provide enough degree of shrinkage for certain data and thus generate a large number of false positive effects as shown in  and as demonstrated in our simulation study.
The prior distributions used by the EBLASSO and RVM methods enable one to estimate the variance components in a closed form, while the EB method generally needs a numerical optimization algorithm to estimate the variance components. This difference has at least two implications: 1) both the EBLASSO and the RVM methods requires much less computation than the EB method to estimate the variance components, and 2) the EBLASSO method and the RVM method can always find the unique optimal estimate of a variance component but the numerical optimization algorithm used by the EB method may not find the optimal value of the variance due to the nonlinearity and non-convexity of the objective function. Another main factor that makes the EBLASSO method and RVM more efficient is an automatic variable selection procedure resulting from the process of estimating variance components, because the variables whose precision is infinity or equivalently whose variance is zero are excluded from the model. This results in an efficient formula in (12) for calculating the inverse of the covariance matrix of the data. This is especially beneficial when the number of samples is relatively large. On the other hand, the EB method in principle can be applied to a linear regression model with any prior distribution for the variances of regression coefficients. Since the prior distribution may play an important role in estimation of the QTL effects, the EB method has its value when one tries to explore different prior distributions.
To get the best performance, the EBLASSO method needs to properly choose values of hyperparameters a and b. In this paper, we selected the values of a and b that gave the smallest average PE resulting from ten-fold cross validation. Ideally, we need to find PEs for a large set of values for a ≥ -1 and b > 0 and then identify the best values for a and b. In our simulation study, we found a two-step cross validation procedure could significantly reduce the number of values to be tested without missing the best values, thereby reducing computational time. In this two-step procedure, we first run cross-validation for the following set of values: a = b = 0.001, 0.01, 0.05, 0.1, 0.5, 1. We identify the values (denoted as a* and b*) from this set of values that yields the smallest PE. We then fixed b to be b* and run cross-validation for several other values of a greater or less than a*. The final best values of a and b are the ones that yield the smallest PE.
The EBLASSO algorithm may still be improved. In the analysis of simulated data with both main and epistatic effects, although the EBLASSO method detected 8 more true effects than the EB method without any false positive effects, it missed three effects that the EB method detected. It is unclear how this discrepancy occurred. One possible reason is the different prior distributions used in the two methods. Although it is difficult for the EBLASSO method to use the scaled inverse chi-square distribution that is used by the EB method, other prior distributions may worth investigation. Another possible reason may be the greedy method used to select the variable to include in or to exclude from the model. In the current algorithm, we choose the variable that gives the largest increase in the likelihood to add to or delete from the model. It may be better to simultaneously add or delete more than one variables. The EBLASSO method presented in this paper assumes continuous quantitative traits. It can also be extended to handle binary or polychotomous traits and the algorithm is under development. The algorithm is currently implemented in Matlab. We are developing programs in C++ to implement the algorithm, which is expected to be much faster and to be capable of running in R and SAS environments.
We have developed a fast empirical Bayesian LASSO method for multiple QTL mapping that can deal with a large number of effects possibly including main and epistatic QTL effects, environmental effects and the effects of environment and gene interactions. Our simulation studies demonstrated that the EBLASSO algorithm needed about 3.4 minutes of CPU time, running in Matlab on a PC with 2.4 GHz Intel Core2 CPU and 2 Gb memory running Windows XP, to analyze a QTL model with more than 105 possible effects, whereas the EB method took more than 2,000 minutes to analyze the same model on the same computer. Our simulation studies also showed that the EBLASSO method could detect more true effects with almost the same false positive rate comparing to the EB method. Our real data analysis demonstrated that the EBLASSO method could output more reasonable effects than the EB method. Comparing with the LASSO, our simulation showed that the current version of the EBLASSO implemented in Matlab was slightly slower than the LASSO implemented with glmnet in Fortran, and that the EBLASSO detected the same number of true effects as the LASSO but a much smaller number of false positive effects. In conclusion, the EBLASSO method will be a useful tool in multiple QTL mapping.
This work was supported by the National Science Foundation (NSF) under NSF CAREER Award no. 0746882 to XC and by the Agriculture and Food Research Initiative (AFRI) of the USDA National Institute of Food and Agriculture under the Plant Genome, Genetics and Breeding Program 2007-35300-18285 to SX.
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